A real symplectic manifold is a symplectic manifold endowed with an involution which is anti-symplectic. Given a real symplectic manifold, we may ask: are there any anti-invariant symplectic forms which are cohomologous but not isotopic within anti-invariant forms? In this talk, we will show that such disconnectivity indeed appears for certain real quadric surfaces.

We introduce some classical problems in graph Ramsey Theory. We also discuss new results on a less-classical problem, namely, multicolor Ramsey numbers for some triple system paths of length three. The latter is joint work with Tom Bohman. Come for the tikzpictures, stay for the Ramsey Theory (or vice versa)!

High dimensional non-Gaussian data are increasingly encountered in a wide range of applications. It poses new challenges to traditional statistical tools. In this talk, we will present some recent development on methodologies and theories for the analysis of fat-tailed data as well as some high dimensional estimation and inference problems.

The persistence diagram is an increasingly useful shape descriptor from Topological Data Analysis, but its use alongside typical machine learning techniques requires mathematical finesse. We will describe in this talk a mathematical framework for featurization of said descriptors, and we show how it addresses the problem of approximating continuous functions on compact subsets of the space of persistence diagrams. We will also show how these techniques can be applied to problems in semi-supervised learning where these descriptors are relevant.

I will discuss the Attractor Conjecture for Calabi-Yau
varieties, which was formulated by Moore in the nineties, highlighting
the difference between Calabi-Yau varieties with and without Shimura
moduli. In the Shimura case, I show that the conjecture holds and gives
rise to an explicit parametrization of CM points on certain Shimura
varieties; in the case without Shimura moduli, I'll present
counterexamples to the conjecture using unlikely intersection theory.
\\
\\
Part of this is joint work with Arnav Tripathy.

Let $\Gamma$ be an Anosov subgroup of a connected semisimple real linear Lie group $G$. For a maximal horospherical subgroup $N$ of $G$, we show that the space of all non-trivial $NM$-invariant ergodic and $A$-quasi-invariant Radon measures on $\Gamma \backslash G$, up to proportionality, is homeomorphic to $\mathbb{R}^{\mathrm{rank} (G)-1}$, where $A$ is a maximal real split torus and $M$ is a maximal compact subgroup which normalizes $N$.
\\
\\
This is joint work with Hee Oh.

Kronheimer-Mrowka recently proved that the Dehn twist along a 3-sphere in the neck of $K3\#K3$ is not smoothly isotopic to the identity. This provides a new example of self-diffeomorphisms on 4-manifolds that are isotopic to the identity in the topological category but not smoothly so. (The first such examples were given by Ruberman.)
\\
\\
In this talk, we study the Bauer-Furuta invariant as an element in the Pin(2)-equivariant stable homotopy group of spheres. We use it to show that this Dehn twist is not smoothly isotopic to the identity even after a single stabilization (connected summing with the identity map on S2 cross S2). This gives the first example of exotic phenomena on simply-connected smooth 4-manifolds that do not disappear after a single stabilization. In particular, it implies that one stabilization is not enough in the diffeomorphism isotopy problem for 4-manifolds. It gives an interesting comparison with Auckly-Kim-Melvin-Ruberman-Schwartz's theorem that one stabilization is enough in the surface isotopy problem.

In this talk things will get extreme, but not with number fields or schemes. Instead we'll focus on combinatorics, though we won't discuss clever tricks. We speak only of glorious theorems, as well as some notorious problems. And for all those that attend, there is a surprise at the end!

We give a quick and friendly introduction to projective space and then introduce and explore some of the most elementary and fundamental examples in algebraic geometry: hypersurfaces in projective space (especially cubic hypersurfaces).

Anisotropic energies are functionals defined by integrating over a generalized surface (such as a current or a varifold) an integrand depending on the tangent plane to the surface. In the case of a constant positive integrand, one obtains the area functional, and hence one can see anisotropic energies as a generalization of it. A long standing question in geometric measure theory is to establish regularity properties of critical points to such functionals. In this talk, I will discuss some recent developments on this theory, addressing in particular the question of rectifiability of stationary points and regularity of stationary Lipschitz graphs.
\\
\\
The talk is based on joint work with Antonio De Rosa.

The inhomogeneous l-TASEP is an interacting particle process wherein particles stochastically enter, unidirectionally traverse, and finally exit a one-dimensional lattice segment at rates that may depend on a particle's location within the lattice. Its homogeneous version is known to exhibit various phase transitions in macroscopic observables like particle density and current, with fluctuations governed by what is known as the KPZ equation. In this talk, we begin to extend such results to the inhomogeneous setting by developing the so-called hydrodynamic limit, which governs the system dynamics on an LLN-type scale. If time permits, we apply our results to elucidate the key determinants of protein synthesis, which motivated the introduction of TASEP fifty years ago.
\\
\\
This is based on joint work with Khanh Dao Duc and Yun S. Song.

In neuroscience, learning and memory are usually associated to long-term changes of neuronal connectivity. Synaptic plasticity refers to the set of mechanisms driving the dynamics of neuronal connections, called synapses and represented by a scalar value, the synaptic weight. Spike-Timing Dependent Plasticity (STDP) is a biologically-based model representing the time evolution of the synaptic weight as a functional of the past spiking activity of adjacent neurons.
\\
\\
In this talk we present a new, general, mathematical framework to study synaptic plasticity associated to different STDP rules. The system composed of two neurons connected by a single synapse is investigated and a stochastic process describing its dynamical behavior is presented and analyzed. We show that a large number of STDP rules from neuroscience and physics can be represented by this formalism. Several aspects of these models are discussed and compared to canonical models of computational neuroscience. An important sub-class of plasticity kernels with a Markovian formulation is also defined and investigated via averaging principles.
\\
\\
Joint work with Gaetan Vignoud

Locally analytic representations of $p$-adic analytic groups
have played a crucial role in many areas of arithmetic and
representation theory (including in $p$-adic local Langlands program)
since their introduction by Schneider and Teitelbaum. In this talk we
will briefly review some aspects of the theory of locally analytic
representations. Then, for a locally analytic representation $V$ of
$GL_2(F)$ we will construct a coefficient system attached to the
Bruhat-Tits tree of $Gl_2(F)$. Finally we will use this coefficient
system to construct a resolution for locally analytic principal series
of $GL_2(F)$.

This will be the continuation to Calum's talk. The plan,
building on what Calum explained, is to discuss some recent work
building towards the birational classification of holomorphic foliations
on projective varieties (particularly 3folds) in the spirit of the
Minimal Model program.
\\
\\
We will explain some applications of these ideas to the study of the
dynamics and geometry of foliations and foliation singularities.
\\
\\
Features works of C. Spicer and P. Cascini and joint work with C. Spicer.

We discuss the relationship between the Borel measurable / Baire measurable combinatorics of the action of a finitely generated group on its Bernoulli shift and the discrete combinatorics of the multiplication action of that group on itself. Our focus is on various chromatic numbers of graphs generated by these actions. We show that marked groups with isomorphic Cayley graphs can have Borel measurable / Baire measurable chromatic numbers which differ by arbitrarily much. In the Borel two-ended, Baire measurable, and measurable hyperfinite settings, we show our constructions are nearly best possible (up to only a single additional color), and we discuss prospects for improving our constructions in the general Borel setting. Along the way, we will get tightness of some bounds of Conley and Miller on Baire measurable and measurable chromatic numbers of locally finite Borel graphs.

This is joint work with Asaf Horev and Inbar Klang. Factorization homology theories are invariants of $n$-manifolds with coefficients in suitable $E_n$-algebras. Let $G$ be a finite group and $V$ be a finite dimensional $G$-representation. The equivariant factorization homology for $V$-framed $G$-manifolds have $E_V$-algebra as coefficients. We show that when coefficient algebra $A$ is the Thom spectrum of an $E_{V+W}$-map for a large enough representation $W$, the factorization homology of $A$ can be computed by a certain Thom spectrum. With nonabelian Poincar\'e duality theorem, we are able to simplify the result in some cases. In particular, we compute $\mathrm{THR}(\mathrm{H}\mathbb{F}_{2})$, $\mathrm{THR}(\mathrm{H}\mathbb{Z}_{(2)})$, $\mathrm{THH}_{C_2}(\mathrm{H}\mathbb{F}_2)$.

Nonlocal continuum models are in general integro-differential equations in place of the conventional partial differential equations. While nonlocal models show their effectiveness in modeling a number of anomalous and singular processes in physics and material sciences, for example, the peridynamics model of fracture mechanics, they also come with increased difficulty in computation with nonlocality involved. In this talk, we will give a review of the asymptotically compatible schemes for nonlocal models with a parameter dependence. Such numerical schemes are robust under the change of the nonlocal length parameter and are suitable for multiscale simulations where nonlocal and local models are coupled. Some open questions will also be discussed.

The Kurosh problem can be seen as an analogue of the Burnside problem for algebras. It asks whether or not a finitely generated algebra over a field has finite dimension. While the answer is negative in general, you don't have to go all the way to \textbf{Chinatown} to find a class of algebras for which the answer is affirmative.
\\
\\
In this talk, we will show that for algebras satisfying a polynomial identity (PI algebras), the Kurosh problem is true. Along the way, we will have a \textbf{(The) Conversation} about the basics of the theory of PI algebras, discussing their properties, constructing specific identities for classes of algebras, and looking at their structure. Time permitting, before we say our \textbf{(Long) Goodbyes} we will also look at another type of \textbf{Nice (Guys)} identities: central polynomials. Additionally, we will use them to prove a not so \textbf{Small (Town Crime)} result, Rowan's theorem.
\\
\\
Hopefully, after this talk, as \textbf{Twilight} turns into \textbf{Night (Moves)}, you will go to \textbf{(The Big) Sleep} dreaming about PI algebras.

I will introduce and discuss some recent development of a fundamental problem in topology - the classification of continuous maps between spheres up to homotopy. These mathematical objects are called homotopy groups of spheres. I will start with some geometric background - its connection to the Generalized Poincare Conjecture for example. I will then introduce some classical and new methods of doing such computations, using certain spectral sequences. If time permits, I will discuss some recent development using motivic homotopy theory, a theory that was designed to use algebraic topology to study algebraic geometry, but has now been applied successfully in the reverse direction. Old and new open problems will be mentioned along the discussion.

In this talk, I will discuss when two probability measures on the unit sphere can be transported to one another using the Gauss map of a convex body. Here, a convex body is a compact convex subset of the Euclidean n-space with non-empty interior. Notice that the boundary of a convex body might not be smooth---in general, it can even contain a fractal structure. This problem can be viewed as the problem of reconstructing a convex body using partial data regarding its Gauss map. When smoothness is assumed, it reduces to a Monge-Ampere type equation on the sphere. However, in this talk, we will work with generic convex bodies and talk about how variational argument can work in this setting.
\\
\\
This is joint work with K\'aroly B"{o}r"{o}czky, Erwin Lutwak, Deane Yang, and Gaoyong Zhang.

In the point location problem one is given a hyperplane arrangement and an unknown point. By making linear queries about that point one wants to determine which cell of the hyperplane arrangement it lies in. This problem has an unexpected connection to the problem in machine learning of actively learning a halfspace. We discuss these problems and their relationship and provide a new and nearly optimal algorithm for solving them.

We give an asymptotic formula for the number of partitions of an integer n where the sums of the kth powers of the parts are also fixed, for some collection of values k. To obtain this result, we reframe the counting problem as an optimization problem, and find the probability distribution on the set of all integer partitions with maximum entropy among those that satisfy our restrictions in expectation (in essence, this is an application of Jaynes' principle of maximum entropy). This approach leads to an approximate version of our formula as the solution to a relatively straightforward optimization problem over real-valued functions. To establish more precise asymptotics, we prove a local central limit theorem using an equidistribution result of Green and Tao.
\\
\\
A large portion of the talk will be devoted to outlining how our method can be used to re-derive a classical result of Hardy and Ramanujan, with an emphasis on the intuitions behind the method, and limited technical detail. This is joint work with Marcus Michelen and Will Perkins.

Classically over the rational numbers, the exponential and
logarithm series converge $p$-adically within some open disc of
$\mathbb{C}_p$. For function fields, exponential and logarithm series
arise naturally from Drinfeld modules, which are objects constructed by
Drinfeld in his thesis to prove the Langlands conjecture for
$\mathrm{GL}_2$ over function fields. For a ``finite place'' $v$ on such a
curve, one can ask if the exp and log possess similar $v$-adic
convergence properties. For the most basic case, namely that of the
Carlitz module over $\mathbb{F}_q[T]$, this question has been long
understood. In this talk, we will show the $v$-adic convergence for
Drinfeld-(Hayes) modules on elliptic curves and a certain class of
hyperelliptic curves. As an application, we are then able to obtain a
formula for the $v$-adic $L$-value $L_v(1,\Psi)$ for characters in these
cases, analogous to Leopoldt's formula in the number field case.

Waiting in a queue (or a line) for some type of service is a commonplace experience. People do this at a grocery store, bank, amusement park or DMV, to give just a few examples. These customer service systems feature inherent randomness, which impacts performance. Modern computer and communications systems, manufacturing processes, transportation systems and even biological networks experience similar stochastic effects. Stochastic network theory is the study of the performance and optimal control of such systems. At the beginning of the talk, I will show a few simple examples of where mathematics plays an integral role in illuminating system behavior. Following that I will discuss some of the mathematical challenges associated with analyzing the performance of modern networks.
\\
\\
Finally, I will end by discussing work-in- progress related to modern call centers that is joint with Amy Ward (U. Chicago Booth) and Yueyang Zhong (U. Chicago Booth).

Gopakumar-Vafa type invariants on Calabi-Yau 4-folds (which
are non-trivial only for genus zero and one) are defined by
Klemm-Pandharipande from Gromov-Witten theory, and their integrality
is conjectured. In this talk, I will explain how to give a sheaf
theoretical interpretation of them using counting invariants on moduli
spaces of one dimensional stable sheaves.
\\
\\
Based on joint works with D. Maulik and Y. Toda.

Let $A$ be a $C^*$-algebra, $f \colon \mathbb{R} \to \mathbb{C}$ be a continuous function, and $\tilde{f} \colon A_{\text{sa}} \to A$ be the functional calculus map $A_{\text{sa}} \ni a \mapsto f(a) \in A$. It is elementary to show that $\tilde{f}$ is continuous, so it is natural to wonder how the differentiability properties of $f$ relate/transfer to those of $\tilde{f}$. This turns out to be a delicate, complicated problem. In this talk, I introduce a rich class $NC^k(\mathbb{R}) \subseteq C^k(\mathbb{R})$ of noncommutative $C^k$ functions $f$ such that $\tilde{f}$ is $k$-times differentiable. I shall also discuss the interesting objects, called multiple operator integrals, used to express the derivatives of $\tilde{f}$.

Bilevel optimization problem is a two-level optimization problem, where a subset of its variables is constrained in the optimizer set of another optimization problem parameterized by the remaining variables. In this talk, we introduce a Lagrange multiplier expression method for bilevel polynomial optimization based on polynomial optimization relaxations. Each relaxation is obtained from the Kurash-Kuhn-Tucker (KKT) conditions for the lower level optimization and the exchange technique for semi-infinite programming. The global convergence of the method is proved under some general assumptions. And some numerical examples will be given to show the efficiency of the method.

Report on work in progress, joint with Paul Arne Oestvaer.
\\
\\
A cellular motivic invariant is a special type of functor from the
category of commutative rings (or the opposite of schemes, say) to
spectra. Examples include algebraic K-theory, motivic cohomology, \'e{}tale
cohomology and algebraic cobordism. Dwyer-Friedlander observed that for
2-adic \'e{}tale K-theory and certain related invariants, the value on
Z[1/2] can be described in terms of a fiber square involving the values
on the real numbers, the complex numbers, and the field with three elements.
\\
\\
I will explain a generalization of this result to arbitrary 2-adic
cellular motivic invariants. As an application, we show that $\pi_0$ of the
motivic sphere spectrum over Z[1/2] is given by the Grothendieck-Witt
ring of Z[1/2], up to odd torsion.

Consider a town of $n$ people and suppose this town wants to impose the following rules on its clubs (formally subsets of the towns $n$ residents). First, each club must have an odd number of members. Second, each distinct pair of clubs in the town must have an even number of members in common. What is the maximum number of clubs this town can have while adhering to the rules?

The last decade has seen tremendous progress in applying random matrix methods to adjacency matrices or Laplacians of random graphs, in order to understand their spectra and be able to apply the new results to algorithms in machine learning, coding theory, data science, etc. Nevertheless, many problems remain. I will present some of the most interesting tools and new results and mention some (still) open problems.

We find a family of 3d steady gradient Ricci solitons that are flying wings. This verifies a conjecture by Hamilton. For a 3d flying wing, we show that the scalar curvature does not vanish at infinity. The 3d flying wings are collapsed. For dimension $n \geq 4$, we find a family of $Z2$ $\times$ $O(n - 1)$-symmetric but non-rotationally symmetric n-dimensional steady gradient solitons with positive curvature operator. We show that these solitons are non-collapsed.

I will talk about some results and open questions related
to the moduli of maps of curves to K3 surfaces, sheaves
on K3 surfaces, and moduli of K3 surfaces themselves.

In an ongoing work, we introduce the notion of isometric orbit equivalence for probability measure preserving actions of marked groups. This notion asks the Schreier graphings defined by the actions of the marked groups to be isomorphic. In the first part of the talk, we will prove that pmp actions of a marked group whose Cayley graph has a discrete automorphisms group are rigid up to isometric orbit equivalence. In a second time, we will explain how to construct pmp actions of the free group that are isometric orbit equivalent but not conjugate.

The Invariant Subspace Problem is one of the most famous problems in Operator Theory, and is concerned with the search of non-trivial, closed, invariant subspaces for bounded operators acting on a separable Banach space. Considerable success has been achieved over the years both for the existence of such subspaces for many classes of operators, as well as for non-existence of invariant subspaces for particular examples of operators. However, for the most important case of a separable Hilbert space, the problem is still open.
\\
\\
A natural, related question deals with the existence of invariant subspaces for perturbations of bounded operators. These types of problems have been studied for a long time, mostly in the Hilbert space setting. In this talk I will present a new approach to these ``perturbation'' questions, in the more general setting of a separable Banach space. I will focus on the recent history, presenting several new results that were obtained along the way with this new approach, and examining their connection and relevance to the Invariant Subspace Problem.

Efficient optimization has become one of the major concerns in data analysis. There has been a lot of focus on first-order optimization algorithms because of their low cost per iteration. In 1983, Nesterov's Accelerated Gradient method (NAG) was shown to converge in $O(1/k^2)$ to the minimum of the convex objective function $f(x)$, improving on the $O(1/k)$ convergence rate exhibited by the standard gradient descent methods, which is the phenomenon referred to as acceleration. It was shown that NAG limits to a second order ODE, as the time step goes to 0, and that the objective function $f(x(t)$) converges to its optimal value at a rate of $O(1/t^2)$ along the trajectories of this ODE. In this talk, we will discuss how the convergence of $f(x(t))$ can be accelerated in continuous time to an arbitrary convergence rate $O(1/t^p)$ in normed spaces, by considering flow maps generated by a family of time-dependent Bregman Lagrangian and Hamiltonian systems which is closed under time resca
ling. We will then discuss how this variational framework can be exploited together with the time-invariance property of the family of Bregman Lagrangians using adaptive geometric integrators to design efficient explicit algorithms for symplectic accelerated optimization. Finally, we will discuss briefly the generalization from normed spaces to Riemannian manifolds.

In the 1960's, Kervaire and Milnor boiled down the problem of counting smooth structure on spheres of dimension greater than 4 to the computation of stable homotopy groups of spheres and the Kervaire invariant one problem. In the following decades, the elements of Kervaire invariant one whose dimension are less or equals to 62 are shown to exist, and finally, Hill, Hopkins and Ravenel in their 2016 paper show that the Kervaire invariant one elements doesn't exist for dimension larger or equals to 254, leaving the 126-dimensional case open.
\\
\\
The $C_2$-equivariant Real bordism spectrum and its norms are crucial in HHR's solution, and computation of them is a central topic in equivariant stable homotopy theory. In this talk, I will explore two aspects of Real bordism and its norms:
\\
\\
1. How computation in Real bordism helps us to understand Lubin-Tate E-theories at p = 2. In particular, we can understand almost all actions of finite subgroups of the Morava stablizer groups on E-theories in homotopy.
\\
\\
2. The relation between Real bordism and the Segal conjecture. This relation allows us to bring new tools and perspective into this equivariant computation, and we will show how a spectral sequence based on (Real) topological Hochschild homology can help in understanding Real bordism and its norms.
\\
\\
This talk is based on joint work with Beaudry, Hill, Lawson, Meier and Shi.

One of the largest problems in Quantum Computation is how you deal with errors. Alexei Kitaev invented a method whereby we can use the discretization of a manifold to encode logical information in a subspace of the Hilbert space that corresponds to the homology of the Manifold itself. This has been vastly generalized, but we will restrict to looking at his original example of the toric code. We will go through how it can be used as an error-correcting code and several methods on how to actually compute on such a code. If time permits we can discuss recent results where expander graphs (and in general combinatorial methods) are used to construct a very good code which is then used to construct a manifold that solves a certain problem in differential topology.

There has been a growing interest in the study of nonlocal models as more general and sometimes more realistic alternatives to the conventional PDE models. We will give an introduction to nonlocal models in this talk. In particular, we will focus on the nonlocal models with a finite range of nonlocal interactions, which serve as bridges connecting the classical PDEs, nonlocal discrete models and the fractional differential equations. This talk will cover topics including nonlocal modeling, nonlocal calculus and numerical analysis for the nonlocal models.

Modern machine learning methods (i.e. neural networks) have been very successful in tasks such as image classification and speech recognition, but have been shown to be extremely brittle to small, adversarially-chosen perturbations of their inputs. This is a critical issue in many deep learning applications (e.g. object detection, robotic perception, ranking and recommendation, etc.). In this talk, I will provide an overview of the problem of adversarial robustness, formally introduce some general principles (what we know and what we don't know about this phenomenon), and discuss heuristic solutions (methods that appear to work in practice) and recent certification techniques (how do we provably - and efficiently - guarantee robustness?).

Lojasiewicz inequalities are a popular tool for studying the stability of geometric structures. For mean curvature flow, Schulze used Simon's reduction to the classical Lojasiewicz inequality to study compact tangent flows. For round cylinders, Colding and Minicozzi instead used a direct method to prove Lojasiewicz inequalities. We'll discuss similarly explicit Lojasiewicz inequalities and applications for other shrinking cylinders and Clifford shrinkers.

We provide a new general upper bound for the minimal L2-worst-case recovery error in the framework of RKHS, where only n function samples are allowed. This quantity can be bounded in terms of the singular numbers of the compact embedding into the space of square integrable functions. It turns out that in many relevant situations this quantity is asymptotically only worse by square root of log(n) compared to the singular numbers. The algorithm which realizes this behavior is a weighted least squares algorithm based on a specific set of sampling nodes which works for the whole class of functions simultaneously. These points are constructed out of a random draw with respect to distribution tailored to the spectral properties of the reproducing kernel (importance sampling) in combination with a sub-sampling procedure coming from the celebrated proof of Weaver's conjecture, which was shown to be equivalent to the Kadison-Singer problem. For the above multivariate setting, it is still a fundamental open problem whether sampling algorithms are as powerful as algorithms allowing general linear information like Fourier or wavelet coefficients. However, the gap is now rather small. As a consequence, we may study well-known scenarios where it was widely believed that sparse grid sampling recovery methods perform optimally. It turns out that this is not the case for dimensions d greater than 2.
\\
\\
This is joint work with N. Nagel and M. Schaefer from TU Chemnitz.

Given an elliptic curve E, Kolyvagin used CM points on
modular curves to construct a system of classes valued in the Galois
cohomology of the torsion points of E. Under the conjecture that not
all of these classes vanish, he gave a description for the Selmer group
of E. This talk will report on recent work proving new cases of
Kolyvagin's conjecture. The methods follow in the footsteps of Wei
Zhang, who used congruences between modular forms to prove Kolyvagin's
conjecture under some technical hypotheses. We remove many of these
hypotheses by considering congruences modulo higher powers of p. The
talk will explain the difficulties associated with higher congruences of
modular forms and how they can be overcome. I will also provide an
introduction to the conjecture and its consequences, including a `converse theorem': algebraic rank one implies analytic rank one.

Our work was motivated by the analysis projects using the linked US SEER-Medicare database to study mortality in men of age 65 years or older who were diagnosed with prostate cancer. Such data sets contain up to 100,000 human subjects and over 20,000 claim codes. For studying mortality the number of deaths are the ``effective'' sample size, so here we are in the situation of p is greater than n which is referred to as having high-dimensional predictors. In addition, a patient might die of cancer, or of other causes such as heart disease etc. These are referred to as competing risks. How to best perform prediction which inevitably involves variable selection for this type of complex survival data had not been previously investigated. Interest may also lie in comparing treatments such as radical prostatectomy versus conservative treatment. In this case the data were obviously not randomized with regard to the treatment assignments, and confounding most likely exists, possibly even beyond the commonly captured clinical variables in the SEER database. We will showcase research work done by our former PhD students from the UCSD Math Dept to account for such unobserved confounding, as well as efforts to make use of the high dimensional claims codes which have been shown to contain rich information about the patients survival.

Numerical algebraic geometry studies methods to approach problems in algebraic geometry numerically. Especially, finding roots of systems of equations using theory in algebraic geometry involves symbolic algorithm which requires expensive computations. However, numerical techniques often provides faster methods to tackle these problems. We establish numerical techniques to approximate roots of systems of equations and ways to certify its correctness.
\\
\\
As techniques for approximating roots of systems of equations, homotopy continuation method will be introduced. Since numerical approaches rely on heuristic method, we study how to certify numerical roots of systems of equations. Krawczyk method from interval arithmetic and Smale's alpha theory will be used as main paradigms for certification. Furthermore, as an approach for multiple roots, we establish the local separation bound of a multiple root. For a regular quadratic multiple zero, we give their local separation bound and study how to certify an approximation of such multiple roots.

This talk, the first on the third paper https://arxiv.org/abs/2009.03243 of a series, is partly a continuation of talks given in the fall.
See: http://www.math.ucsd.edu/\~{}benchow/cc-seminar.html
\\
\\
Some review will be given to make the talks more self-contained.

Our settings are one-dimensional compact symbolic systems. We discuss the flexibility of the pressure function of a continuous potential (observable) with respect to a parameter regarded as the inverse temperature. The points of non-differentiability of this function are of particular interest in statistical physics since they correspond to qualitative changes of the characteristics of a dynamical system referred to as phase transitions. It is well known that the pressure function is convex, Lipschitz, and has an asymptote at infinity. We show that these are the only restrictions. We present a method to explicitly construct a continuous potential whose pressure function coincides with any prescribed convex Lipschitz asymptotically linear function starting at a given positive value of the parameter.
\\
\\
This is based on joint work with Anthony Quas.

For a non-amenable group $G$, there may be many (exotic) group $C^*$-algebras that lie naturally between the universal and the reduced $C^*$-algebra of $G$. Let $G$ be a simple Lie group or an appropriate locally compact group acting on a tree. I will explain how the $L^p$-integrability properties of different spherical functions on $G$ (relative to a maximal compact subgroup) can be used to distinguish between different (exotic) group $C^*$-algebras. This recovers results of Samei and Wiersma. Additionally, I will explain that under certain natural assumptions, the aforementioned exotic group $C^*$-algebras are the only ones coming from $G$-invariant ideals in the Fourier-Stieltjes algebra of $G$.
\\
\\
This is based on joint work with Dennis Heinig and Timo Siebenand.

A classical question in differential topology is the following: Classify all simply-connected, closed, smooth (2n)-manifolds whose only non-trivial homology groups are $H_0, H_n$ and $H_{2n}$.
In this talk I will survey the history of the high dimensional side of this question and how its resolution requires a surprisingly deep understanding of the Adams spectral sequence computing the stable homotopy groups of spheres. Time permitting, I will then discuss how the situation changes as we relax our topological restrictions on the manifold (for example allowing $H_{n-e}$, $H_{n-e+1}$, ... $H_{n+e}$ to be non-trivial for a small number e).
\\
\\
This talk represents joint work with Jeremy Hahn and Andy Senger.

Interior Point methods and Sequential Quadratic Programming (SQP) methods have become two of the most crucial methods for solving large-scale nonlinear optimization problems. The two methods take very different approaches to solving the same problem. SQP methods find approximate solutions to a sequence of linearly constrained quadratic subproblems in which a quadratic model of the Lagrangian is minimized subject to a linear model of the constraints. Typically, the QP subproblems are solved using an active-set method, giving the problem a major-minor iteration pattern in which each iteration of the active-set method solves an indefinite system. In contrast, interior point methods follow a continuous path towards the optimal solution by perturbing the first-order optimality conditions of the problem. In this talk, we discuss a shifted primal dual interior point method and its potential applicability in solving the QP subproblem of an SQP method. We also discuss some of
the potential issues with this approach that we hope to overcome.

Set theory has long been considered the foundation of all mathematical thought. However, people who prove theorems on computers for a living don't use set theory anymore--and now some suggest that mathematicians should do the same. We'll discuss the problems people have with Set Theory and its main alternative, Type Theory.

How many ways can a positive integer be written as the sum of four squares? There is a simple formula for the number of ways, which goes back to Jacobi. I'll introduce modular forms and sketch how they provide an answer to this question.

I present new stochastic multi-particle models for global optimization of nonconvex functions on the sphere. These models belong to the class of Consensus-Based Optimization methods. In fact, particles move over the manifold driven by a drift towards an instantaneous consensus point, computed as a combination of the particle locations weighted by the cost function according to Laplace's principle. The consensus point represents an approximation to a global minimizer. The dynamics is further perturbed by a random vector field to favor exploration, whose variance is a function of the distance of the particles to the consensus point. In particular, as soon as the consensus is reached, then the stochastic component vanishes. In the first part of the talk, I present the well-posedness of the model on the sphere and we derive rigorously its mean-field approximation for large particle limit.
\\
\\
In the second part I address the proof of convergence of numerical schemes to global minimizers provided conditions of well-preparation of the initial datum. The proof combines the mean-field limit with a novel asymptotic analysis, and classical convergence results of numerical methods for SDE. We present several numerical experiments, which show that the proposed algorithm scales well with the dimension and is extremely versatile. To quantify the performances of the new approach, we show that the algorithm is able to perform essentially as good as ad hoc state of the art methods in challenging problems in signal processing and machine learning, namely the phase retrieval problem and the robust subspace detection.
\\
\\
Joint work with H. Huang, L. Pareschi, and P. S"{u}nnen

A generalization of Mazur's theorem states that there are 26
possibilities for the torsion subgroup of an elliptic curve over a
quadratic extension of $\mathbb{Q}$. If $G$ is one of these groups, we
count the number of elliptic curves of bounded naive height whose
torsion subgroup is isomorphic to $G$ in the case of imaginary quadratic
fields.

A projective manifold is algebraically hyperbolic if the
degree of any curve is bounded from above by its genus times a
constant, which is independent from the curve. This is a property which
follows from Kobayashi hyperbolicity. We prove that hyperkahler
manifolds are not algebraically hyperbolic when the Picard rank is at
least 3, or if the Picard rank is 2 and the SYZ conjecture on existence
of Lagrangian fibrations is true. We also prove that if the automorphism
group of a hyperkahler manifold is infinite, then it is algebraically
non-hyperbolic.
\\
\\
These results are a joint work with Misha Verbitsky.

Krieger’s generator theorem says that all free ergodic measure preserving actions (under natural entropy constraints) can be modelled by a full shift. Recently, in a sequence of two papers Mike Hochman noticed that this theorem can be strengthened: He showed that all free homeomorphisms of a Polish space (under entropy constraints) can be Borel embedded into the full shift. In this talk we will discuss some results along this line from a recent paper with Tom Meyerovitch and ongoing work with Spencer Unger.
\\
\\
With Meyerovitch, we established a condition called flexibility under which a large class of systems are almost Borel universal, meaning that such systems can model any free Z\^{}d action on a Polish space up to a null set. The condition of flexibility covered a large class of examples including those of domino tilings and the space of proper 3-colourings and answered questions by Robinson and Sahin. However extending the embedding to include the null set is a daunting task and there are many partial results towards this. Using tools developed by Gao, Jackson, Krohne and Seward, along with Spencer Unger we were able to get Borel embedding of symbolic systems (as opposed to all Borel systems) under some very similar assumptions which still covered all the examples that we were interested in. This answered questions by Gao and Jackson and recovered results announced by Gao, Jackson, Krohne and Seward.

Given a smooth Jordan curve and a cyclic quadrilateral (a cyclic quadrilateral is a quadrilateral that can be inscribed in a circle) we show that there exist four points on the Jordan curve forming the vertices of a quadrilateral similar to the one given. The smoothness condition cannot be dropped (since not all cyclic quadrilaterals can be inscribed in all triangles), while the cyclicity is necessary (since the circle is itself a smooth Jordan curve). The proof involves some results in symplectic topology. No prior knowledge assumed.
\\
\\
Joint work with Josh Greene.

I'll discuss joint with Jekel-Nelson-Sinclair. We give the first free entropy proof of Popa's famous result that the generator MASA in a free group factor is maximal amenable, and we partially recover Houdayer's results on amenable absorption and Gamma stability. Moreover, we give a unified approach to all these results using 1-bounded entropy. The main techniques are concentration of measure on unitary groups as well as Voiculescu's asymptotic freeness theorem.

Consider the following statement: If $G$ is a finitely generated group, and all elements of $G$ have finite order, then $G$ is a finite group. Is it true? Nope. We'll construct a counterexample (the Grigorchuk group), and then talk a little about the properties and representations of any such counterexample.

In ergodic theory, one often studies measure-preserving actions of countable groups on probability spaces. Bernoulli shifts are a class of such actions that are particularly simple to define, but despite several decades of study some elementary questions about them still remain open, such as how they are classified up to isomorphism. Progress in understanding Bernoulli shifts has historically gone hand-in-hand with the development of a tool known as entropy. In this talk, I will review classical concepts and results, which apply in the case where the acting group is amenable, and then I will discuss recent developments that are beginning to illuminate the case of non-amenable groups.

I will show how to construct a very large family of new examples of convex ancient and translating solutions to mean curvature flow in all dimensions. At $t=-\infty$, these examples resemble a family of standard Grim hyperplanes of certain prescribed orientations. The existence of such examples has been suggested by Hamilton and Huisken—Sinestrari. Our examples include solutions with symmetry group $D\times \mathbb Z_2$, where $D$ is the symmetry group of any given regular polytope, and, surprisingly, many examples which admit only a single reflection symmetry. We also exhibit a family of eternal solutions which do not evolve by translation, settling a conjecture of Brian White in the negative. Time permitting, I will present further structure and partial classification results for this class of solutions, as well as some open questions and conjectures.
\\
\\
Joint with T. Bourni and G. Tinaglia.

The success of deep learning is due, to a large extent, to the remarkable effectiveness of gradient-based optimization methods applied to large neural networks. In this talk I will discuss some general mathematical principles allowing for efficient optimization in over-parameterized non-linear systems, a setting that includes deep neural networks. Remarkably, it seems that optimization of such systems is "easy". In particular, optimization problems corresponding to these systems are not convex, even locally, but instead satisfy locally the Polyak-Lojasiewicz (PL) condition allowing for efficient optimization by gradient descent or SGD. We connect the PL condition of these systems to the condition number associated to the tangent kernel and develop a non-linear theory parallel to classical analyses of over-parameterized linear equations.
\\
\\
In a related but conceptually separate development, I will discuss a new perspective on the remarkable recently discovered phenomenon of transition to linearity (constancy of NTK) in certain classes of large neural networks. I will show how this transition to linearity results from the scaling of the Hessian with the size of the network.
\\
\\
Joint work with Chaoyue Liu and Libin Zhu.

To a cuspidal automorphic representation of GSp(4) over
$\mathbb Q$, one can associate a compatible system of Galois
representations $\{\rho_p\}_{p \; \mathrm{prime}}$. For $p$ sufficiently
large, the deformation theory of the mod-$p$ reduction $\overline
\rho_p$ is expected to be unobstructed -- meaning the universal
deformation ring is a power series ring. The global obstructions to
deforming $\overline \rho_p$ is controlled by certain adjoint Bloch-Kato
Selmer groups, which are expected to be trivial for $p$ large enough.
\\
\\
I will talk about some recent results showing that there are no local
obstructions to the deformation theory of $\overline \rho_p$ for almost
all $p$.
\\
\\
This is joint work with M. Broshi, C. Sorensen, and T. Weston.

In complex algebraic geometry, the decomposition
theorem asserts that semisimple geometric objects remain semisimple
after taking direct images under proper algebraic maps. This was
conjectured by Kashiwara and is proved by Mochizuki and Sabbah in a
series of long papers via harmonic analysis and D-modules.
In this talk, I would like to explain a simpler proof in the case of
semisimple local systems using a more geometric approach. As a
byproduct, we recover a weak form of Saito's decomposition theorem for
variations of Hodge structures.
\\
\\
Joint work in progress with Chuanhao Wei.

This talk, the first on the third paper https://arxiv.org/abs/2009.03243 of a series, is partly a continuation of talks given in the fall. See: http://www.math.ucsd.edu/\~{}benchow/cc-seminar 20.html

This talk will be about a pair of related matrix integrals, the Harish-Chandra/Itzykson-Zuber integral and the Brezin-Gross-Witten integral, which play an important role in random matrix theory, representation theory, and mathematical physics. While these integrals cannot be exactly evaluated, an old conjecture says that they admit asymptotic expansions whose coefficients are themselves generating functions for some unspecified combinatorial invariants of compact Riemann surfaces (or smooth projective curves).

When $G$ is a Polish group, one way of knowing that it has nice
dynamics is to show that $M(G)$, the universal minimal flow of $G$, is
metrizable. For non-Polish groups, this is not the relevant dividing
line: the universal minimal flow of the symmetric group of a set of
cardinality $\kappa$ is the space of linear orders on $\kappa$---not
a metrizable space, but still nice---, for example.
In this talk, we present a set of equivalent properties of topological
groups which characterize having nice dynamics. We show that the class
of groups satisfying such properties is closed under some topological
operations and use this to compute the universal minimal flows of some
concrete groups, like $\mathrm{Homeo}(\omega_{1})$.
\\
\\
This is joint work with Andy Zucker.

We investigate some structural properties of C*-algebras generated by quasi-regular representations of stabilizers of boundary actions of discrete groups G. Our main tool is the notion of (noncommutative) boundary maps, namely G-equivariant unital positive maps from G-C*algebras to C(B), where B is the Furstenberg boundary of G. We completely describe the tracial structure and characterize the simplicity of these C*-algebras. As an application, we show that the C*-algebra generated by the quasi-regular representation associated to Thompson's groups $F < T$ does not admit traces and is simple.
\\
\\
This is joint work with Eduardo Scarparo.

Levenberg-Marquardt (LM) algorithms are a class of methods that add a regularization term to a Gauss-Newton method to promote better convergence properties. This talk presents three works on this class of methods. The first discusses a new method that simultaneously achieves all types of state of the art convergence guarantees for unconstrained problems. Stochastic LM is discussed next, which is an algorithm to handle noisy data. An example is presented on data assimilation. Finally, a LM method is presented to handle equality constraints, with examples from inverse problems in PDEs.

The fundamental group is one of the most powerful invariants to distinguish closed three-manifolds. One measure of the non-triviality of a three-manifold is the existence of non-trivial SU(2)-representations. In this talk I will show that if an integer homology three-sphere contains an embedded incompressible torus, then its fundamental group admits irreducible SU(2)-representations.
\\
\\
This is joint work with Tye Lidman and Raphael Zentner.

Real Grassmannians' uses in geometry are manifold, but in general, their integral (co)homology groups were unknown. Until now. I won't Stiefel myself any longer, and I will (co)change your views on this class of manifolds. At the end of this talk, you should be able to differentiate Grassmannian manifolds and feel right at (co)home with them, K? This talk will Blow Up your mind, Link some ideas you may not have heard of, and you'll take an Exit-Path out with your mind (po)set straight.

In this talk, we consider Riemannian manifolds with almost non-negative scalar curvature and Perelman entropy. We establish an $\epsilon$-regularity theorem showing that such a space must be close to Euclidean space in a suitable sense. Interestingly, such a result is false with respect to the Gromov-Hausdorff and Intrinsic Flat distances, and more generally the metric space structure is not controlled under entropy and scalar lower bounds. Instead, we introduce the notion of the $d_p$ distance between (in particular) Riemannian manifolds, which measures the distance between $W^{1,p}$ Sobolev spaces, and it is with respect to this distance that the $\epsilon$ regularity theorem holds. We will discuss various applications to manifolds with scalar curvature and entropy lower bounds, including a compactness and limit structure theorem for sequences, a uniform $L^\infty$ Sobolev embedding, and a priori $L^p$ scalar curvature bounds for $p<1$.
\\
\\
This is joint work with Man-Chun Lee and Aaron Naber.

We study the benign overfitting phenomenon induced by simple optimization algorithms in deep learning.
Oftentimes the neural network is over-parameterized in the sense that the number of parameters exceeds the
training data size, but the obtained solution generalizes well to unseen data. The generalization stems from
an implicit regularization of the optimization algorithm. We present the recent theoretical development of
over-parameterization for linear/non-linear models, together with some numerical experiments.

We briefly overview the current developments of rigorous constructions in "stochastic quantization” - an active field linking quantum field theory with stochastic PDE.
We then focus on stochastic quantization of the Yang-Mills model in 2 and 3 space dimensions.
This includes constructing the Langevin dynamic for the formal Yang-Mills measure, defining the state space of gauge orbits, proving gauge equivariance of the dynamic, and making sense of Wilson loop observables in this context. We will also discuss some future directions.
\\
\\
The talk is based on several works mostly joint with A.Chandra, I.Chevyrev, and M.Hairer.

Sadly, I won't have time to prove the Riemann hypothesis in
this talk. However, I do hope to outline recent work in a record
partial-verification of RH.
\\
\\
This is joint work with Dave Platt, in
Bristol, UK.

Epistasis is a situation when the effect of one mutation changes as other mutations are introduced into the genome. Epistasis is used in genetics to probe functional relationships between genes, and it also plays an important role in evolution. However, there is no theory to understand how functional relationships at the molecular level translate into epistasis at the level of whole-organism phenotypes, such as fitness. I will present a simple model of a hierarchical metabolic network with first-order kinetics which helps us gain some intuition in this problem. I will derive two rules for how epistasis between mutations with small effects propagates from lower- to higher-level phenotypes and how such epistasis depends on the topologyof the network. Most importantly, weak epistasis at a lower level may be distorted as it propagates to higher levels. These results suggest that pairwise inter-gene epistasis should be common and it should generically depend on the genetic background and environment. Furthermore, the epistasis coefficients measured for high-level phenotypes may not be sufficient to fully infer the underlying functional relationships.

The effectiveness of a graduate school admissions ultimately
rests on the quality of the information collected and the decision
making process that is used to arrive at a decision. Admissions relies
on faculty judgment combining a variety of tools including grades, test
scores, letters of recommendation, interviews, and student essays to
identify the best candidates. Unfortunately, current practice often
falls short of well established best practices leading to lower quality
decisions and the possible introduction of bias. In this talk, I will
make the case that improvement is urgently needed and then lay out both
a short and long term place for modernizing graduate school admissions.

I will introduce and discuss an invariant of
hypersurface singularities, Saito's minimal exponent (a.k.a. Arnold
exponent in the case of isolated singularities). This can be considered
as a refinement of the log canonical threshold, which is interesting in
the case of rational singularities. I will focus on recent work on this
invariant and remaining open problems.

With every Coxeter system one can associate a family of algebras considered as deformation of its group algebra. These are so-called Hecke algebras, which are classical objects of study in combinatorics and representation theory. Complex Hecke algebras admit a natural *-structure and a *-representation on Hilbert space. Taking the norm- and SOT-closure in such representation, one obtains Hecke operator algebras, which have recently seen increased attention.
\\
\\
In this talk, I will introduce Hecke operator algebras from scratch, focusing on the case of right-angled Coxeter groups. This case is particularly interesting from an operator algebraic perspective, thanks to its description by iterated amalgamated free products. I will survey known results on the structure of Hecke operator algebras, before I describe recent work that clarified the factor decomposition of Hecke von Neumann algebras. Two applications to representation theory will be presented. I will finish with some results on the scope and limits of K-theoretic classification of right-angled Hecke C*-algebras.
\\
\\
This is joint work with Adam Skalski.

I discuss an optimization framework for solving problems with group sparsity inducing regularization. Such regularizers include Lasso (L1), group Lasso, and latent group Lasso. The framework computes iterates by optimizing over small dimensional subspaces, thus keeping the cost per iteration relatively low. Theoretical convergence results and numerical tests on various learning problems will be presented.

In this talk we consider a canonical clustering problem where one receives unlabeled samples drawn from a balanced mixture of two elliptical distributions and aims for a classifier to estimate the labels. Many popular methods including PCA and k-means require individual components of the mixture to be somewhat spherical, and perform poorly when they are stretched. To overcome this issue, we propose a non-convex program seeking for an affine transform to turn the data into a one-dimensional point cloud concentrating around -1 and 1, after which clustering becomes easy. Our theoretical contributions are two-fold: (1) we show that the non-convex loss function exhibits desirable geometric properties when the sample size exceeds some constant multiple of the dimension, and (2) we leverage this to prove that an efficient first-order algorithm achieves near-optimal statistical precision without good initialization. We also propose a general methodology for clustering with flexible choices of feature transforms and loss objectives.

In this talk I will describe pathways to invite and retain underrepresented minorities and women local to San Diego and Tijuana in mathematics and sciences at UCSD. I will describe Math Jams, which I pioneered at San Diego City College, and how I can lead Math Jams at Living Learning Communities (LLCs) such as the African Black Diaspora LLC at Sixth College and Raza LLC at Eleanor Roosevelt College. I will discuss the Hesabu Circle, a safe space for black students of all ages pre-K to post-doc, and how math circles can be created for underrepresented minorities and women at UCSD supporting undergraduate and graduate students. I will discuss student success statistics with my Umoja and Puente students at San Diego City College and Upward Bound students.

Deep Learning is much publicized and has had great empirical success on challenging problems in learning. Yet there is no quantifiable proof of performance and certified guarantees for these methods. This talk will give an overview of Deep Learning from the viewpoint of mathematics and numerical computation.

The classical problem of counting elliptic curves with a
rational N-isogeny can be phrased in terms of counting rational points
on certain moduli stacks of elliptic curves. Counting points on stacks
poses various challenges, and I will discuss these along with a few ways
to overcome them. I will also talk about the theory of heights on stacks
developed in recent work of Ellenberg, Satriano and Zureick-Brown and
use it to count elliptic curves with an $N$-isogeny for certain $N$. The
talk assumes no prior knowledge of stacks and is based on joint work
with Brandon Boggess.

Kodaira classified the singular elliptic fibers
occurring on relatively minimal elliptic surfaces (over C). I will
explain a birational Kodaira's classifications for higher dimensional
elliptic fibrations. (Based on work in collaboration with T. Weigand)

The FFT problem, which was inspired by work of Guldemond, can be stated as follows: how can you fill a 3x3 grid with F's and T's such that it contains as many copies of the word "FFT" as possible? For example, the following two grids each contain 5 copies of the word FFT (we allow the word to be written forwards or backwards, and to appear in rows, columns, or diagonals):
\\
\\
\[\begin{matrix} F F T\\ F F T\\ F F T\end{matrix}\hspace{30pt} \begin{matrix} F T F\\ T F F\\ F F T\end{matrix}\]
\\
\\
Grubb claimed that there exists a grid containing 6 copies of FFT. Eight minutes later he claimed that actually, the best you could do is 5. He offered no proof of either claim. In this talk we consider a generalization of the FFT problem. Namely, given a word $w$ of length $n$ and a grid $G$ of letters, let $f(w,G)$ be the number of times $w$ appears in $G$, and let $f(w)=\max_G f(w,G)$. We determine $f(w)$ for a number of words, and in particular we determine $f(FFT)$, solving the FFT problem. I, Sam Spiro, will be the only person talking for the entire hour that the talk is given. Absolutely nothing out of the ordinary will happen during the talk.

Let $X=\mathrm{SL}_3(\mathbb{R})/\mathrm{SL}_3(\mathbb{Z})$ and $a(t)=\mathrm{diag}(t^2,t^{-1},t^{-1})$. The expanding horospherical group $U^+$ is isomorphic to $\mathbb{R}^2$. A result of Shah tells us that the $a(t)$-translates of a non-degenerate real-analytic curve in a $(U^+)$-orbit get equidistributed in $X$. It remains to study degenerate curves, i.e. planar lines $y=ax+b$. In this talk, we give a Diophantine condition on the parameter $(a,b)$ which serves as a necessary and sufficient condition for equidistribution.
\\
\\
Joint work with Kleinbock, Saxcé and Shah. If time permits, I will also talk about generalisations to $\mathrm{SL}_n(\mathbb{R})/\mathrm{SL}_n(\mathbb{Z})$. Joint work with Shah.

Given a nontrivial band sum of two knots, we may add full twists to the band to obtain a family of knots indexed by the integers. In this talk, I'll show that the knots in this family have the same knot Floer homology, the same instanton homology, but distinct Khovanov homology, generalizing a result of M. Hedden and L. Watson. A key component of the argument is a proof that each of the three knot homologies detects the trivial band. The main application is a verification of the generalized cosmetic crossing conjecture for split links.

A quiver is defined as a directed graph with an attitude towards representation theory. In this talk, I will introduce quiver representations and discuss a fundamental classification result due to Gabriel. If time permits, I will also discuss one possible answer to the question, ``Why are quivers?'' There are no prerequisites, and there will be many examples.

Prognostic models in survival analysis are aimed at understanding the relationship between patients' covariates and the distribution of survival time. Traditionally, semi-parametric models, such as the Cox model and the Additive Hazards model, have been assumed. In this talk I will derive a measure of explained variation under the Additive Hazards model showing its properties. Moreover I will describe the development of a new flexible method for survival prediction: DeepHazard, a neural network for time-varying risks. I will show its performance on popular real datasets.

I will discuss the problem of understanding the collapsing behavior of Ricci-flat Kahler metrics on a Calabi-Yau manifold that admits a holomorphic fibration structure, when the Kahler class degenerates to the pullback of a Kahler class from the base. I will present new work with Hans-Joachim Hein where we obtain a priori estimates of all orders for the Ricci-flat metrics away from the singular fibers, as a corollary of a complete asymptotic expansion.

The University of Minnesota Talented Youth Mathematics Program
(UMTYMP) is a selective, five-year accelerated mathematics program for
students in grades 6-12. During the course of the program, students take
advanced mathematics courses on University of Minnesota campuses,
starting with algebra and continuing through logic and proofs, linear
algebra, and multivariable calculus. The majority of UMTYMP students come
from one of three demographic groups: White/Caucasian, East Asian/East
Asian American, and South Asian/South Asian American. We use the term Asian/Asian American to describe students in the latter two demographic groups.
\\
\\
The model minority stereotype (MMS) is the classification of Asian/Asian
American students as gifted, ``academic whizzes'' who outperform their
peers (Choi \& Lahey, 2006). In 2020, we initiated an IRB-approved study
to understand the impact of MMS on Asian/Asian American students who are
labelled as ``gifted" and/or ``talented." In this talk, I will
discuss the process and results of the study, propose best practices for
instructors who interact with students navigating MMS, and suggest ideas
for follow-up studies on this topic.

We consider the Popularity Adjusted Block model (PABM) introduced by Sengupta and Chen (2018).
We argue that the main appeal of the PABM is the flexibility of the spectral properties of the graph
which makes the PABM an attractive choice for modeling networks that appear in biological sciences.
We expand the theory of PABM to the case of an arbitrary number of communities which possibly
grows with a number of nodes in the network and is not assumed to be known. We produce estimators
of the probability matrix and the community structure and provide non-asymptotic upper bounds for the
estimation and the clustering errors. We use the Sparse Subspace Clustering (SSC) approach for
partitioning the network into communities, the approach that, to the best of our knowledge, has not been
used for clustering network data. The theory is supplemented by a simulation study. In addition, we
show advantages of the PABM for modeling a butterfly similarity network and a human brain functional
network.

We explain how Hodge theory unifies three a priori very different
types of deformations of K3 surfaces: twistor spaces, Brauer (or Tate-Shafarevich)
families and Dwork families. All three share the property of transporting
complex multiplication from one fibre in the Noether-Lefschetz locus to
any other. This phenomenon is at the moment observed in all three cases but
geometrically only explained for Brauer families. The motivation comes
from the Hodge conjecture for squares of K3 surfaces which is still open.

We will discuss joint work with R. Bamler, B. Chow, Y. Deng, and Y. Zhang on 4-dimensional steady Ricci soliton singularity models with 3-cylindrical tangent flows at infinity, as well as mention the somewhat parallel work with Y. Zhang on the existence of asymptotic shrinkers on steady solitons with $Ric \geq 0$.

This presentation contributes to the theoretical background for a new quadratic prediction method for time series. We develop a theory of polyspectral factorization, providing new mathematical results for polyspectral densities. New bijections between a restricted space of higher-dimensional cepstral coefficients (where the restrictions are induced by the symmetries of the polyspectra) and the auto-cumulants are derived. Applications to modeling are developed; in particular, it is shown that semi-parametric nonlinear time series modeling can be accomplished by approximation of the cepstral representation of polyspectra.

Consider an ``ideal" optimization problem where constraints and objective function are defined in terms of expectations over some distribution P. The sample average approximation (SAA) -- a fundamental idea in stochastic optimization -- consists of replacing the expectations by an average over a sample from P. A key question is how much the solutions of the SAA differ from those of the original problem. Results by Shapiro from many years ago consider what happens asymptotically when the sample size diverges, especially when the solution of the ideal problem lies on the boundary of the feasible set. In joint work with Philip Thompson (Purdue), we consider what happens with finite samples. As we will see, our results improve upon the nonasymptotic state of the art in various ways: we allow for heavier tails, unbounded feasible sets, and obtain bounds that (in favorable cases) only depend on the geometry of the feasible set in a small neighborhood of the optimal solution. Our results combine ``localization" and ``fixed-point" type arguments inpired by the work of Mendelson with chaining-type inequalities. One of our contributions is showing what can be said when the SAA constraints are random.

We find a family of 3d steady gradient Ricci solitons that are flying wings. This verifies a conjecture by Hamilton. For a 3d flying wing, we show that the scalar curvature does not vanish at infinity. The 3d flying wings are collapsed. For dimension $n \geq 4$, we find a family of $\mathbb{Z}_2 \times O(n − 1)$-symmetric but non-rotationally symmetric n-dimensional steady gradient solitons with positive curvature operator. We show that these solitons are non-collapsed.

Random feature methods have been successful in various machine learning tasks, are easy to compute, and come with theoretical accuracy bounds. They serve as an alternative approach to standard neural networks since they can represent similar function spaces without a costly training phase. However, for accuracy, random feature methods require more measurements than trainable parameters, limiting their use for data-scarce applications or problems in scientific machine learning. This paper introduces the sparse random feature method that learns parsimonious random feature models utilizing techniques from compressive sensing. We provide uniform bounds on the approximation error for functions in a reproducing kernel Hilbert space depending on the number of samples and the distribution of features. The error bounds improve with additional structural conditions, such as coordinate sparsity, compact clusters of the spectrum, or rapid spectral decay. We show that the sparse random feature method outperforms shallow networks for well-structured functions and applications to scientific machine learning tasks.

In many scientific areas, a deterministic model (e.g., a differential equation) is equipped with parameters. In practice, these parameters might be uncertain or noisy, and so an honest model should provide a statistical description of the quantity of interest. Underlying this computational question is a fundamental one - If two ``similar" functions push-forward the same measure, are the new resulting measures close, and if so, in what sense? I will first show how the probability density function (PDF) can be approximated, using spectral and local methods, and present applications to nonlinear optics. We will then discuss the limitations of PDF approximation, and present an alternative Wasserstein-distance formulation of this problem, which yields a much simpler theory.

A self-similar k-graph is a pair consisting of a discrete group and a k-graph, such that the group acts on the k-graph self-similarly. For such a pair, one can associate it a universal C*-algebra, called the self-similar k-graph C*-algebra. This class of C*-algebras embraces many important and interesting C*-algebras, such as the higher rank graph C*-algebras of Kumjian-Pask, the Katsura algebra, the Nekrashevych algebra, and the Exel-Pardo algebra. In this talk, I will present some results about those C*-algebras, which are based on joint work with Hui Li.

Given a graph $G$, one can compute the eigenvalues of its adjacency matrix $A_G$. Remarkably, these eigenvalues can tell us quite a bit about the structure $G$. More generally, spectral graph theory consists of taking a graph $G$, associating to it a matrix $M_G$, and then using algebraic properties of $M_G$ to recover combinatorial information about $G$. In this talk we discuss some of the more common applications of spectral graph theory, as well as a very simple proof of the sensitivity conjecture due to Huang.

According to Perelman's work on Ricci flow with surgeries in dimension 3, we know that it is important to understand at least qualitative behaviors of singularity formation in order to perform surgeries. The situation in dimension 4 is much more complicated as some new types of singularity models may arise and the classification of the singularity models is far from complete. We expect that the singularity models should be solitons, self-similar solutions to the Ricci flow, and we expect that most of singularity models are shrinking gradient solitons with possible singularities by the recent work of Richard Bamler. Steady gradient Ricci solitons may also arise as singularity models and they are related to shrinking solitons with quadratic curvature growth. In a recent joint work with R. Bamler, B. Chow, Y. Deng, and Y. Zhang, we managed to classify tangent flows at infinity which can be viewed as a blow-down of 4 dimensional steady gradient Ricci soliton singularity models. When the tangent flow at infinity is 3-cylindrical, we can give very good qualitative characterization of such steady solitons. We will also mention the somewhat parallel work with Y. Zhang on the existence of asymptotic shrinkers on steady solitons with nonnegative Ricci curvature.

In 2002 Malle conjectured an asymptotic formula for the number
of $G$-extensions of a number field $K$ with discriminant bounded by
$X$. In this talk I will discuss recent joint work with Etienne Fouvry
on this conjecture. Our main result proves Malle's conjecture in the
special case of nonic Heisenberg extensions.

In this talk, we offer an entirely ``white box'' interpretation of deep (convolution) networks from the perspective of data compression (and group invariance). In particular, we show how modern deep layered architectures, linear (convolution) operators and nonlinear activations, and even all parameters can be derived from the principle of maximizing rate reduction (with group invariance). All layers, operators, and parameters of the network are explicitly constructed via forward propagation, instead of learned via back propagation. All components of so-obtained network, called ReduNet, have precise optimization, geometric, and statistical interpretation. There are also several nice surprises from this principled approach: it reveals a fundamental tradeoff between invariance and sparsity for class separability; it reveals a fundamental connection between deep networks and Fourier transform for group invariance – the computational advantage in the spectral domain (why spiking neurons?); this approach also clarifies the mathematical role of forward propagation (optimization) and backward propagation (variation). In particular, the so-obtained ReduNet is amenable to fine-tuning via both forward and backward (stochastic) propagation, both for optimizing the same objective.
\\
\\
This is joint work with students Yaodong Yu, Ryan Chan, Haozhi Qi of Berkeley, Dr. Chong You now at Google Research, and Professor John Wright of Columbia University.

Cubic hypersurfaces are the zero sets of homogeneous polynomials of degree 3. They have been, still are, and probably will be for quite some time, the subject of a lot of research. I will survey a few well-known and fun facts about cubic hypersurfaces and will also mention some open problems.

In the classical pointwise ergodic theorem for a probability measure preserving (pmp) transformation T, one takes averages of a given integrable function over the intervals {x, T(x), T2(x),..., Tn(x)} in front of the point x. We prove a "backward" ergodic theorem for a countable-to-one pmp T, where the averages are taken over subtrees of the graph of T that are rooted at x and lie behind x (in the direction of T-1). Surprisingly, this theorem yields forward ergodic theorems for countable groups, in particular, one for pmp actions of free groups of finite rank, where the averages are taken along subtrees of the standard Cayley graph rooted at the identity. This strengthens Bufetov's theorem from 2000, which was the most general result in this vein.
\\
\\
This is joint work with Anush Tserunyan.

One well-known strategy for distinguishing smooth structures on closed 4-manifolds is to produce a knot $K$ in $S^3$ which is (smoothly) slice in one smooth filling $W$ of $S^3$ but not slice in some homeomorphic smooth filling $W’$. There are many techniques for distinguishing smooth structures on complicated closed 4-manifolds, but this strategy stands out for it’s potential to work for 4-manifolds $W$ with very little algebraic topology. However, this strategy had never actually been used in practice, even for complicated $W$. I’ll discuss joint work with Manolescu and Marengon which gives the first application of this strategy. I’ll also discuss joint work with Manolescu which gives a systematic approach towards using this strategy to produce exotic definite closed 4-manifolds.

Logarithmic Sobolev inequalities (LSI) were first introduced by
Gross in the 70's, and later found rich connections to geometry,
probability, graph theory, optimal transport as well as information
theory. In recent years, logarithmic Sobolev inequalities for quantum
Markov semigroups have attracted a lot of attention and found
applications in quantum information theory and quantum many-body
system. For classical Markov semigroup on a probability space, an
important advantage of log-Sobolev inequalities is the tensorization
property that if two semigroups satisfies LSI, so does their tensor
product semigroup. Nevertheless, tensoraization property fails for LSI
in the quantum cases. In this talk, I'll present some recent progress
on tensor stable log-Sobolev inequalities for quantum Markov
semigroups.
\\
\\
This talk is based on joint works with Michael Brannan,
Marius Junge, Nicholas LaRacuente, Haojian Li and Cambyse Rouze.

Algebraic Geometry is known for its abstract nonsense and its towers of abstraction (one might even say skyscraper sheaves of abstraction). But for this talk, we'll forget about all of that -- we'll explore toric varieties, an extremely concrete way of constructing examples of algebraic varieties. We'll even see examples that are *gasp* not complex manifolds.

In this talk, I will describe a new class of domains in complex Euclidean space which is defined in terms of the existence of Kaehler metrics with good geometric properties. By definition, this class is invariant under biholomorphism. It also includes many well-studied classes of domains such as strongly pseudoconvex domains, finite type domains in dimension two, convex domains, homogeneous domains, and embeddings of Teichmuller space. Analytic problems are also tractable for this class, in particular we show that compactness of the dbar-Neumann operator on (0,q)-forms is equivalent to a growth condition of the Bergman metric. This generalizes an old result of Fu-Straube for convex domains.

Stochastic optimization algorithms have become indispensable in modern machine learning. An important question in this area is the difference between with-replacement sampling and without-replacement sampling -- does the latter have superior convergence rate compared to the former? A paper of Recht and Re reduces the problem to a noncommutative analogue of the arithmetic-geometric mean inequality where n positive numbers are replaced by n positive definite matrices. If this inequality holds for all n, then without-replacement sampling (also known as random reshuffling) indeed outperforms with-replacement sampling in some important optimization problems. In this talk, We will explain basic ideas and techniques in polynomial optimization and the theory of noncommutative Positivstellensatz, which allows us to reduce the conjectured inequality to a semidefinite program and the validity of the conjecture to certain bounds for the optimum values. Finally, we show that Recht--Re conjecture is false as soon as $n = 5$.
\\
\\
This is a joint work with Lek-Heng Lim.

I will report on my joint work with Samit Dasgupta on the
Brumer-Stark conjecture proving existence of the Brumer-Stark units and
on a conjecture of Dasgupta giving a p-adic analytic formula for these
units. I will present a sketch of our proof of the Brumer-Stark
conjecture and also mention applications to Hilbert's 12th problem, or
explicit class field theory.

Despite breakthrough performance, modern learning models are known to be highly vulnerable to small adversarial perturbations in their inputs. While a wide variety of recent adversarial training methods have been effective at improving robustness to perturbed inputs (robust accuracy), often this benefit is accompanied by a decrease in accuracy on benign inputs (standard accuracy), leading to a tradeoff between often competing objectives. Complicating matters further, recent empirical evidence suggests that a variety of other factors (size and quality of training data, model size, etc.) affect this tradeoff in somewhat surprising ways. In this talk we will provide a precise and comprehensive understanding of the role of adversarial training in the context of linear regression with Gaussian features and binary classification in a mixture model. We precisely characterize the standard/robust accuracy and the corresponding tradeoff achieved by a contemporary mini-max adversarial training approach in a high-dimensional regime where the number of data points and the parameters of the model grow in proportion to each other. Our theory for adversarial training algorithms also facilitates the rigorous study of how a variety of factors (size and quality of training data, model overparametrization etc.) affect the tradeoff between these two competing accuracies.

Virtual counting of coherent sheaves has seen recently a large
development in complex dimension four, where it was defined for
Calabi--Yau fourfolds by Borisov--Joyce and Oh--Thomas. I will focus
on invariants for Hilbert schemes of points as they have not been well
understood before. The only known result expressed integrals of top
Chern classes of tautological vector bundles associated to smooth
divisors in terms of the MacMahon function and Cao--Kool conjectured
this holds for any line bundle. To address these questions I discuss
the conjectural wall-crossing formulae of Joyce and discuss how to
relate them to the conjectures on Hilbert schemes. On the other hand,
Arbesfeld--Johnson--Lim--Oprea--Pandharipande studied Quot-schemes on
surfaces and their virtual integrals giving explicit expressions for
their generating series. Interestingly, these satisfy similar
wall-crossing formulae as Hilbert schemes in the fourfold case when
the curve class is zero. As a consequence their general invariants
share a large similarity. Computing explicitly virtual fundamental
classes and integrals on both, we can firstly recover the results in
the five author paper from a small piece of data. Moreover, we obtain
a universal transformation comparing integrals on Hilbert schemes on
fourfolds and elliptic surfaces.

Flow polytopes are an important class of polytopes in combinatorics whose lattice points and volumes have interesting properties and relations to other parts of geometric and algebraic combinatorics. These polytopes were recently related to (multiplex) juggling sequences of Butler, Graham and Chung. The Chan-Robbins-Yuen (CRY) polytope is a flow polytope with normalized volume equal to the product of consecutive Catalan numbers. Zeilberger proved this by evaluating the Morris constant term identity, but no combinatorial proof is known. There is a refinement of this formula that splits the largest Catalan number into Narayana numbers, which Mészáros gave an interpretation as the volume of a collection of flow polytopes. In this talk we will talk about the connection between juggling and flow polytopes and introduce a new refinement of the Morris identity with combinatorial interpretations both in terms of lattice points and volumes of flow polytopes.
\\
\\
The first part of the talk is based on joint work with Benedetti, Hanusa, Harris and Simpson and the second part is based on joint work with William Shi.

In these two talks, we will first discuss Cheeger and Naber's quantitative estimates of singular sets on manifolds with lower Ricci curvature and also review some recent developments. As consequences, we will discuss the hessian estimates of harmonic functions on Einstein manifolds.

The multisymplectic structure of Lagrangian and Hamiltonian PDEs is a covariant generalization of the field-theoretic symplectic structure and encodes many important physical conservation laws. Multisymplectic integrators are a class of numerical methods which, at the discrete level, preserve the multisymplectic structure of a Lagrangian or Hamiltonian field theory. By preserving this structure at the discrete level, a multisymplectic integrator admits discrete analogs of the conservation laws encoded by multisymplecticity. Such methods have been used, for example, for structure-preserving modeling of nonlinear wave phenomena and for stable discretizations of plasma physics problems. There have also been recent investigations into the application of discrete multisymplectic structures for lattice quantum field theory.
\\
\\
Traditionally, such multisymplectic integrators have been constructed from the variational perspective in the Lagrangian framework, or from directly discretizing the equations of motion in the Hamiltonian framework and subsequently determining the conditions for the discretization method to be multisymplectic in an ad hoc manner. In this talk, after discussing the necessary background material, I will discuss a systematic framework for constructing multisymplectic Hamiltonian integrators variationally utilizing the notion of a Type II generating functional. This framework only requires a choice of finite-dimensional function space and quadrature, so it is applicable to unstructured meshes, whereas traditional Hamiltonian multisymplectic integrators require rectangular meshes. As an application of this framework, I will derive the class of multisymplectic partitioned Runge--Kutta methods and show that, in this framework, discretizing via a tensor product partitioned Runge--Kutta expansion in spacetime is well-defined if and only if the partitioned Runge--Kutta methods are symplectic in space and time. This is joint work with Prof. Melvin Leok.
\\
\\
Time permitting, I will discuss future research directions, such as applications to lattice quantum field theory.

We introduce notions of finite presentation which serve as analogues of finite-dimensionality for operator modules over completely contractive Banach algebras. With these notions we then introduce analogues of the local lifting property, nuclearity, and semi-discreteness. For a large class of operator modules, we show that the local lifting property is equivalent to flatness, generalizing the operator space result of Kye and Ruan. We pursue applications to abstract harmonic analysis, where, for a locally compact group G, we show that A(G)-nuclearity of the inclusion $C*_r(G) \to C^*_r(G)**$ and $A(G)$-semi-discreteness of $VN(G)$ are both equivalent to amenability of $G$. We also present the equivalence between $A(G)$-injectivity of the crossed product $G\bar{\ltimes}M$, $A(G)$-semi-discreteness of $G\bar{\ltimes} M$, and amenability of W*-dynamical systems $(M,G,\alpha)$ with $M$ injective.

Consider a Conway two-sphere S intersecting a knot K in 4 points, and thus decomposing the knot into two 4-ended tangles, T and T’. We will first interpret Khovanov homology Kh(K) as Lagrangian Floer homology of a pair of specifically constructed immersed curves C(T) and C'(T’) on the dividing 4-punctured sphere S. Next, motivated by several tangle-replacement questions in knot theory, we will describe a recently obtained structural result concerning the curve invariant C(T), which severely restricts the types of curves that may appear as tangle invariants. The proof relies on the matrix factorization framework of Khovanov-Rozansky, as well as the homological mirror symmetry statement for the 3-punctured sphere.
\\
\\
This is joint work with Liam Watson and Claudius Zibrowius.

Point-free measure theory is an approach to measure theory in a more abstract viewpoint. Specifically, it forgets the notion of ``points" and tries to recover the whole measure theory only with the notion of measurable sets. We will briefly see why this viewpoint has a potential to liberate us from the agony that we feel all the time when an uncountable collection pops up in measure theory. We will also talk about how to define ``measurable functions" in the point-free way.

Let $u$ be a harmonic function in $\Omega \subset \mathbb{R}^d.$ It is known that in the interior, the singular set $\mathcal{S}(u) = \{u=|\nabla u|=0 \}$ is $(d-2)$-dimensional, and moreover $\mathcal{S}(u)$ is $(d-2)$-rectifiable and its Minkowski content is bounded (depending on the frequency of $u$). We prove the analogue near the boundary for $C^1$-Dini domains: If the harmonic function $u$ vanishes on an open subset $E$ of the boundary, then near $E$ the singular set $\mathcal{S}(u) \cap \overline{\Omega}$ is $(d-2)$-rectifiable and has bounded Minkowski content. Dini domain is the optimal domain for which $\nabla$ u is continuous towards the boundary, and in particular every $C^{1,\alpha}$ domain is Dini. The main difficulty is the lack of monotonicity formula near the boundary of a Dini domain.
\\
\\
This is joint work with Carlos Kenig.

We discuss some recent results on model-based methods for online reinforcement learning (RL). The goal of online RL is to adaptively explore an unknown environment and learn to act with provable regret bounds. In particular, we focus on finite-horizon episodic RL where the unknown transition law belongs to a generic family of models. We propose a model based `value-targeted regression' RL algorithm that is based on optimism principle: In each episode, the set of models that are `consistent' with the data collected is constructed. The criterion of consistency is based on the total squared error of that the model incurs on the task of predicting values as determined by the last value estimate along the transitions. The next value function is then chosen by solving the optimistic planning problem with the constructed set of models. We derive a bound on the regret, for arbitrary family of transition models, using the notion of the so-called Eluder dimension proposed by Russo \& Van Roy (2014).

In these two talks, we will first discuss Cheeger and Naber's quantitative estimates of singular sets on manifolds with lower Ricci curvature and also review some recent developments. As consequences, we will discuss the hessian estimates of harmonic functions on Einstein manifolds.

In a seminal paper, Chatterjee and Varadhan derived an LDP for the dense Erd\H{o}s-R\'{e}nyi random graph, viewed as a random graphon. This directly provides LDPs for continuous functionals such as subgraph counts, spectral norms, etc. In contrast, very little is understood about this problem if the underlying random graph is \emph{inhomogeneous} or \emph{constrained}.
\\
\\
In this talk, we will explore large deviations for dense random graphs, beyond the ``mean-field'' setting. In particular, we will study large deviations for uniform random graphs with given degrees, and a family of dense block model random graphs. We will establish the LDP in each case, and identify the rate function. In the block model setting, we will use this LDP to study the upper tail problem for homomorphism densities of regular sub-graphs. Our results establish that this problem exhibits a symmetry/symmetry-breaking transition, similar to one observed for Erd\H{o}s-R\'{e}nyi random graphs.
\\
\\
Based on joint works with Christian Borgs, Jennifer Chayes, Souvik Dhara, Julia Gaudio and Samantha Petti.

Toeplitz operators are fundamental and ubiquitous in signal processing and information theory as models for convolutional (filtering) systems. Due to the fact that any practical system can access only signals of finite duration, however, time-limited restrictions of Toeplitz operators are also of interest. In the discrete-time case, time-limited Toeplitz operators are simply Toeplitz matrices. In this talk we survey existing and present new bounds on the eigenvalues (spectra) of time-limited Toeplitz operators, and we discuss applications of these results in various signal processing contexts. As a special case, we discuss time-frequency limiting operators, which alternatingly limit a signal in the time and frequency domains. Slepian functions arise as eigenfunctions of these operators, and we describe applications of Slepian functions in spectral analysis of multiband signals, super-resolution SAR imaging, and blind beamforming in antenna arrays.
\\
\\
This talk draws from joint work with numerous collaborators including Zhihui Zhu from the University of Denver.

Fix a prime integer $p$. Set $R$ the completed valuation ring
of the maximal unramified extension of $\mathbb{Q}_p$. For $X :=
X_1(N)$ the modular curve with $N$ at least 4 and coprime to $p$,
Buium-Poonen in 2009 showed that the locus of canonical lifts enjoys
finite intersection with preimages of finite rank subgroups of $E(R)$
when $E$ is an elliptic curve with a surjection from $X$. This is done
using Buium's theory of arithmetic ODEs, in particular interesting
homomorphisms $E(R) \to R$ which are arithmetic analogues of Manin maps.
\\
\\
In this talk, I will review the general idea behind this result and
other applications of arithmetic jet spaces to Diophantine geometry and
discuss a recent analogous result for quasi-canonical lifts, i.e., those
curves with Serre-Tate parameter a root of unity. Here the ODE Manin
maps are insufficient, so we introduce a PDE version of Buium's theory
to provide the needed maps. All of this is joint work with A. Buium.

I will state the question of uniqueness for centers of
blow ups and blow downs of birational maps, explain what is currently
known and give two applications. The first is to the structure of
Cremona groups, namely their nongeneration by involutions in dimension
$>$= 3. The second application is for the Grothendieck ring of
varieties, of dimension $<$= 2, over perfect fields.
\\
\\
Based on joint work with H.-Y. Lin, and with H.-Y. Lin and S. Zimmermann.

Let $F$ be a finite subset of $\mathbb{Z}^d$. We say that $F$ is a translational tile of $\mathbb{Z}^d$ if it is possible to cover $\mathbb{Z}^d$ by translates of $F$ without any overlaps. The periodic tiling conjecture, which is perhaps the most well-known conjecture in the area, suggests that any translational tile admits at least one periodic tiling. In the talk, we will motivate and discuss the study of this conjecture. We will also present some new results, joint with Terence Tao, on the structure of translational tilings in lattices and introduce some applications.

We discuss how knot Floer homology can be used to study equivariant knots. We introduce some large-surgery correction terms that obstruct equivariant sliceness (and more generally, bound equivariant genus, following work of Juhasz and Zemke). We describe some crossing-change inequalities for these invariants. We also describe an amusing application to distinguishing (up to isotopy rel boundary) pairs of slice disks related by symmetries of a knot.
\\
\\
This is work in progress with Abhishek Mallick and Matthew Stoffregen.

We present a perspective on neural networks stemming from quiver representation theory. This point of view emphasizes the symmetries inherent in neural networks, interacts nicely with gradient descent, and has the potential to improve training algorithms. As an application, we prove an analogue of the QR decomposition for radial neural networks, which leads to a dimensional reduction result. We assume a basic machine learning background, while explaining all necessary representation theory concepts from first principles.
\\
\\
The talk is based on joint work-in-progress with Robin Walters.

After discussing the motivation behind the talk and some necessary preliminaries, we will consider the ``stabilization" of a countable ergodic p.m.p. equivalence relation which is not Schmidt, i.e. admits no central sequences in its full group. Using a new local characterization of the Schmidt property, we show that this always gives rise to a so-called stable equivalence relation with a unique stable decomposition, providing the first non-strongly ergodic such examples. We will also discuss some new structural results for product equivalence relations, which we will obtain using von Neumann algebraic techniques.

The Generalized Nash Equilibrium Problem (GNEP) is a kind of games to find strategies for a group of players such that each player’s objective function is optimized, given other players’ strategies. If all the objective and constraining functions involved are polynomials, we call the problem a Generalized Nash Equilibrium Problem of Polynomials (GNEPP). When the constraining functions of each player are independent of other player’s strategies, the GNEP is called a (standard) Nash Equilibrium Problem (NEP). The GNEP is said to be convex if each player’s optimization is a convex optimization problem, given other players’ strategies.
\\
\\
For nonconvex Nash equilibrium problems that are given by polynomial functions, we formulate efficient polynomial optimization problems for computing Nash equilibria. We show that under generic assumptions, the method can find one or even all Nash equilibria if they exist, or detect nonexistence of Nash equilibria. For convex GNEPPs, we introduce rational and parametric expressions for Lagrange multipliers to formulate polynomial optimization for computing Generalized Nash Equilibria (GNEs). We prove that under some specific assumptions, the method can find a GNE if there exists one, or detect nonexistence of GNEs. Numerical experiments are presented to show the efficiency of the methods. The Moment-SOS hierarchy of semidefinite relaxations are used to solve the polynomial optimization.
\\
\\
Moreover, we study the Gauss-Seidel method for solving the nonconvex GNEPPs. We give a certificate for a class of GNEPPs such that the Gauss-Seidel method is guaranteed to converge, and the numerical experiments show that the Gauss-Seidel method can solve many GNEPPs efficiently.

Fermat's Last Theorem that $x^n + y^n = z^n$ has no positive integer solutions for n$>$2 was first written down by Fermat himself in the early 17th century and resisted proof until Andrew Wiles' monumental 1994 paper. During the 300 years in between, many others tried their hand and along the way developed a lot of interesting number theory. We will discuss more classical topics in algebraic number theory - cyclotomic fields, higher reciprocity laws, class field theory, etc. - in the context of historical attempts to prove the theorem. We will be able to verify (the first case of) Fermat's Last Theorem for pretty high prime exponents (p $<$= 156,442,236,847,241,729).

The well-known Simons's cone suggests that minimal hypersurfaces could be possibly singular in a Riemannian manifold with dimension greater than 7, unlike the lower dimensional case. Nevertheless, it was conjectured that one could perturb away these singularities generically. In this talk, I will discuss how to perturb them away to obtain a smooth minimal hypersurface in an 8-dimension closed manifold, by induction on the ``capacity" of singular sets. This result generalizes the previous works by N. Smale and by Chodosh-Liokumovich-Spolaor to any 8-dimensional closed manifold.
\\
\\
This talk is based on joint work with Zhihan Wang.

MemComputing [1, 2] is a novel physics-based approach to computation that employs time non-locality (memory) to both process and store information on the same physical location. Its digital version [3, 4] is designed to solve combinatorial optimization problems. A practical realization of digital memcomputing machines (DMMs) can be accomplished via circuits of non-linear dynamical systems with memory engineered so that periodic orbits and chaos can be avoided. A given logic problem is first mapped into this type of dynamical system whose point attractors represent the solutions of the original problem. A DMM then finds the solution via a succession of elementary instantons whose role is to eliminate solitonic configurations of logical inconsistency (``logical defects") from the circuit [5, 6]. I will discuss the physics behind memcomputing and show many examples of its applicability to various combinatorial optimization and Machine Learning problems demonstrating its advantages over traditional approaches [7, 8]. Work supported by DARPA, DOE, NSF, CMRR, and MemComputing, Inc.
\\
\\
{[1]} M. Di Ventra and Y.V. Pershin, Computing: the Parallel Approach, Nature Physics 9, 200 (2013).
\\
{[2]} F. L. Traversa and M. Di Ventra, Universal Memcomputing Machines, IEEE Transactions on Neural Networks and Learning Systems 26, 2702 (2015).
\\
{[3]} M. Di Ventra and F.L. Traversa, Memcomputing: leveraging memory and physics to compute efficiently, J. Appl. Phys. 123, 180901 (2018).
\\
{[4]} F. L. Traversa and M. Di Ventra, Polynomial-time solution of prime factorization and NP-complete problems with digital memcomputing machines, Chaos: An Interdisciplinary Journal of Nonlinear Science 27, 023107 (2017).
\\
{[5]} M. Di Ventra, F. L. Traversa and I.V. Ovchinnikov, Topological field theory and computing with instantons, Annalen der Physik 529,1700123 (2017).
\\
{[6]} M. Di Ventra and I.V. Ovchinnikov, Digital memcomputing: from logic to dynamics to topology, Annals of Physics 409, 167935 (2019).
\\
{[7]} F. L. Traversa, P. Cicotti, F. Sheldon, and M. Di Ventra, Evidence of an exponential speed-up in the solution of hard optimization problems, Complexity 2018, 7982851 (2018).
\\
{[8]} F. Sheldon, F.L. Traversa, and M. Di Ventra, Taming a non-convex landscape with dynamical long-range order: memcomputing Ising benchmarks, Phys. Rev. E 100, 053311 (2019).

Nori proved in 2002 that given a complex algebraic variety
$X$, the bounded derived category of the abelian category of constructible sheaves on $X$ is
equivalent to the usual triangulated category $D(X)$ of bounded constructible complexes on $X$.
He moreover showed that given any constructible sheaf $\mathcal F$ on
$\mathbb{A}^n$, there is an injection $\mathcal F\hookrightarrow\mathcal G$ with
$\mathcal G$ constructible and $H^i(\mathbb{A}^n,\mathcal G)=0$ for $i>0$.
\\
\\
In this talk, I'll discuss how to extend Nori's theorem to the case of a
variety over an algebraically closed field of positive characteristic, with
Betti constructible sheaves replaced by $\ell$-adic sheaves.
This is the case $p=0$ of the general problem which asks whether the bounded
derived category of $p$-perverse sheaves is equivalent to $D(X)$, resolved
affirmatively for the middle perversity by Beilinson.

In this talk I will discuss various topological and geometric consequences of the existence of zeros of global holomorphic 1-forms on smooth projective varieties. Such consequences have been indicated by a plethora of results. I will present some old and new results in this direction. One highlight of the topic is an interesting connection between two sets of such 1-forms, one that arises out of the generic vanishing theory and the other that falls out of Hodge theory of algebraic maps.
\\
\\
This is joint work with Feng Hao and Yongqiang Liu.

In this talk, after discussing the necessary background material, I will discuss the main results of my recent work \emph{Variational Structures in Cochain Projection Based Discretizations of Lagrangian PDEs} and \emph{Multisymplectic Hamiltonian Variational Integrators}; this is joint work with Prof. Melvin Leok.
\\
\\
Building on these ideas, I will conclude by discussing future research directions.

In 1987, Margulis solved an old conjecture of Oppenheim which states that for a nondegenerate, indefinite and irrational quadratic form $Q$ in $n \geq 3$ variables, $Q(\mathbb{Z}^n)$ is dense in $\mathbb{R}$. Following this, Dani and Margulis proved the simultaneous density at integer points for a pair consisting of quadratic and linear form in $3$ variables when certain conditions are satisfied. We prove an analogue of this for the case of an inhomogeneous quadratic form and a linear form.
\\
\\
This is based on joint work with Anish Ghosh.

A parallel cut-cell algorithm is described to solve the free boundary problem of the Grad-Shafranov equation.
The algorithm reformulates the free-boundary problem in an irregular bounded domain and its important aspects include a
searching algorithm for the magnetic axis and separatrix, a surface integral along the irregular boundary to determine the
boundary values, an approach to optimize the coil current based on a targeting plasma shape, Picard iterations with Aitken's
acceleration for the resulting nonlinear problem and a Cartesian grid embedded boundary method to handle the complex
geometry. The algorithm is implemented in parallel using a standard domain-decomposition approach and a good parallel
scaling is observed. Numerical results verify the accuracy and efficiency of the free-boundary Grad-Shafranov solver.

An $n$-tangle is a proper embedding of the disjoint union of n arcs into a $3$-ball, in such a way that the endpoints are mapped to $2n$ marked points on the ball’s boundary. In 1967, Conway developed the theory of a special class of $2$-tangles, called ``rational tangles," leading to important results on the classification of knots. Rational tangles themselves have an elegant classification which relates to continued fraction expansions of rational numbers. We explore this connection in the context of a fun activity, which was developed by Conway in order to demonstrate some aspects of the theory.

This talk establishes a concentration phenomenon on the empirical eigenvalue distribution (EED) of the principal submatrix in a random hermitian matrix whose distribution is invariant under unitary conjugacy. More precisely, if the EED of the whole matrix converges to some deterministic probability measure 𝔪, then its difference from the EED of its principal submatrix, after a rescaling, concentrates at the Rayleigh measure (in general, a Schwartz distribution) associated with 𝔪 by the Markov-Krein correspondence. For the proof, we use the moment method with Weingarten calculus and free probability. At some stage of calculations, the proof requires a relation between the moments of the Rayleigh measure and free cumulants of 𝔪. This formula is more or less known, but we provide a different proof by observing a combinatorial structure of non-crossing partitions.
\\
\\
This is a joint work with Katsunori Fujie.

This is joint work with Will Wylie. The goal is to classify, if possible, the homogeneous geometric solitons. Here a geometric soliton is the soliton for a geometric flow. The Ricci flow is the most prominent example of such a flow, but there are man others where the Ricci tensor is replaced with some other tensor that depends in a natural way on the Riemannian structure. We will also consider some more general problems showing that our techniques can be used for other geometric problems.

We develop a new empirical Bayesian inference algorithm for solving a linear inverse problem given multiple measurement vectors (MMV) of under-sampled and noisy observable data. Specifically, by exploiting the joint sparsity across the multiple measurements in the sparse domain of the underlying signal or image, we construct a new support informed sparsity promoting prior. Several applications can be modeled using this framework. Our numerical experiments demonstrate that using this new prior not only improves accuracy of the recovery, but also reduces the uncertainty in the posterior when compared to standard sparsity producing priors.
\\
\\
This is joint work with Theresa Scarnati formerly of the Air Force Research Lab Wright Patterson and now working at Qualis Corporation in Huntsville, AL, and Jack Zhang, recent bachelor degree recipient at Dartmouth College and now enrolled at University of Minnesota’s PhD program in mathematics.

We give a non-asymptotic bound on the spectral norm of a $d\times d$ matrix $X$ with centered jointly Gaussian entries in terms of the covariance matrix of the entries. In some cases, this estimate is sharp and removes the $\sqrt{\log d}$ factor in the noncommutative Khintchine inequality.
\\
\\
Joint work with Afonso Bandeira

Simpson conjectured that for a reductive group $G$, rigid
$G$-local systems on a smooth projective complex variety are integral. I
will discuss a proof of integrality for cohomologically rigid $G$-local
systems. This generalizes and is inspired by work of Esnault and
Groechenig for $GL_n$. Surprisingly, the main tools used in the proof
(for general $G$ and $GL_n$) are the work of L. Lafforgue on the
Langlands program for curves over function fields, and work of Drinfeld
on companions of $\ell$-adic sheaves. The major differences between
general $G$ and $GL_n$ are first to make sense of companions for
$G$-local systems, and second to show that the monodromy group of a
rigid G-local system is semisimple.
\\
\\
All work is joint with Stefan Patrikis.

It is known that there are factors of IID for the free Ising model on the d-regular tree when it has a unique Gibbs measure and not when reconstruction holds (when it is not extremal). We construct a factor of IID for the free Ising model on the d-regular tree in (part of) its intermediate regime, where there is non-uniqueness but still extremality. The construction is via the limit of a system of stochastic differential equations.
\\
\\
This is a joint work with Danny Nam and Allan Sly.

Let $X$ be a compact metric space, and let $h$ be a homeomorphism
of $X$. The mean dimension of $h$ is an invariant invented by people
in topological dynamics, with no consideration of C*-algebras. The
shift on the product of copies of $[0, 1]^d$ indexed by $\mathbb{Z}$ has mean
dimension $d$.
\\
\\
The radius of comparison of a C*-algebra $A$ is an invariant
introduced with no consideration of dynamics, and originally applied
to C*-algebras which are not given as crossed products. It is a
numerical measure of bad behavior in the Cuntz semigroup of $A$, and
its original use was to distinguish counterexamples to the original
formulation of the Elliott conjecture.
\\
\\
It is conjectured that if $h$ is a minimal homeomorphism of a compact
metric space, then the radius of comparison of $C* (Z, X, h)$ is equal
to half the mean dimension of $h$. There is a generalization to
countable amenable groups. Considerable progress has been made on
proving that the radius of comparison of $C* (Z, X, h)$ is at most
half the mean dimension; in particular, this is known in full
generality for minimal homeomorphisms. We give the first systematic
results for the opposite inequality. We do not get the exact expected
lower bound, but, for many known examples of actions of amenable
groups with large mean dimension, we come close.
\\
\\
The methods depend on ``mean cohomological independence dimension,"
Cech cohomology, and the Chern character.
\\
\\
This is joint work with Ilan Hirshberg.

Multi-agent systems are ubiquitous in science, from the modeling of particles in Physics to prey-predator in Biology, to opinion dynamics in economics and social sciences, where the interaction law between agents yields a rich variety of collective dynamics. We consider the following inference problem for a system of interacting particles or agents: given only observed trajectories of the agents in the system, can we learn what the laws of interactions are? We would like to do this without assuming any particular form for the interaction laws, i.e. they might be ``any" function of pairwise distances.
\\
\\
In this talk, we consider this problem in the case of a finite number of agents, with an increasing number of observations. We cast this as an inverse problem, and study it in the case where the interaction is governed by an (unknown) function of pairwise distances. We discuss when this problem is well-posed, and we construct estimators for the interaction kernels with provably good statistical and computational properties. We measure their performance on various examples, that include extensions to agent systems with different types of agents, second-order systems, and stochastic systems. We also conduct numerical experiments to test the large time behavior of these systems, especially in the cases where they exhibit emergent behavior.
\\
\\
This talk is based on the joint work with Fei Lu, Mauro Maggioni, Jason Miller, and Ming Zhong.

The classical Gauss's Theorema Egregium and the Uniformization theorem for the Riemann surfaces are illustrations of a prominent theme in geometry -- control of the global topology/geometry of a manifold through the bounds on its curvature. In the last decades, with the development of new analytic tools (Yamabe equation, mean-curvature flow etc), this idea has found numerous applications in classification problems. Application of geometric flows (specifically the Ricci flow) turned out to be particularly fruitful in the context of Kaehler (and projective algebraic) geometry. At the same time there are very few efficient analytic methods available in non-Kahler complex geometry. In this talk we will introduce the Hermitian Curvature Flow on an arbitrary compact complex manifold. We will prove a delicate version of the maximum principle for tensors along this flow and present applications to the classification problems for the complex/algebraic manifolds admitting a compatible metric with ``semipositive curvature."

I will explain the a priori estimates for the cscK equation on a compact manifold, and how to use these estimates to obtain existence when the associated energy functional is ``coercive." If time permits, I will also explain how we can hope to get existence from a more ``algebraic" condition, which might be easier to check in practice.

In a matrix completion problem, one has access to a subset of entries of a matrix and wishes to determine the missing entries subject to some constraint (e.g. a rank bound). Such problems appear in computer vision, collaborative filtering, and many other applications. In this talk, I will discuss how certain notions from nonlinear algebra can be used to understand the structure underlying various types of matrix completion problems.

Let X be a compact, smooth manifold and Diff(X) the diffeomorphism group. The topology of Diff(X) and of the classifying space BDiff(X) are of great interest. For instance, the k-th homotopy group of BDiff(X) corresponds to smooth families over the k-sphere with fibres diffeomorphic to X. By a recent result of Bustamante, Krannich and Kupers, if X has even dimension not equal to 4 and finite fundamental group, then the homotopy groups of BDiff(X) are all finitely generated. In contrast we will show that when X is a K3 surface, the second homotopy group of BDiff(X) contains a free abelian group of countably infinite rank as a direct summand. Our families are constructed using the moduli space of Einstein metrics on K3. Their non-triviality is detected using families Seiberg--Witten invariants.

Unitary Shimura varieties are moduli spaces of abelian
varieties with an action of a quadratic imaginary field, and extra
structure. In this talk, we'll discuss specific examples of unitary
Shimura varieties whose supersingular loci can be concretely described
in terms of Deligne-Lusztig varieties. By Rapoport-Zink uniformization,
much of the structure of these supersingular loci can be understood by
studying an associated moduli space of p-divisible groups (a
Rapoport-Zink space). We'll discuss the geometric structure of these
associated Rapoport-Zink spaces as well as some techniques for studying
them.

Service platforms must determine rules for matching heterogeneous demand (customers) and supply (workers) that arrive randomly over time and may be lost if forced to wait too long for a match. We show how to balance the trade-off between making a less good match quickly and waiting for a better match, at the risk of losing impatient customers and/or workers. When the objective is to maximize the cumulative value of matches over a finite-time horizon, we propose discrete-review matching policies, both for the case in which the platform has access to arrival rate parameter information and the case in which the platform does not. We show that both the blind and nonblind policies are asymptotically optimal in a high-volume setting. However, the blind policy requires frequent re-solving of a linear program. For that reason, we also investigate a blind policy that makes decisions in a greedy manner, and we are able to establish an asymptotic lower bound for the greedy, blind policy that depends on the matching values and is always higher than half of the value of an optimal policy. Next, we develop a fluid model that approximates the evolution of the stochastic model and captures explicitly the nonlinear dependence between the amount of demand and supply waiting and the distribution of their patience times. We establish a fluid limit theorem and show that the fluid limit converges to its equilibrium. Based on the fluid analysis, we propose a policy for a more general objective that additionally penalizes queue build-up.

In this talk, I will explain a mixed characteristic
analogue of Frobenius regularity and how it can be used to establish the
Minimal Model Program for threefolds in mixed characteristic.
\\
\\
This is based on a joint work with Bhargav Bhatt, Linquan Ma, Zsolt Patakfalvi,
Karl Schwede, Kevin Tucker, and Joe Waldron.

We will start by reviewing recent developments in random walks on homogeneous spaces. In a second part, we will discuss the notion of a $H$-expanding probability measure on a connected semisimple Lie group $H$. As we shall see, for a $H$-expanding $\mu$ with $H < G$, on the one hand, one can obtain a description of $\mu$-stationary probability measures on the homogeneous space $G / \Lambda$ ($G$ Lie group, $\Lambda$ lattice) using the measure classification results of Eskin-Lindenstrauss, and on the other hand, the recurrence techniques of Benoist-Quint and Eskin-Mirzakhani-Mohammadi can be adapted to this setting. With some further work, these allow us to deduce equidistribution and orbit closure description results simultaneously for a class of subgroups which contains Zariski-dense subgroups and further epimorphic subgroups of $H$. If time allows, we will see how, utilizing an idea of Simmons-Weiss, these also allow us to deduce Birkhoff genericity of a class of fractal measures with respect to certain diagonal flows, which, in turn, has applications in diophantine approximation problems. \\
\\
Joint work with Roland Prohaska and Ronggang Shi.

We consider the problem of approximating high dimensional functions using shallow neural networks, and more generally by sparse linear combinations of elements of a dictionary. We begin by introducing natural spaces of functions which can be efficiently approximated in this way. Then, we derive the metric entropy of the unit balls in these spaces, which allows us to calculate optimal approximation rates for approximation by shallow neural networks. This gives a precise measure of how large this class of functions is and how well shallow neural networks overcome the curse of dimensionality. Finally, we describe an algorithm which can be used to solve high-dimensional PDEs using this space of functions.

Suppose L is a link in S\^{}3, we show that $\pi_1(S^3-L)$ admits an irreducible meridian-traceless representation in SU(2) if and only if L is not the unknot, the Hopf link, or a connected sum of Hopf links. As a corollary, $\pi_1(S^3-L)$ admits a (not necessarily meridian-traceless) irreducible representation in SU(2) if and only if L is neither the unknot nor the Hopf link. This result generalizes a theorem of Kronheimer and Mrowka to the case of links. The proof is based on singular instanton Floer theory and an observation about finite simple graphs.
\\
\\
This is joint work with Yi Xie.

We present a direct unified analytic proof of the classical Boundary Harnack Principle for solutions to linear uniformly elliptic equations in either divergence or non-divergence form. The proof extends also to (appropriate) H\"older domains. The strategy also applies to the parabolic context. Applications of the BHP to free boundary problems are discussed."

From polynomial to deep neural network approximation in high dimensions approximating multi-dimensional functions from pointwise samples is a ubiquitous task in data science and scientific computing. This task is made intrinsically difficult by the presence of four contemporary challenges: (i) the target function is usually defined over a high- or infinite-dimensional domain; (ii) generating samples can be very expensive; (iii) samples are corrupted by unknown sources of errors; (iv) the target function might take values in a function space. In this talk, we will show how these challenges can be substantially overcome thanks to the ``blessings" of sparsity and Monte Carlo sampling.
\\
\\
First, we will consider the case of sparse polynomial approximation via compressed sensing. Focusing on the case where the target function is smooth (e.g., holomorphic), but possibly highly anisotropic, we will show how to obtain sample complexity bounds only mildly affected by the curse of dimensionality, near-optimal accuracy guarantees, stability to unknown errors corrupting the data, and rigorous convergence rates of algebraic and exponential type. Then, we will illustrate how the mathematical toolkit of sparse polynomial approximation via compressed sensing can be employed to obtain a practical existence theorem for Deep Neural Network (DNN) approximation of high-dimensional Hilbert-valued functions. This result shows not only the existence of a DNN with desirable approximation properties, but also how to compute it via a suitable training procedure in order to achieve best-in-class performance guarantees. We will conclude by discussing open research questions.
\\
\\
The presentation is based on joint work with Ben Adcock, Casie Bao, Nick Dexter, Sebastian Moraga, and Clayton G. Webster.

The growth of random surfaces has attracted a lot of attention in probability theory in the last ten years, especially in the context of the Kardar-Parisi-Zhang (KPZ) equation. Most of the available results are for exactly solvable one-dimensional models. In this talk I will present some recent results for models that are not exactly solvable. In particular, I will talk about the universality of deterministic KPZ growth in arbitrary dimensions, and if time permits, a necessary and sufficient condition for superconcentration in a class of growing random surfaces.

A classical transcendence result of Schneider on the modular
$j$-invariant states that, for $\tau \in \mathbb{H}$, both $\tau$ and
$j(\tau)$ are in $\overline{\mathbb{Q}}$ if and only if $\tau$ is
contained in an imaginary quadratic extension of $\mathbb{Q}$. The space
$\mathbb{H}$ has a natural interpretation as a parameter space for
$\mathbb{Q}$-Hodge structures (or, in this case, elliptic curves), and
from this perspective the imaginary quadratic points are distinguished
as corresponding to objects with maximal symmetry. This result has been
generalized by Cohen and Shiga-Wolfart to more general moduli of Hodge
structures (corresponding to abelian-type Shimura varieties), and by
Ullmo-Yafaev to higher dimensional loci of extra symmetry (special
subvarieties), where bialgebraicity is intimately connected with the
Pila-Zannier approach to the Andre-Oort conjecture.
\\
\\
In this talk, I will discuss work in progress with Christian Klevdal on
an analogous bialgebraicity characterization of special subvarieties in
Scholze's local Shimura varieties and more general diamond moduli of
$p$-adic Hodge structures.

This talk is about optimizing polynomial functions under constraints. A general method is to apply the Moment-SOS hierarchy of semidefinite programming relaxations. The convergence is based on various Positivstellensatz. Closely related polynomial optimization is convex algebraic geometry.
\\
\\
It concerns geometric properties of convex semialgebraic sets through semidefinite programming. We are going to review basic results for these topics.

Dating back to the foundational paper by John von Neumann, a fundamental theme in ergodic theory is the \emph{isomorphism problem} to classify invertible measure-preserving transformations (MPT's) up to isomorphism. In a series of papers, Matthew Foreman, Daniel Rudolph and Benjamin Weiss have shown in a rigorous way that such a classification is impossible. Besides isomorphism, Kakutani equivalence is the best known and most natural equivalence relation on ergodic MPT's for which the classification problem can be considered. In joint work with Marlies Gerber we prove that the Kakutani equivalence relation of ergodic MPT's is not a Borel set. This shows in a precise way that the problem of classifying such transformations up to Kakutani equivalence is also intractable.

A central problem in low-dimensional topology asks which homology 3-spheres bound contractible 4-manifolds and homology 4-balls. In this talk, we address this problem for plumbed 3-manifolds and we
present the classical and new results together. Our approach is based on
Mazur’s famous argument which provides a unification of all results in a
fairly simple way.

A central sequence in a $C^*$-algebra is a sequence (x\_n) of elements such that [x\_n, a] converges to zero, for any element a of the $C^*$-algebra. In von Neumann algebra setting one typically means the convergence with respect to tracial norms, while in $C^*$-theory it is with respect to the $C^*$-norm. In this talk we will consider the $C^*$-theory version of central sequences. We will discuss properties of central sequence algebras and in particular address a question of J. Phillips and of Ando and Kirchberg of which separable $C^*$-algebras have abelian central sequence algebras.
\\
\\
Joint work with Dominic Enders.

In this talk, we study the property of the solution of semidefinite programs with multi-dimensional perturbation variables using the Davidenko di erential equations. Under the assumptions of strict complementary and non-degeneracy, we aim to find the a priori unknown maximal convex permissible perturbation set where the semidefinite program has a unique optimum and the optimum is analytic. A sweeping euler numerical method is developed to approximate this a priori unknown perturbation set and solve the semidefinite program within this set. We prove local and global error bounds for this second-order sweeping Euler scheme and demonstrate results on several examples.

The thin obstacle problem is a classical free boundary problem arising from the study of an elastic membrane resting on a lower-dimensional obstacle. Concerning the behavior of the solution near a contact point between the membrane and the obstacle, many important questions remain open. In this talk, we discuss a unified method that leads to a rate of convergence to `tangent cones' at contact points with integer frequencies.
\\
\\
This talk is based on a recent joint work with Ovidiu Savin.

Metric algebraic geometry is a term proposed for the study of properties of real algebraic varieties that depend on a distance metric. The distance metric can be the Euclidean metric in the ambient space or a metric intrinsic to the variety. In this talk, we introduce metric algebraic geometry through a discussion of Voronoi cells, bottlenecks, offset hypersurfaces, and the reach of an algebraic variety. We also show applications to the computational study of the geometry of data with nonlinear models.

Let $\Pi$ be a cuspidal automorphic representation for GL(3) over a
number field. We fix an integer $k \geq 2$ and we assume that the
symmetric $m$th power lifts of $\Pi$ are automorphic for $m \leq k$,
cuspidal for $m < k$, and that certain associated Rankin–Selberg
products are automorphic. In this setting, we bound the number of
cuspidal isobaric summands in the $k$th symmetric power lift. In
particular, we show it is bounded above by 3 for $k \geq 7$, and bounded
above by 2 when $k \geq 19$ with $k$ congruent to 1 mod 3. We will also
discuss the analogous problem for GL(4).

The talk will be an overview of Comparison Methods for Stochastic Models and Risks by Muller and Stoyan, and Stochastic Orderings for Markov Processes on Partially Ordered Sets, by William Massey.

Let $X$ be a smooth projective variety whose algebra of
holomorphic forms is generated in degree at most $2$, for instance an
abelian variety or a hyperK\"ahler manifold. A recent conjecture of Voisin

How do we make mathematicians? How do we become them ourselves? And why is it rare that we teach that way? In this talk, we will discuss the pedagogy of inquiry-based learning (IBL) and how it can make for a more holistic, equitable, and successful classroom. Bring questions and an open mind - there may be a quiz!

We will study complete Ricci-flat manifolds with Euclidean volume growth. In the case when a tangent cone at infinity of the manifold has smooth cross section, the Green function for the Laplace equation can be used to define a functional which measures how fast the manifold converges to the tangent cone. Using the Łojasiewicz inequality of Colding-Minicozzi for this functional, we describe how two arbitrarily far apart scales in the manifold can be identified in a natural way. I will also discuss a matrix Harnack inequality for the Green function when there is an additional condition on sectional curvature, which is an analogue of various matrix Harnack inequalities obtained by Hamilton and Li-Cao in time-dependent settings.

Short-firm video (SFV) has been exploding on digital platforms recently. The vast amount of videos and fast-evolving trends on digital platforms pose technical challenges in making personalized recommendations. In this work, we introduce a new pure exploration problem on SFV platforms. We propose an adaptive data acquisition method, called Adaptive Acquisition Tree (AAT), to jointly account for heterogeneity in user preferences and high-dimensional product characteristics. We adaptively divide users based on preference similarity and then learn a personalized transductive bandit policy that can be used on partially or even unobserved arms to accommodate the fast-evolving and emerging trends on SFV platforms. We analytically characterize the prediction error, which is determined by both the sample size and the impurity of parameters within a group. We further derive the sample complexity for identifying an optimal set for a single user and for all users. We evaluate the algorithm via numerical experiments on data collected from the NetEase platform. Our result demonstrates that the proposed policy, compared with several state-of-the-art benchmarks, performs significantly better in four transductive scenarios for both spotlight recommendation (i.e., best-arm identification) and top-K recommendations. With the potential to improve the expected view time by 20-25\%, our method pertains to both academic and practical values, given the increasing popularity of short-form videos and, more broadly, online user-content generation platforms.

Dynamical sampling is a term describing an emerging set of problems related to recovering signals and evolution operators from space-time samples. For example, one problem is to recover a function f from space-time samples $\{(A_{t_i}f)(x_i)\}$ of its time evolution $f_t = (A_t f)(x)$, where $\{A_t\}_{t \in T}$ is a known evolution operator and $(x_i,t_i) \in R^d \times R^+$.
\\
\\
Another example is a system identification problem when both $f$ and the evolution family $\{A_t\}_{t\in T}$ are unknown. Applications of dynamical sampling include inverse problems in sensor networks, and source term recovery from physically driven evolutionary systems. Dynamical sampling problems are tightly connected to frame theory as well as more classical areas of mathematics such as approximation theory, and functional analysis. In this talk, I will describe a few problems in dynamical sampling, some results and open problems.

L\'{e}vy matrices are symmetric matrices whose entries are random variables with infinite variance; they are governed by a parameter $\alpha \in (0, 2)$ dictating the power law decay of their entries. For $\alpha <1$, they are believed to serve as one of the few examples of a matrix model exhibiting a mobility edge, also called an Anderson transition, that separates chaotic (GOE) eigenvalue spacing statistics from ordered (Poisson) ones. In this talk we describe results concerning the statistics for the eigenvalue spacings and eigenvector entries of L\'{e}vy matrices. In particular, for $\alpha \in (1, 2)$ their eigenvalue statistics asymptotically follow those of the GOE throughout the spectrum, and for $\alpha < 1$ the same statement holds around small eigenvalues.
\\
\\
These describe joint works with Patrick Lopatto, Jake Marcinek, and Horng-Tzer Yau.

Coclasses in a Galois cohomology group $H^1(K, T)$ parametrize
extensions of a number field with certain Galois group. It is natural to
want to count these coclasses with general local conditions and with
respect to a discriminant-like invariant. In joint work with Brandon
Alberts, I present a novel tool for studying this: harmonic analysis on
adelic cohomology, modeled after the celebrated use of harmonic analysis
on the adeles in Tate's thesis. This leads to a more illuminating
explanation of a fact previously noticed by Alberts, namely that the
Dirichlet series counting such coclasses is a finite sum of Euler
products; and we nail down the asymptotic count of coclasses in
satisfying generality.

The goal will be to talk about Dasgupta and Kakde's recent results on understanding the maximal abelian extensions for totally real number fields.

I will give a brief overview of sofic groups, some of their equivalent characterizations, and their relation to the surjunctivity conjecture. Time permitting, I will discuss other similar groups, such as hyperlinear or linearly sofic groups.

This talk will discuss moduli spaces of algebraic varieties and will introduce K-stability as a tool to construct moduli spaces of Fano varieties.

Let G be a connected reductive group with maximal torus T, and let V
and E be two representations of G. Then E defines a vector bundle on
the orbifold V//G; let X//G be the zero locus of a regular section.
The quasimap I-function of X//G encodes the geometry of maps from $P^1$
to X//G and is related to Gromov-Witten invariants of X//G. By
directly analyzing these maps from $P^1$, we explain how to relate the
I-function of X//G to that of V//T. Our formulas prove a mirror
symmetry conjecture of Oneto-Petracci that relates the quantum period
of X//G to a certain Laurent polynomial defined by a Fano polytope.
Finally, we describe a large class of examples to which our formulas
apply, examples that are the orbifold analog of quiver flag varieties.
Question for the audience: what else can one investigate with these
examples?

We discuss periods in the classical sense, with motivation from transcendental number theory, and relate them to the study of differential equations and values of L-functions.

The Mozes-Shah phenomenon on homogeneous spaces of Lie groups asserts that the space of ergodic measures under the action by subgroups generated by unipotents is closed. A key input to their work is Ratner's fundamental rigidity theorems. Beyond its intrinsic interest, this result has many applications to counting problems in number theory. The problem of counting saddle connections on flat surfaces has motivated the search for analogous phenomena for horocycle flows on moduli spaces of flat structures. In this setting, Eskin, Mirzakhani, and Mohammadi showed that this property is enjoyed by the space of ergodic measures under the action of (the full upper triangular subgroup of) $\mathrm{SL}(2,\mathbb{R})$. We will discuss joint work with Jon Chaika and John Smillie showing that this phenomenon fails to hold for the horocycle flow on the stratum of genus two flat surfaces with one cone point. As a corollary, we show that a dense set of horocycle flow orbits are not generic for any measure; in contrast with Ratner's genericity theorem.

Hamiltonian simulation is a basic task in quantum computation. The accuracy of such simulation is usually measured by the error of the unitary evolution operator in the operator norm, which in turn depends on certain norm of the Hamiltonian. For unbounded operators, after suitable discretization, the norm of the Hamiltonian can be very large, which significantly increases the simulation cost. However, the operator norm measures the worst-case error of the quantum simulation, while practical simulation concerns the error with respect to a given initial vector at hand. We demonstrate that under suitable assumptions of the Hamiltonian and the initial vector, if the error is measured in terms of the vector norm, the computational cost may not increase at all as the norm of the Hamiltonian increases using Trotter type methods. In this sense, our result outperforms all previous error bounds in the quantum simulation literature. Our result extends that of [Jahnke, Lubich, BIT Numer. Math. 2000] to the time-dependent setting. We also clarify the existence and the importance of commutator scalings of Trotter and generalized Trotter methods for time-dependent Hamiltonian simulations.

A bold conjecture due to Connes (1980) predicts that the group von Neumann algebra of an i.c.c. property (T) group completely remembers the group. The strong form of Connes' conjecture, due to Popa (2005) predicts that the factors arising from property (T) groups have trivial fundamental group. In this talk I shall discuss recent progress towards these conjectures, and present the first examples of property (T) factors with trivial fundamental group. This talk is based on a joint work with Ionut Chifan, Cyril Houdayer, and Krishnendu Khan.

Random graphs play a central role across several branches of mathematics and applied sciences.
The phase transition of random graphs, where the component structure suddenly changes from `many small' components to `one giant' component, is an intriguing phenomenon with rich connections to percolation theory and mathematical physics.
In computer science and extremal combinatorics, random graphs also provide strong probabilistic guarantees for hard-to-answer deterministic questions, such as the construction of interesting combinatorial objects.
In this talk we illustrate these two complementary aspects of random graphs, highlighting that their analysis often brings together tools and techniques from different areas (including combinatorics, probability and differential equations).
\\
\\
We will first focus on Achlioptas processes, which are variants of classical Erdos-Renyi random graphs that are difficult to analyze. Settling a number of conjectures and open problems, we show that the phase transition of so-called `bounded-size' Achlioptas processes have the same key features as the Erd\"o{}s-R\'{e}nyi reference model (which in the language of mathematical physics means that they are in the same `universality class').

We show that the sparse polynomial interpolation problem reduces to a discrete super-resolution problem on the n-dimensional torus. Therefore, the semidefinite programming approach initiated by Candès and Fernandez-Granda (Commun. Pure Appl. Math. 67(6) 906---956, 2014) in the univariate case can be applied. We extend their result to the multivariate case, i.e., we show that exact recovery is guaranteed provided that a geometric spacing condition on the supports holds and evaluations are sufficiently many (but not many). It also turns out that the sparse recovery LP-formulation of ℓ1-norm minimization is also guaranteed to provide exact recovery provided that the evaluations are made in a certain manner and even though the restricted isometry property for exact recovery is not satisfied. (A naive sparse recovery LP approach does not offer such a guarantee.) Finally, we also describe the algebraic Prony method for sparse interpolation, which also recovers the exact decomposition but from less point evaluations and with no geometric spacing condition. We provide two sets of numerical experiments, one in which the super-resolution technique and Prony's method seem to cope equally well with noise, and another in which the super-resolution technique seems to cope with noise better than Prony's method, at the cost of an extra computational burden (i.e., a semidefinite optimization).

In this talk, we study a new recovery procedure for non-harmonic signals, or more generally for extended exponential sums y(t), which are determined by a finite number of parameters. For the reconstruction we employ a finite set of classical Fourier coefficients of y(t). Our new recovery method is based on the observation that the Fourier coefficients of y(t) possess a special rational structure. We apply the recently proposed AAA algorithm by Nakatsukasa et al. (2018) to recover this rational structure in a stable way. If a sufficiently large set of Fourier coefficients of y(t) is available, then our recovery method automatically detects the correct number of terms M of the exponential sums y(t) and reconstructs all unknown parameters of the signal model. Our method provides a new stable alternative to the known numerical approaches for the recovery of exponential sums that are based on Prony's method.
\\
\\
These results have been obtained jointly with Markus Petz and Nadiia Derevianko.

The directed landscape is a random `directed metric' on the spacetime plane that arises as the scaling limit of integrable models of last passage percolation. It is expected to be the universal scaling limit for all models in the KPZ universality class for random growth. In this talk, I will describe its construction in terms of the Airy line ensemble via an isometric property of the Robinson-Schensted-Knuth correspondence, and discuss some surprising Brownian structures that arise from this construction.
\\
\\
Based on joint work with M. Nica, J. Ortmann, B. Virag, and L. Zhang.

Let $K$ be a number field of degree $n$ and $\mathcal{O}_K$ be
its ring of integers. An order in $\mathcal{O}_K$ is a finite index
subring that contains the identity. A major open question in arithmetic
statistics asks for the asymptotic growth of orders in $K$. In this
talk, we will give the best known lower bound for this asymptotic
growth. The main strategy is to relate orders in $\mathcal{O}_K$ to
subrings in $\mathbb{Z}^n$ via zeta functions. Along the way, we will
give lower bounds for the asymptotic growth of subrings in
$\mathbb{Z}^n$ and for the number of index $p^e$ subrings in
$\mathbb{Z}^n$. We will also discuss analytic properties of these zeta
functions.

Stochastic networks arise in a variety of real world applications including telecommunications, service systems such as call centers, computer networks, health care services and biological systems. Leaving aside some very simple examples, it is usually infeasible to perform an exact analysis of these networks. A useful alternative is instead to identify a suitable approximation that provides insight into performance and can be shown to be accurate in a relevant asymptotic regime. This talk looks at two different types of scaling limits of two classes of many-server stochastic networks. In both cases, the scaling limits are infinite-dimensional and require the development of new techniques for their analysis.

S. Griffin introduced a quotient $R_{n,\lambda}$ of the polynomial ring $\mathbb{Q}$ with $\mathbb{Q}$. It simultaneously generalizes the Delta Conjecture coinvariant rings of Haglund-Rhoades-Shimozono and the cohomology rings of Springer fibers studied by Tanisaki and Garsia-Procesi. We describe the space $V_{n,\lambda}$ of harmonics attached to $R_{n,\lambda}$ and produce a harmonic basis of $R_{n,\lambda}$ indexed by certain ordered set partitions $\mathcal{OP}_{n,\lambda}$. Our description of $V_{n,\lambda}$ involves injective tableaux and Vandermonde determinants. Combinatorics of our harmonic basis is governed by a new extension of the Lemher code.
\\
\\
This is a joint work with Brendon Rhoades and Zehong Zhao.

Stochastic models of complex networks with limited resources arise in a wide variety of applications in science and engineering, e.g., in manufacturing, transportation, telecommunications, computer systems, customer service facilities, and systems biology. Bottlenecks in such networks cause congestion, leading to queueing and delay. Sharing of resources can lead to entrainment effects. Understanding the dynamic behavior of such modern stochastic networks present challenging mathematical problems.
\\
\\
This talk will describe some developments in this area. A key feature will be dimension reduction, resulting from entrainment due to resource sharing. An example of bandwidth sharing in a data network will be featured.

I will try to convince you that there is an opposite of a vector space! This talk will involve additive combinatorics and fourier analysis on finite, not infinite groups...

Recently there has been great progress in constructing moduli spaces of K-stable Fano Varieties, and there are many questions to ask about the geometry of these moduli spaces. The question I am considering is whether they are uniruled. My goal is to show that there are components of these moduli spaces that are general type, answering that question in the negative. I wish to do this by studying moduli spaces of vector bundles on curves, showing they are K-stable, and studying the components of the moduli space of K-stable Fano varieties that parametrize them.

This is a piece of beautiful math involving basic theory of algebraic number fields and L-functions. Aimed for those who did not take the 204 Series.

Isogeny-based cryptography is becoming an increasingly well-established subject within the post-quantum cryptography landscape. In this talk, we introduce supersingular isogeny graphs and study the hard problems in them. We also introduce some modern-day isogeny-based cryptosystems.

As an analyst, I used to think I didn't like groups. And when studying generalized Bernoulli actions, I came across an example of a horrible combinatorial argument to prove a fact about groups. Luckily, there is another way: functional analysis, which I'd like to share with you. Now I realize I just don't like finite groups, which usually aren't amenable to analytic methods.

The Kardar-Parisi-Zhang (KPZ) equation is the stochastic partial differential equation that models interface growth. In the talk I will present the construction of a stationary measure for the KPZ equation on a bounded interval with general inhomogeneous Neumann boundary conditions. Along the way, we will encounter classical orthogonal polynomials, the asymmetric simple exclusion process, and precise asymptotics of q-Gamma functions.
\\
\\
This construction is a joint work with Ivan Corwin.

Classical algorithms typically provide "one size fits all" performance, and do not leverage properties or patterns in their inputs. A recent line of work aims to address this issue by developing algorithms that use machine learning predictions to improve their performance. In this talk I will present two examples of this type, in the context of streaming and sampling algorithms. In particular, I will show how to use machine learning predictions to improve the performance of (a) low-memory streaming algorithms for frequency estimation (ICLR’19), and (b) sampling algorithms for estimating the support size of a distribution (ICLR’21). Both algorithms use an ML-based predictor that, given a data item, estimates the number of times the item occurs in the input data set.
\\
\\
The talk will cover material from papers co-authored with T Eden, CY Hsu, D Katabi, S Narayanan, R Rubinfeld, S Silwal, T Wagner and A Vakilian.

Many data in real-world applications are in a high-dimensional space but exhibit low-dimensional structures. In mathematics, these data can be modeled as random samples on a low-dimensional manifold. Our goal is to estimate a target function or learn an optimal policy using neural networks. This talk is based on an efficient approximation theory of deep ReLU networks for functions supported on a low-dimensional manifold. We further establish the sample complexity for regression and off-policy learning with finite samples of data. When data are sampled on a low-dimensional manifold, the sample complexity crucially depends on the intrinsic dimension of the manifold instead of the ambient dimension of the data. These results demonstrate that deep neural networks are adaptive to low-dimensional geometric structures of data sets.
\\
\\
This is a joint work with Minshuo Chen, Haoming Jiang, Liu Hao, Tuo Zhao at Georgia Institute of Technology.

Applications such as climate science and transportation require learning complex dynamics from large-scale spatiotemporal data. Existing machine learning frameworks are still insufficient to learn spatiotemporal dynamics as they often fail to exploit the underlying physics principles. Representation theory can be used to describe and exploit the symmetry of the dynamical system. We will show how to design neural networks that are equivariant to various symmetries for learning spatiotemporal dynamics. Our methods demonstrate significant improvement in prediction accuracy, generalization, and sample efficiency in forecasting turbulent flows and predicting real-world trajectories. This is joint work with Robin Walters, Rui Wang, and Jinxi Li.

We consider the problem of recovering a structured signal x that lies close to a subset of interest T in $R^n$, from its random noisy linear measurements y = B A x + w, i.e., a generalization of the classic compressed sensing problem. Above, B is a fixed matrix and A has independent sub-gaussian rows. By varying B, and the sub-gaussian distribution of A, this gives a family of measurement matrices which may have heavy tails, dependent rows and columns, and singular values with large dynamic range. Typically, generalized compressed sensing assumes a random measurement matrix with nearly uniform singular values (with high probability), and asks: How many measurements are needed to recover x? In our setting, this question becomes: What properties of B allow good recovery? We show that the “effective rank'' of B may be used as a surrogate for the number of measurements, and if this exceeds the squared Gaussian complexity of T-T then accurate recovery is guaranteed. We also derive the optimal dependence on the sub-gaussian norm of the rows of A, to our knowledge this was not known previously even in the case when B is the identity. We allow model mismatch and inexact optimization with the aim of creating an easily accessible theory that specializes well to the case when T is the range of an expansive neural net.

The coronavirus disease 2019 (COVID-19) pandemic placed epidemic modeling at the forefront of worldwide public policy making. Nonetheless, modeling and forecasting the spread of COVID-19 remain a challenge. This talk begins with a review of the historical use of epidemic models and addresses the challenges of choosing a model in the early stages of a worldwide pandemic. The spread of COVID-19 has illustrated the heterogeneity of disease spread at different population levels. With finite size populations, chance variations may lead to significant departures from the mean. In real-life applications, finite size effects arise from the variance of individual realizations of an epidemic course about its fluid limit. I will illustrate how to model this variance with a martingale formulation consisting of a deterministic and a stochastic component, along with estimates for the size of the variance compared to real world data and simulations. Another cause of heterogeneities is the differing attitudes at the population level for control measures such as mask-wearing and physical distancing. Often, individuals form opinions about their behaviors from social network opinions. I will show some results from a two-layer multiplex network for the coupled spread of a disease and conflicting opinions. We model each process as a contagion. We develop approximations of mean-field type by generalizing monolayer pair approximations to multilayer networks; these approximations agree well with Monte Carlo simulations for a broad range of parameters and several network structures. We find that lengthening the duration that individuals hold an opinion may help suppress disease transmission, and we demonstrate that increasing the cross-layer correlations or intra-layer correlations of node degrees may lead to fewer individuals becoming infected with the disease.

In this talk, we will introduce first introduce Toeplitz structured random matrices and review some asymptotic results of these matrices. We will briefly show the limiting Toeplitz law and recent result of central limit theorem for linear statistics of a specific strucetured Toeplitz matrix based on moment methods. Some challenges, future directions and applications of random Toeplitz matrices will also be mentioned in this talk. Secondly, we will introduce the nonlinear random matrices in random neural networks. In the linear proportional regime, the limiting eigenvalue distributions of conjugate matrices and empirical neural tangent kernels at initial have been studied via Stieltjes transform, which will help us better understand the deep neural networks. We will finally present a recent result of nonlinear random matrix theory beyond linear regime, where a deformed a Wigner law will appear.

The Kaczmarz algorithm is an iterative method for solving linear systems of equations of the form Ax=y. Owing to its low memory footprint, the Kaczmarz algorithm has gained popularity for its practicality in applications to large-scale data, acting only on single rows of A at a time. In this talk, we discuss selecting rows of A randomly (Randomized Kaczmarz), selecting rows in a greedy fashion (Motzkin's Method), and selecting rows in a partially greedy fashion (Sampling Kaczmarz-Motzkin algorithm). Despite their variable computational costs, these algorithms have been proven to have the same theoretical upper bound on the convergence rate. Here we present an improvement upon previous known convergence bounds of the Sampling Kaczmarz-Motzkin algorithm, capturing the benefit of partially greedy selection schemes. Time permitting, we also will discuss an extension of the Kaczmarz algorithm to the setting where data takes on the form of a tensor and make connections between the new Tensor Kaczmarz algorithm and previously established algorithms. \\ \\ This presentation contains joint work with Jamie Haddock and Denali Molitor.

Semidefinite programming (SDP) is a powerful framework from convex optimization that has striking potential for data science applications. This talk describes a provably correct randomized algorithm for solving large, weakly constrained SDP problems by economizing on the storage and arithmetic costs. Numerical evidence shows that the method is effective for a range of applications, including relaxations of MaxCut, abstract phase retrieval, and quadratic assignment problems. Running on a laptop equivalent, the algorithm can handle SDP instances where the matrix variable has over $10^{14}$ entries. This talk will highlight the ideas behind the algorithm in a streamlined setting. The insights include a careful problem formulation, design of a bespoke optimization method, and use of randomized matrix computations. Joint work with Alp Yurtsever, Olivier Fercoq, Madeleine Udell, and Volkan Cevher. Based on arXiv 1912.02949 (Scalable SDP, SIMODS 2021) and other papers (SketchyCGM in AISTATS 2017, Nystr\"{o}m sketch in NeurIPS 2017)."

We will give an introduction to local entropy theory and we will trace the descriptive complexity of different families of topological dynamical systems with completely positive entropy (CPE) and uniform positive entropy (UPE). Joint work with Udayan B. Darji.

The cohomology of classifying spaces is an important classical topic in algebraic topology. However, much less is known in the equivariant setting, where one wants to know the RO(G)-graded cohomology of classifying G-spaces. The problem is that RO(G)-graded cohomology is notoriously difficult to compute even when G is cyclic.In this talk, I will explain my computations in the case of cyclic 2-groups G while keeping technical details to a minimum. The main goal is to understand rational equivariant characteristic classes, but I will also discuss some mod 2 computations and their relevance to the equivariant dual Steenrod algebra.

The Connes Embedding Problem (CEP) is arguably one of the most famous open problems in operator algebras. Roughly, it asks if every tracial von Neumann algebra can be approximated by matrix algebras. Earlier this year, a group of computer scientists proved a landmark result in complexity theory called MIP*=RE, and, as a corollary, gave a negative solution to the CEP. However, the derivation of the negative solution of the CEP from MIP*=RE involves several very complicated detours through C*-algebra theory and quantum information theory. In this talk, I will present joint work with Bradd Hart where we show how some relatively simple model-theoretic arguments can yield a direct proof of the failure of the CEP from MIP*=RE while simultaneously yielding a stronger, Gödelian-style refutation of CEP as well as the existence of “many” counterexamples to CEP. No prior background in any of these areas will be assumed.

Many of the questions asked during the birth of probability (e.g. what's the probability of getting a certain hand in poker?) are equivalent to basic counting problems, and since then there have been numerous applications of combinatorics to probability (e.g. moment method proofs for the semi-circle law). In this talk, probability strikes back with a vengeance by solving some (non-trivial) counting problems.

Dirac proved that every $n$-vertex graph with minimum degree at least $n/2$ contains a hamiltonian cycle. We prove an analogue for hypergraphs: we give exact bounds for the minimum degree of a uniform hypergraph that implies the existence of hamiltonian Berge cycles. \\ \\ This is joint work with Alexandr Kostochka and Grace McCourt.

A common task in many data-driven applications is to find a low dimensional manifold that describes the data accurately. Estimating a manifold from noisy samples has proven to be a challenging task. Indeed, even after decades of research, there is no (computationally tractable) algorithm that accurately estimates a manifold from noisy samples with a constant level of noise. In this talk, we will present a method that estimates a manifold and its tangent in the ambient space. Moreover, we establish rigorous convergence rates, which are essentially as good as existing convergence rates for function estimation. This is a joint work with Barak Sober.

Every continuous action of a countable group on a Polish space induces a Borel equivalence relation. We are interested in the problem of realizing (i.e. finding a Borel isomorphic copy of) these equivalence relations as continuous actions on compact spaces. We provide a number of positive results for variants of this problem, and we investigate the connection to subshifts.

A novel approach to simulate simple protein-­ligand systems at large time­ and length­-scales is to couple Markov state models (MSMs) of molecular kinetics with particle-­based reaction­-diffusion (PBRD) simulations; this approach is named MSM/RD. Current formulations of MSM/RD lack an underlying mathematical framework to derive coupling schemes; they are limited to protein-­ligand systems, where the ligand orientation and conformation switching are not taken into account; and they lack multi­particle extensions. In this work, we develop a general MSM/RD framework by coarse­-graining molecular dynamics into hybrid switching diffusion processes, a class of stochastic processes that integrate continuous dynamics and discrete events into the same process. With this MSM/RD framework, it is possi­ble to derive MSM/RD coupling schemes as discretizations of the underlying equations. It also allows conformation switching and the inclusion of all the rotational degrees of freedom. Given enough data to parametrize the model, it is capable of modeling protein­-protein interactions over large time­ and length­-scales, and it can be extended to han­dle multiple molecules. We derive the MSM/RD framework, and we implement and verify it for two protein­-protein benchmark systems and one multiparticle implementation to model the formation of pentameric ring molecules.

For any $C_2$-equivariant spectrum, we can functorially assign two non-equivariant spectra - the underlying spectrum and the geometric fixed point spectrum. They both induce maps from $RO(C_2)$-graded cohomology to classical cohomology. In this talk, I will compare the $RO(C_2)$-graded squaring operations with the classical squaring operations along the induced maps. This is joint work with Prasit Bhattacharya and Bertrand Guillou.

I will talk about two recent projects on solving inverse problems for kinetic models, where a change of perspective between Lagrangian and Eulerian is highly beneficial. In the first project, we are interested in recovering the initial temperature of the nonlinear Boltzmann equation given macroscopic quantities observed at a later time. With the problem formulated as constrained optimization, our proposed adjoint DSMC method, together with the well-known (forward) DSMC method, makes it possible to evaluate Boltzmann-constrained gradient within seconds, independent of the size of the parameter. In the second project, we are interested in calibrating the parameter in the chaotic dynamic system. Transforming the long-time trajectories to an invariant measure significantly improves the inverse problem's ill-posedness. It also turns the original ODE model into a PDE model (continuity equation or Fokker-Planck equation), allowing efficient gradient calculation for the resulting PDE-constrained optimization problem.

Abstract: Strong 1-boundedness is a notion introduced by Kenley Jung which captures (among Connes-embeddable von Neumann algebras) the property of having "a small amount of matrix approximations". Some examples are diffuse hyperfinite von Neumann algebras, vNa's with Property Gamma, vNa's that are non prime, vNa's that have a diffuse hyperfinite regular subalgebra and so on. This is a von Neumann algebra invariant, and was used to prove in a unified approach remarkable rigidity theorems. One such is the following: A non trivial free product of von Neumann algebras can never be generated by two strongly 1-bounded von Neumann subalgebras with diffuse intersection. This result cannot be recovered by any other methods for even hyperfinite subalgebras. In this talk, I will present joint work with David Jekel and Ben Hayes, settling an open question from 2005, whether Property (T) II$_1$ factors are strongly 1-bounded, in the affirmative. Time permitting, I will discuss some insights and future directions.

In an $n$-resident town, Oddtown, all of their clubs must satisfy the following properties: all clubs must have an odd number of members and amongst any two distinct clubs, there must be an even number of residents in common. The classical oddtown theorem states that any such town can have at most $n$ clubs. In this talk, we explore how the residents can have $n+1$ clubs of odd size and minimize the chance of the town catching them. That is, we'd like to minimize the number of pairs of clubs with an odd number of members in common. We will also explore a similar problem with Eventown.

The letters D and O are topologically indistinguishable (both are circles). However, after superimposing each symbol with their reflections across several axes, one \emph{can} distinguish between them. The curvature in the O results in a different evolution in first homology when compared to the angled D. In this talk we will expand on this idea by explaining a white box classification algorithm which classifies an image as one of the 26 letters in the (capitalized) English alphabet. The driving force is the theory of persistent homology, as implemented in the Ripser package. This technique is less powerful than traditional techniques of machine learning (such as a neural net), but it is much more explainable.

An independent set $I$ of a graph $G$ is said to be a maximal independent set (MIS) if it is maximal with respect to set inclusion. Nielsen proved that the maximum number of MIS's of size $k$ in an $n$-vertex graph is asymptotic to $(n/k)^k$, with the extremal construction being a disjoint union of $k$ cliques with sizes as close to $n/k$ as possible. In this talk we study how many MIS's of size $k$ an $n$-vertex graph $G$ can have if $G$ does not contain a clique $K_t$. We prove for all fixed $k$ and $t$ that there exist such graphs with $n^{\lfloor\frac{(t-2)k}{t-1}\rfloor-o(1)}$ MIS's of size $k$ by utilizing recent work of Gowers and B. Janzer on a generalization of the Ruzsa-Szemer\'edi problem. We prove that this bound is essentially best possible for triangle-free graphs when $k\le 4$. \medskip This is joint work with Xiaoyu He and Jiaxi Nie.

The Monge-Ampere equation det$(D^2u) = 1$ arises in prescribed curvature problems and in optimal transport. An interesting feature of the equation is that it admits singular solutions. We will discuss new examples of convex functions on $R^n$ that solve the Monge-Ampere equation away from finitely many points, but contain polyhedral and Y-shaped singular structures. Along the way we will discuss geometric motivations for constructing such examples, as well as their connection to a certain obstacle problem.

The method of moments is a statistical method for density estimation that equates sample moments to moment equations for a given family of densities. When the underlying distribution is assumed to be a convex combination of Gaussian densities, the resulting moment equations are polynomial in the density parameters. We examine the asymptotic behavior of the variety stemming from these equations as the number of components and the dimension of each component increases. This is joint work with Jose Israel Rodriguez and Carlos Amendola.

A Cayley diagram is a labeling of a graph $G$ that encodes an action of a group which induces $G$. For instance, a $d$-edge coloring of a $d$-regular tree is a Cayley diagram for the group $(\mathbb{Z}/2\mathbb{Z})^{*d}$. In this talk, we will investigate when a Cayley graph $G=(\Gamma, E)$ admits an $\operatorname{Aut}(G)$-f.i.i.d. Cayley diagram and show that $\Gamma$-f.i.i.d. solutions to local labeling problems for such graphs lift to $\operatorname{Aut}(G)$-f.i.i.d. solutions.

We describe several recent results on orders of abelian varieties over $\mathbb{F}_2$: every positive integer occurs as the order of an ordinary abelian variety over $\mathbb{F}_2$ (joint with E. Howe); every positive integer occurs infinitely often as the order of a simple abelian variety over $\mathbb{F}_2$; the geometric decomposition of the simple abelian varieties over $\mathbb{F}_2$ can be described explicitly (joint with T. D'Nelly-Warady); and the relative class number one problem for function fields is reduced to a finite computation (work in progress). All of these results rely on the relationship between isogeny classes of abelian varieties over finite fields and Weil polynomials given by the work of Weil and Honda-Tate. With these results in hand, most of the work is to construct algebraic integers satisfying suitable archimedean constraints.

The moduli space of stable curves parameterizes tuples $(C,p_1,...,p_n)$ of a compact, complex curve $C$ together with distinct marked points $p_1,\dots, p_n$. Inside this moduli space, there are natural subsets, called the strata of $k$-differentials, defined by the condition that there exists a meromorphic $k$-differential on $C$ with zeros and poles of some fixed multiplicities at the points $p_i$. I will discuss basic properties of these strata and explain a conjecture relating their fundamental class to the so-called double ramification cycles on the moduli space. I explain the idea of the proof of this conjecture and some ongoing work with Costantini and Sauvaget on how to use this relation to compute intersection numbers of the strata with $\psi$-classes on the moduli of curves.

Calculating the coefficients of equivariant generalized cohomology theories has been a fundamental question for equivariant homotopy theory. In this talk, I will talk about some calculations when the group is nonabelian. Examples include $RO(G)$-graded Eilenberg-MacLane cohomology of a point with constant coefficient when $G$ is a dihedral group of order $2p$ or the quaternion group $Q_8$, and coefficient ring of $\Sigma_3$-equivariant complex cobordism. I will discuss techniques in such computations: isotropy separation, cellular structures and dualities. This is joint work with Po Hu and Igor Kriz.

I will present a recent result showing that the direct product group $\Gamma = \mathbb F_2 \times \mathbb F_2$ is not Hilbert-Schmidt stable. Specifically, $\Gamma$ admits a sequence of asymptotic homomorphisms (with respect to the normalized Hilbert-Schmidt norm) which are not perturbations of genuine homomorphisms. As we will explain, while this result concerns unitary matrices, its proof relies on techniques and ideas from the theory of von Neumann algebras. We will also explain how this result can be used to settle in the negative a natural version of an old question of Rosenthal concerning almost commuting matrices. More precisely, we derive the existence of contraction matrices $A,B$ such that $A$ almost commutes with $B$ and $B^*$ (in the normalized Hilbert-Schmidt norm), but there are no matrices $A’,B’$ close to $A,B$ such that $A’$ commutes with $B’$ and $B’^*$.

For computations with many variables in optimization or solving large systems in numerical linear algebra, developing efficient methods is highly desirable. This talk introduces an approach for large-scale optimization with sparse linear equality constraints that exploits computationally efficient orthogonal projections. For approximately solving large linear systems, (randomized) sketching methods are becoming increasingly popular. By recursively augmenting a deterministic sketching matrix, we develop a method with a finite termination property that compares favorably to randomized methods. Moreover, we describe the construction of logical linear systems that can be used in e.g., COVID-19 pooling tests, and a nonlinear least-squares method that addresses large data sizes in machine learning.

We study long time dynamics of combustive processes in random media, modeled by reaction-diffusion equations with random ignition reactions. One expects that under reasonable hypotheses on the randomness, large scale dynamics of solutions to these equations is almost surely governed by a homogeneous Hamilton-Jacobi equation. While this was previously shown in one dimension as well as for radially symmetric reactions in several dimensions, we prove this phenomenon in the general non-isotropic multidimensional setting. We also show that the rate of convergence of solutions to the Hamilton-Jacobi dynamics is at least algebraic in the relevant space-time scales when the initial data is close to an indicator function of a convex set. This talk is based on joint work with Andrej Zlato\v{s}.

We define a Fibonacci walk to be any sequence of positive integers satisfying the recurrence $w_{k+2}=w_{k+2}=w_{k+1}+w_k$, and we say that a sequence is an $n$-Fibonacci walk if $w_k=n$ for some $k$. Note that every $n$ has a number of (boring) $n$-Fibonacci walks, e.g. the sequence starting $n,n,2n,\ldots$. To make things interesting, we consider $n$-Fibonacci walks which have $w_k=n$ with $k$ as large as possible, and we call this an $n$-slow Fibonacci walk. For example, the two 6-slow Fibonacci walks start 2, 2, 4, 6 and 4, 1, 5, 6. In this talk we discuss a number of properties about $n$-slow Fibonacci walks, such as the number of slow walks a given $n$ can have, as well as how many $n$ have a given number of walks. We also discuss slow walks that follow more general recurrence relations. This is joint work with Fan Chung and Ron Graham.

Have you ever thought to yourself ``Homological Algebra is great, but I wish there were more technicalities and operations to keep track of"? Do you ever worry that the chain complex feels left out after you take cohomology? Does it keep you up at night that elements whose products are 0 get less of a say in the cohomology ring's structure? Has your topologist friend ever ignored nullhomotopic maps, making them feel excluded and unheard? In this talk, Max will be explaining two closely related constructions, in homological algebra and in homotopy, that arise only when other operations give a trivial output. These so called ``secondary operations" play a large role in computational algebraic topology, and can be an indispensable tool for proving results both highly abstract (see Tyler Lawson's ``BP is not $E_\infty$") and very concrete (see Massey's ``Higher Order Linking Numbers").

We establish the conjecture of Reiner and Yong for an explicit combinatorial formula for the expansion of a Grothendieck polynomial into the basis of Lascoux polynomials. This expansion is a subtle refinement of its symmetric function version due to Buch, Kresch, Shimozono, Tamvakis, and Yong, which gives the expansion of stable Grothendieck polynomials indexed by permutations into Grassmannian stable Grothendieck polynomials. Our expansion is the K-theoretic analogue of that of a Schubert polynomial into Demazure characters, whose symmetric analogue is the expansion of a Stanley symmetric function into Schur functions. This is a joint work with Mark Shimozono.

Let $\Gamma$ be a countably infinite group. Seward and Tucker-Drob proved that every free Borel action of $\Gamma$ on a Polish space $X$ admits a Borel equivariant map $\pi$ to the free part of the Bernoulli shift $k^\Gamma$, for any $k \geq 2$. Our goal in this talk is to generalize this result by putting extra restrictions on the image of $\pi$. For instance, can we ensure that $\pi(x)$ is a proper coloring of the Cayley graph of $\Gamma$ for all $x \in X$? More generally, can we guarantee that the image of $\pi$ is contained in a given subshift of finite type? The main result of this talk is a positive answer to this question in a very broad (and, in some sense, optimal) setting. The main tool used in the proof of our result is a probabilistic technique for constructing continuous functions with desirable properties, namely a continuous version of the Lov\'{a}sz Local Lemma.

This talk is expository. It describes the history of an exciting connection made by physicists between an unsolved problem in combinatorial design theory- the existence of maximal sets of $d^2$ complex equiangular lines in ${\mathbb C}^d$- rephrased as a problem in quantum information theory, and topics in algebraic number theory involving class fields of real quadratic fields. Work of my former student Gene Kopp recently uncovered a surprising, deep (unproved!) connection with the Stark conjectures. For infinitely many dimensions $d$ he predicts the existence of maximal equiangular sets, constructible by a specific recipe starting from suitable Stark units, in the rank one case. Numerically computing special values at $s=0$ of suitable L-functions then permits recovering the units numerically to high precision, then reconstructing them exactly, then testing they satisfy suitable extra algebraic identities to yield a construction of the set of equiangular lines. It has been carried out for $d=5, 11, 17$ and $23$.

The Mondrian process in machine learning is a Markov partition process that recursively divides space with random axis-aligned cuts. This process is used to build random forests for regression and classification as well as Laplace kernel approximations. The construction allows for efficient online algorithms, but the restriction to axis-aligned cuts does not capture dependencies between features. By viewing the Mondrian as a special case of the stable under iterated (STIT) process in stochastic geometry, we resolve open questions about the generalization of cut directions. We utilize the theory of stationary random tessellations to show that STIT processes approximate a large class of stationary kernels and STIT random forests achieve minimax rates for Lipschitz functions (forests and trees) and $C^2$ functions (forests only). This work opens many new questions at the intersection of stochastic geometry and statistical learning theory. Based on joint work with Ngoc Mai Tran.

For a smooth projective surface X, natural objects of study are its moduli spaces of (semi-) stable coherent sheaves. In rank one, their structural invariants are well-understood, starting with G\"{o}ttsche's famous formula for the Betti numbers of the Hilbert schemes of points of X in terms of the Betti numbers of X itself. Even for rank two

There is a deep relationship between deformation theory for symplectic manifolds and quantizing field theories. In this talk, I'll discuss this story for symplectic supermanifolds and supersymmetric mechanics. We will approach these questions using modern descent techniques that work more generally for factorization algebras associated to higher-dimensional field theories. Relations to manifold invariants such as the L-genus will also be discussed. No physics knowledge is required.

We consider the quintic, focusing semilinear wave equation on $\mathbb{R}^{1+3}$, in the radially symmetric setting, and construct infinite time blow-up, relaxation, and intermediate types of solutions. More precisely, we first define an admissible class of time-dependent length scales, which includes a symbol class of functions. Then, we construct solutions which can be decomposed, for all sufficiently large time, into an Aubin-Talentini (soliton) solution, re-scaled by an admissible length scale, plus radiation (which solves the free 3 dimensional wave equation), plus corrections which decay as time approaches infinity. The solutions include infinite time blow-up and relaxation with rates including, but not limited to, positive and negative powers of time, with exponents sufficiently small in absolute value. We also obtain solutions whose soliton component has oscillatory length scales, including ones which converge to zero along one sequence of times approaching infinity, but which diverge to infinity along another such sequence of times. The method of proof is similar to a recent wave maps work of the author, which is itself inspired by matched asymptotic expansions.

The open-source FEniCS Project (https://fenicsproject.org/) has proven to be a popular and successful finite element (FE) automation tool, applicable to many problem domains involving partial differential equations (PDEs). (CCoM seminar regulars may recall a 2017 talk by L. Ridgway Scott on FEniCS and its implications for pedagogy.) The present talk discusses recent work extending FEniCS to numerical methods other than traditional FE methods. The library tIGAr (https://github.com/david-kamensky/tIGAr) extends FEniCS to isogeometric analysis (IGA), where spline-based geometries from design and graphics replace the meshes of traditional FE analysis. This library retains a similar workflow to traditional FE analysis with FEniCS, while using object-oriented abstractions to separate PDE solution from geometry creation. This design permits analysis of many different PDEs, using a wide variety of existing spline types, and provides an interface to add support for future sp line constructions. This talk surveys several example applications of tIGAr, including divergence-conforming IGA of incompressible flow, Kirchhoff--Love shell analysis, and nonlocal contact mechanics. Going further beyond standard FE analysis, we consider immersed-boundary methods, which present more complicated challenges for automation software. Some initial results on combining FEniCS and tIGAr for immersed fluid--structure interaction will be presented, along with recent work coupling tIGAr-based isogeometric shell analysis at intersection curves of separately-parameterized structural components. Lastly, we discuss the ongoing development of general-purpose tools for immersed FE analysis.

Since the seminal work of Voiculescu in the early 90’s, the connection between the asymptotic behavior of random matrices and free probability has been extensively studied. More recently, in relation to the solution of the Kadison-Singer problem, Marcus, Spielman, and Srivastava discovered a deep connection between certain classical polynomial convolutions and free probability. Soon after, this connection was further understood by Marcus, who introduced the notion of finite free probability.

In this talk I will present recent results on finite free probability with applications to the asymptotic analysis of real-rooted polynomials. Our approach is based on a careful combinatorial analysis of the finite free cumulants, and allows us to study the asymptotic dynamics of the root distribution of polynomials after repeated differentiation, as well as the fluctuations of the root distributions of polynomials around their limiting measure. This is joint work with Octavio Arizmendi and Daniel Perales: arXiv:2108.08489.

Selberg’s 3/16 theorem for congruence covers of the modular surface is a beautiful theorem which has a natural dynamical interpretation as uniform exponential mixing. Bourgain-Gamburd-Sarnak's breakthrough works initiated many recent developments to generalize Selberg's theorem for infinite volume hyperbolic manifolds. One such result is by Oh-Winter establishing uniform exponential mixing for convex cocompact hyperbolic surfaces. These are not only interesting in and of itself but can also be used for a wide range of applications including uniform resonance free regions for the resolvent of the Laplacian, affine sieve, and prime geodesic theorems. I will present a further generalization to higher dimensions and some of these immediate consequences.

Let $X$ be a smooth, geometrically irreducible scheme over a finite field of characteristic $p > 0$. With respect to rigid cohomology, $p$-adic coefficient objects on $X$ come in two types: convergent $F$-isocrystals and the subcategory of overconvergent $F$-isocrystals. Overconvergent isocrystals are related to $\ell$-adic etale objects ($\ell\neq p$) via companions theory, and as such it is desirable to understand when an isocrystal is overconvergent. We show (under a geometric tameness hypothesis) that a convergent $F$-isocrystal $E$ is overconvergent if and only if its restriction to all smooth curves on $X$ is. The technique reduces to an algebraic setting where we use skeleton sheaves and crystalline companions to compare $E$ to an isocrystal which is patently overconvergent. Joint with Kiran Kedlaya and James Upton.

In many biological systems, chemical reactions or changes in a physical state are assumed to occur instantaneously. For describing the dynamics of those systems, Markov models that require exponentially distributed inter-event times have been used widely. However, some biophysical processes such as gene transcription and translation are known to have a significant gap between the initiation and the completion of the processes, which renders the usual assumption of exponential distribution untenable. In this talk, we consider relaxing this assumption by incorporating age-dependent random time delays (distributed according to a given probability distribution) into the system dynamics. We do so by constructing a measure-valued Markov process on a more abstract state space, which allows us to keep track of the 'ages' of molecules participating in a chemical reaction. We study the large-volume limit of such age-structured systems. We show that, when appropriately scaled, the stochastic system can be approximated by a system of partial differential equations (PDEs) in the large-volume limit, as opposed to ordinary differential equations (ODEs) in the classical theory. We show how the limiting PDE system can be used for the purpose of further model reductions and for devising efficient simulation algorithms. To describe the ideas, we will use a simple transcription process as a running example.

In studying complex Chern-Simons theory on a Seifert manifold, Gukov-Pei proposed an equivariant Verlinde formula, a one-parameter deformation of the celebrated Verlinde formula. It computes, among many things, the graded dimension of the space of holomorphic sections of (powers of) a natural determinant line bundle over the Hitchin moduli space. Gukov-Pei conjectured that the equivariant Verlinde numbers are equal to the equivariant quantum K-invariants of a non-compact (Kahler) quotient space studied by Hanany-Tong. In this talk, I will explain the setup of this conjecture and its proof via wall-crossing of moduli spaces of (parabolic) Bradlow-Higgs triples. It is based on work in progress with Wei Gu and Du Pei.

Classical Steenrod algebra is one of the most fundamental algebraic gadgets in stable homotopy theory. It led to the theory of characteristic classes, which is key to some of the most celebrated applications of homotopy theory to geometry. The G-equivariant Steenrod algebra is not known beyond the group of order 2. In this talk, I will recall a geometric construction of the classical Steenrod algebra and generalize it to construct G-equivariant Steenrod operations. Time permitting, I will discuss potential applications to equivariant geometry.

We consider the classical framework of the famous De-Giorgi-Nash-Moser theorem: $div(A(x)\nabla u)=f$, where $A(x)$ is a symmetric, elliptic matrix field, $f$ is given and $u:U\subset \mathbb{R}^n\to\mathbb{R}$ is the unknown. N. Trudinger was the first one to relax the assumptions on the coefficients matrix $A(x)$. He was able to derive boundedness results if the matrix is barely integrable in the right spaces. In particular he was able to show that if $\lambda(x)|\xi|^2\leq \xi\cdot A(x)\xi\leq \Lambda(x)|\xi|^2,\quad \forall x$ and the $\lambda^{-1}\in L^p, \Lambda\in L^q$ satisfying $\frac{1}{p}+\frac{1}{q}<\frac{2}{n}$. The integrability condition had been considerably improved by P. Bella and M. Schaffner in the framework of the Moser-iteration to $\frac{1}{p}+\frac{1}{q}<\frac{2}{n-1}$. A counterexample had been constructed by Franchi, Serapioni, and Serra Cassano under $\frac{1}{p}+\frac{1}{q}>\frac{2}{n}$. The aim of this talk is to revisit De Giorgi’s original approach having in mind the question concerning the optimal integrability assumption on the coefficient field. We will present how this question is surprisingly linked to a question in linear programming with an infinite horizon. This talk will be about my ongoing project with M. Schaffner, hence about work in progress.

Gaussian measures have long been recognized as the appropriate measures to use in infinite-dimensional analysis. Their regularity properties have allowed the development of a calculus on these measure spaces that has become an invaluable tool in the analysis of stochastic processes and their applications. Gaussian measures arise naturally in the context of random diffusions, specifically as the end point distribution of Brownian motion, and one may see their regularity as arising from nice properties of the generator of the diffusion. More particularly, in finite dimensions, hypoellipticity of the generator is a standard assumption required for regularity of the associated measure. However, in infinite dimensions it has remained elusive to demonstrate that hypoellipticity is a sufficient condition for regularity. Using techniques first developed by Bruce Driver and Masha Gordina, there has been some recent success in proving regularity for some natural infinite-dimensional hypoelliptic models. These techniques rely on establishing uniform bounds on coefficients appearing in certain functional analytic inequalities for semi-groups on finite-dimensional approximations. We will discuss some of these successful applications, including more recent work studying models satisfying only a weak notion of hypoellipticity. This includes joint works with Fabrice Baudoin, Dan Dobbs, Bruce Driver, Nate Eldredge, and Masha Gordina.

We discuss quasi-Newton methods for minimizing a strictly convex quadratic function that may be subject to errors in the evaluation of the gradients. The methods all give identical behavior in exact arithmetic, generating minimizers of Krylov subspaces of increasing dimensions, thereby having finite termination. In exact arithmetic, the method of conjugate gradients gives identical iterates and has less computational cost. In a framework of small errors, e.g., finite precision arithmetic, the performance of the method of conjugate gradients is expected to deteriorate, whereas a BFGS quasi-Newton method is empirically known to behave very well. We discuss the behavior of limited-memory quasi-Newton methods, balancing the good performance of a BFGS method to the low computational cost of the method of conjugate gradients. We also discuss large-error scenarios, in which the expected behavior is not so clear. In particular, we are interested in the behavior of quasi-Newton matrices that differ from the identity by a low-rank matrix, such as a memoryless BFGS method. Our numerical results indicate that for large errors, a memory-less quasi-Newton method often outperforms a BFGS method. We also consider a more advanced model for generating search directions, based on solving a chance-constrained optimization problem. Our results indicate that such a model often gives a slight advantage in final accuracy, although the computational cost is significantly higher. The talk is based on joint work with Tove Odland, David Ek, Gianpiero Canessa and Shen Peng.

We present all 143 counterexamples in topology appearing in the book ``Counterexamples In Topology", by Lynn Steen and J. Arthur Seebach, Jr. For each such counterexample, we state whether or not it is compact.

Semidefinite optimization is a type of convex optimization problems over the cone of positive semidefinite (PSD) matrices, and sum-of-squares (SOS) optimization is another type of convex optimization problems concerned with the cone of SOS polynomials. Both semidefinite and SOS optimization have found a wide range of applications, including control theory, fluid dynamics, machine learning, and power systems. In theory, they can be solved in polynomial time using interior-point methods, but these methods are only practical for small- to medium-sized problem instances. In this talk, I will introduce decomposition methods for semidefinite optimization and SOS optimization with chordal sparsity, which scale more favorably to large-scale problem instances. It is known that chordal decomposition allows one to equivalently decompose a PSD cone into a set of smaller and coupled cones. In the first part, I will apply this fact to reformulate a sparse semidefinite program (SDP) into an equivalent SDP with smaller PSD constraints that is suitable for the application of first-order operator-splitting methods. The resulting algorithms have been implemented in the open-source solver CDCS. In the second part, I will extend the classical chordal decomposition to the case of sparse polynomial matrices that are positive (semi)definite globally or locally on a semi-algebraic set. The extended decomposition results can be viewed as sparsity-exploiting versions of the Hilbert-Artin, Reznick, Putinar, and Putinar-Vasilescu Positivstellensätze. This talk is based on our work: https://arxiv.org/abs/1707.05058 and https://arxiv.org/abs/2007.11410

In digital signal processing, quantization is the step of converting a signal's real-valued samples into a finite string of bits. As the first step in digital processing, it plays a crucial role in determining the information conversion rate and the reconstruction accuracy. Compared to non-adaptive quantizers, the adaptive ones are known to be more efficient in quantizing bandlimited signals, especially when the bit-budget is small (e.g.,1 bit) and noises are present. However, adaptive quantizers are currently only designed for 1D functions/signals. In this talk, I will discuss challenges in extending it to high dimensions and present our proposed solutions. Specifically, we design new adaptive quantization schemes to quantize images/videos as well as functions defined on 2D surface manifolds and general graphs, which are common objects in signal processing and machine learning. Mathematically, we start from the 1D Sigma-Delta quantization, extend them to high-dimensions and build suitable decoders. The discussed theory would be useful in natural image acquisition, medical imaging, 3D printing, and graph embedding.

In this talk, we discuss effective versions of Ratner’s theorems in the space of affine lattices. For $d \geq 2$, let $Y=ASL_d(\mathbb{R})/ASL_d(\mathbb{Z})$, $H$ be a minimal horospherical group of $SL_d(\mathbb{R})$ embedded in $ASL_d(\mathbb{R})$, and $a_t$ be the corresponding diagonal flow. Then $(a_t)$-push-forwards of a piece of $H$-orbit become equidistributed with a polynomial error rate under certain Diophantine condition of the initial point of the orbit. This generalizes the previous results of Strömbergsson for $d = 2$ and of Prinyasart for $d = 3$.

Quaternionic automorphic representations are one attempt to generalize to other groups the special place holomorphic modular forms have among automorphic representations of $GL_2$. Like holomorphic modular forms, they are defined by having their real component be one of a particularly nice class (in this case, called quaternionic discrete series). We count quaternionic automorphic representations on the exceptional group $G_2$ by developing a $G_2$ version of the classical Eichler-Selberg trace formula for holomorphic modular forms. There are two main technical difficulties. First, quaternionic discrete series come in L-packets with non-quaternionic members and standard invariant trace formula techniques cannot easily distinguish between discrete series with real component in the same L-packet. Using the more modern stable trace formula resolves this issue. Second, quaternionic discrete series do not satisfy a technical condition of being ``regular", so the trace formula can a priori pick up unwanted contributions from automorphic representations with non-tempered components at infinity. Applying some computations of Mundy, this miraculously does not happen for our specific case of quaternionic representations on $G_2$. Finally, we are only studying level-1 forms, so we can apply some tricks of Chenevier and Taïbi to reduce the problem to counting representations on the compact form of $G_2$ and certain pairs of modular forms. This avoids involved computations on the geometric side of the trace formula.

For an advection-reaction PDE model of population, with a non-local boundary condition modeling ``birth", and with a multiplicative input whose nature is the ``harvesting rate", we design a feedback law that stabilizes a desired equilibrium profile (of population density vs. age). Without feedback the system has one eigenvalue at the origin and the remainder of its infinite spectrum has negative real parts, i.e., the systems is, as engineers call it, ``neutrally stable". Hence, a feedback is needed to move one eigenvalue to the left without making any of the other ones spill to the right of the imaginary axis. This control design objective is achieved by transforming the system into a control-theoretic canonical form consisting of a first-order ODE in which the input is present and whose eigenvalue needs to be made negative by feedback, and an infinite-dimensional input-free system called the ``zero dynamics", which we prove to be exponentially stable. The key feature of the overall PDE system and its feedback control law is the positivity of both the population density state and the harvesting rate input, which is a key element of the analysis, captured by a``control Lyapunov functional" which blows up when the population density or control approach zero.

A longstanding conjecture of T. Fujita asserts that if X is a smooth complex projective variety of dimension n and if L is an ample line bundle, then $K_X+mL$ is basepoint free for $m>=n+1$. The conjecture is known up to dimension five by work of Reider, Ein, Lazarsfeld, Kawamata, Ye and Zhu. In higher dimensions, breakthrough work of Angehrn, Siu, Helmke and others showed that the conjecture holds if m is larger than a quadratic function in n. We show that for $n>=2$ the conjecture holds for m larger than $n(loglog(n)+3)$. This is joint work with L. Ghidelli.

I'll describe a new rigidity phenomenon of lattices in higher rank semisimple groups. Specifically, I'll explain why the theories of such groups can't have (finitely generated) deformations, why these groups have a very rich collection of definable subgroups, and finish by discussing a conjecture saying that being a higher rank arithmetic lattice is a first-order property. Based on joint works with Alex Lubotzky and Chen Meiri.

It has been known for a long time that as their size grow to infinity, many models of random matrices behave as free operators. This link was first explicited by Voiculescu in 1991 in a paper in which he proved that the trace of polynomials in independent GUE matrices converges towards the trace of the same polynomial evaluated in free semicircular variables. In 2005, Haagerup and Thorbjornsen proved the convergence of the norm instead of the trace. The main difficulty of their proof was to prove a sharp enough upper bound of the difference between the trace of random matrices and their free limit. They managed to do so with the help of the so-called linearization trick which allows to relate the spectrum of a polynomial of any degree with scalar coefficients with a polynomial of degree 1 with matrix coefficients. A drawback of this method is that it does not give easily good quantitative estimates. In arXiv:1912.04588, we introduced a new strategy to approach those questions which does not rely on the linearization trick and instead is based on free stochastic calculus. In this talk, I will first focus on the paper arXiv:2011.04146, in which we proved an asymptotic expansion for traces of smooth functions evaluated in independent GUE random matrices, whose coefficients are defined through free probability. And then I will talk about arXiv:2005.1383, in which we adapted the previous method to the case of Haar unitary matrices.

In a large class of reaction-diffusion equations, the solution starting from a compactly supported initial datum develops a transition between two rest states, that moves at an asymptotically linear rate in time, and whose thickness remains asymptotically bounded in time. The issue is its precise location in time, that is, up to terms that are o(1) as time goes to infinity. This question is well understood in one space dimension; I will discuss what happens in the less well settled multi-dimensional framework. Joint works with L. Rossi and V. Roussier.

What makes the digits 645751311064590590501615753639260425710 and 1010100101010011111111010100111010010111 so special? These digits look as if they were chosen at random, yet they are entirely deterministic (take the fractional part of the square root of 7). In this talk, I will explore the theory of quasi-randomness, which characterizes ``random-like" sequences, graphs, sets, and many other objects. In particular, I will present a theory of quasi-randomness for Boolean functions and show how random Boolean functions lead to a very challenging open problem: the Inverse Theory of the Gowers Norms.

In this talk, we discuss the problem of finding generalized Nash equilibria (GNE) in the viewpoint of sparse polynomials. To obtain optimality conditions for GNE, we consider the Karush-Kuhn-Tucker (KKT) system using the Lagrange multiplier. We discuss that if all objectives and constraints polynomials are generic, the number of solutions of the KKT system equals its mixed volume, and so the polyhedral homotopy method can be optimal for finding GNEs. Lastly, comparisons with existing methods will be given.

This talk is concerned with the inverse problem of recovering a discrete measure on the torus given a finite number of its noisy Fourier coefficients. We focus on the diffraction limited regime where at least two atoms are closer together than the Rayleigh length. We show that the fundamental limits of this problem and the stability of subspace (algebraic) methods, such as ESPRIT and MUSIC, are closely connected to the minimum singular value of non-harmonic Fourier matrices. We provide novel bounds for the latter in the case where the atoms are located in clumps. We also provide an analogous theory for a statistical model, where the measure is time-dependent and Fourier measurements are collected over at various times. Joint work with Wenjing Liao, Albert Fannjiang, Zengying Zhu, and Weiguo Gao.

What does it mean to pick a ``random'' hyperbolic surface, and how does one even go about ``picking'' one? Mirzakhani gave an inductive answer to this question by gluing together smaller random surfaces along long curves; this is equivalent to studying the equidistribution of certain sets inside the moduli space of hyperbolic surfaces. Starting from first concepts, in this talk I’ll explain a new method for building random hyperbolic surfaces by building random \emph{flat} ones. As time permits, we will also discuss the application of this technique to Mirzakhani’s ``twist torus conjecture.'' This is joint work (in progress) with James Farre.

Given a prime $p$, a finite extension $L/\mathbb{Q}_{p}$, a
connected $p$-adic reductive group $G/L$, and a smooth irreducible
representation $\pi$ of $G(L)$, Fargues-Scholze recently attached a
semisimple Weil parameter to such $\pi$, giving a general candidate for
the local Langlands correspondence. It is natural to ask whether this
construction is compatible with known instances of the correspondence
after semisimplification. For $G = GL_{n}$ and its inner forms,
Fargues-Scholze and Hansen-Kaletha-Weinstein show that the
correspondence is compatible with the correspondence of
Harris-Taylor/Henniart. We verify a similar compatibility for $G =
GSp_{4}$ and its unique non-split inner form $G = GU_{2}(D)$, where $D$
is the quaternion division algebra over $L$, assuming that
$L/\mathbb{Q}_{p}$ is unramified and $p > 2$. In this case, the local
Langlands correspondence has been constructed by Gan-Takeda and
Gan-Tantono. Analogous to the case of $GL_{n}$ and its inner forms, this
compatibility is proven by describing the Weil group action on the
cohomology of a local Shimura variety associated to $GSp_{4}$, using
basic uniformization of abelian type Shimura varieties due to Shen,
combined with various global results of Kret-Shin and Sorensen on Galois
representations in the cohomology of global Shimura varieties associated
to inner forms of $GSp_{4}$ over a totally real field. After showing the
parameters are the same, we apply some ideas from the geometry of the
Fargues-Scholze construction explored recently by Hansen, to give a more
precise description of the cohomology of this local Shimura variety,
verifying a strong form of the Kottwitz conjecture in the process.

Chemical Reaction Network theory is an area of mathematics that analyzes the behaviors of chemical processes. One major problem concerns the stability of steady states of these networks. Does a given chemical reaction network have the capacity for Hopf bifurcations (an important unstable steady-state that yields periodic oscillations)? Our first contribution is a novel procedure for constructing a Hopf bifurcation of a chemical reaction network. This algorithm -- our Newton-polytope method -- gives an easy-to-check condition for the existence of a Hopf bifurcation and explicitly constructs one if it exists. Another important invariant of a chemical reaction network is its maximum number of steady states. This number, however, is in general difficult to compute, as it translates to counting positive real solutions of parametrized polynomial systems. To this end, we introduce an upper bound on this number -- namely, a network's mixed volume -- that is easy to compute. In this talk, we apply our two new tools to an important biological-signaling network, called the ERK network. Rubinstein et al. (2016) showed that the ERK network exhibits multiple steady states, bistability, and periodic oscillations (for a very particular choice of initial conditions). Conradi and Shiu (2015) proved that when certain reactions are omitted, the ERK network reduces to the processive dual-site phosphorylation network, which has a unique, stable steady-state (for any initial conditions). This stark contrast in dynamics prompted Rubinstein et al.'s question, "How are bistability and oscillations lost as reactions are removed from the ERK network?" By analyzing subnetworks of the ERK network, we systematically answer this question and demonstrate that bistability and oscillations persist even after we greatly simplify the model (by making reactions irreversible and removing intermediate species).

Mnev's universality theorem asserts that every singularity type over the ring of integers appears in some thin Schubert cell of the Grassmannian Gr(3,E) for some vector space E. We construct sequential blowups of Gr(3,E) such that certain induced birational transforms of all thin Schubert cells become smooth over prime fields. This implies that every singular variety X defined over a prime field admits local resolutions. For a singular variety X over a general perfect field k, we spread it out and deduce that X/k admits local resolution as well.

The famous Lehmer problem asks whether there is a gap between 1 and the Mahler measure of algebraic integers which are not roots of unity. Asked in 1933, this deep question concerning number theory has since then been connected to several other subjects. After introducing the concepts involved, we will briefly describe a few of these connections with the theory of linear groups. Then, we will discuss the equivalence of a weak form of the Lehmer conjecture and the ``uniform discreteness" of cocompact lattices in semisimple Lie groups (conjectured by Margulis). (Joint work with Lam Pham.)

I will present the dynamical formulation of optimal transport (a.k.a Benamou-Brenier formulation): it consists in writing the optimal transport problem as the minimization of a convex functional under a PDE constraint, and can handle a priori a vast class of cost functions and geometries. It is one of the oldest numerical method to solve the problem, and it is also the basis for a lot of extensions and generalizations of the optimal transport problem.

The optimization problem is then discretized to end up with a finite dimensional convex optimization problem. I will illustrate this method by presenting a discretization when the ground space is a surface. Although much effort has been devoted to solve efficently the discretized problem, the study of convergence under mesh refinement of the solution of the approximate problems has only been tackled recently. I will present an abstract framework guaranteeing convergence under mesh refinement, with no condition on the relative scale of the spacial and temporal mesh sizes, and even if the densities are very singular.

Free information theory is largely concerned with the following question: given a tuple of non-commutative random variables, what regularity properties of the algebra they generate can be inferred from assumptions about their joint distribution? This can include von Neumann algebraic properties, such as factoriality or absence of Cartan subalgebras, and free probabilistic properties, such as a lack of non-commutative rational relations.

After giving some background, I will talk on free Stein dimension, a quantity which measures the ease of defining derivations on a tuple of non-commutative variables which turns out to be a $*$-algebra invariant. I will mention some recent results on its theory, including behaviour in the presence of algebraic relations as well as under direct sum and amplification of algebras. I will also mention some recent attempts to
adapt its utility from polynomial algebras to W*-algebras, and time permitting, some cases where explicit estimates can be found on the Stein dimension of generating tuples of von Neumann algebras. This project is joint work with Brent Nelson.

In this talk, we discuss the construction and invariance of the Gibbs measure for a three-dimensional wave equation with a Hartree-nonlinearity.

In the first part of the talk, we construct the Gibbs measure and examine its properties. We discuss the
mutual singularity of the Gibbs measure and the so-called Gaussian free field. In contrast, the Gibbs
measure for one or two-dimensional wave equations is absolutely continuous with respect to the Gaussian free field.

In the second part of the talk, we discuss the probabilistic well-posedness of the corresponding nonlinear wave equation, which is needed in the proof of invariance. This was the first theorem proving the invariance of a singular Gibbs measure for any dispersive equation.

We provide a brief introduction to the theory of vector bundles and present a few useful and interesting constructions with a geometric flavor.

Stokes matrices (i.e. unipotent upper triangular matrices) and their nonlinear braid group actions arise naturally in a number of geometric and algebraic contexts. Integral Stokes matrices are often of particular interest, motivating their reduction theory. After reviewing classical work of Markoff treating the case of 3-by-3 matrices, we describe joint work with Yu-Wei Fan for the 4-by-4 case by establishing an exceptional connection to SL2--character varieties of surfaces. This will also serve as an opportunity to present our recent work on effective finite generation of integral points on the latter moduli spaces. Time permitting, we finish by presenting new results (and problems) for Stokes matrices of larger dimension.

Quantum graphs are an operator space generalization of classical graphs that have appeared in different branches of mathematics including operator systems theory, non-commutative topology and quantum information theory. In this talk, I will review the different perspectives to quantum graphs and introduce a chromatic number for quantum graphs using a non-local game with quantum inputs and classical outputs. I will then show that many spectral lower bounds for chromatic numbers in the classical case (such as Hoffman’s bound) also hold in the setting of quantum graphs. This is achieved using an algebraic formulation of quantum graph coloring and tools from linear algebra.

In this talk, we present an algorithm created for Challenge 2 of the Grid Optimization Competition from ARPA-E (Advanced Research Projects Agency--Energy). The competition considered a security constrained optimal power flow (SCOPF) problem whose solution determines optimal dispatch and control settings for power generation and grid control equipment and maximizes the market surplus associated with the operation of the grid, subject to pre- and post-contingency constraints. We will discuss the practical issues associated with the challenge and describe the approach and heuristics established to enhance performance of the algorithm submitted to the competition. Results from the competition will also be presented.

This is joint work with Frank E. Curtis (Lehigh University), Daniel Molzahn (Georgia Tech), Andreas W\"{a}chter and Ermin Wei (Northwestern University)."

We will begin by discussing the multisymplectic structure associated to Hamiltonian PDEs as a generalization of the symplectic structure associated to Hamilton's equations in classical mechanics. Subsequently, we will turn to the question of how to computationally model such Hamiltonian PDEs while preserving the multisymplectic structure at the discrete level. This will lead us to the notion of a multisymplectic integrator. The analogue of these integrators in Hamiltonian mechanics are known as symplectic integrators, which are extremely well-studied and have proven to provide extremely robust and physically faithful simulations of mechanical systems. Multisymplectic integration, on the other hand, is still in its relative infancy.

After establishing the necessary background, we will introduce our construction of variational integrators for Hamiltonian PDEs which automatically yield multisymplectic integrators. This construction gives a systematic framework for constructing such multisymplectic integrators, based on the notion of a Type II generating functional. As an application of our framework, we will derive the class of multisymplectic partitioned Runge--Kutta methods and provide a numerical example with the family of sine--Gordon soliton solutions. This is joint work with Prof. Melvin Leok.

Building on work of Stanley and Bj\"{o}rner--Wachs

In this talk I will introduce Hamilton-Jacobi (HJ) Reachability Analysis for dynamic systems. HJ reachability uses level set methods to compute the set of initial conditions from which a dynamical system is guaranteed to reach its goal and/or avoid unsafe regions despite worst-case conditions. I will discuss challenges, recent advances, and applications.

The dynamics of straight-line flows on compact translation surfaces (surfaces formed by gluing Euclidean polygons edge-to- edge via translations) has been widely studied due to connections to polygonal billiards and Teichmüller theory. However, much less is known regarding straight-line flows on non-compact infinite translation surfaces. In this talk we will review work on straight line flows on infinite translation surfaces and consider such a flow on the Mucube – an infinite $\mathbb{Z}^3$ periodic half-translation square-tiled surface – first discovered by Coxeter and Petrie and more recently studied by Athreya-Lee. We will give a complete characterization of the periodic directions for the straight-line flow on the Mucube – in terms of a subgroup of $\mathrm{SL}_2 \mathbb{Z}$. We will use the latter characterization to obtain the group of derivatives of affine diffeomorphisms of the Mucube. This is joint work (in progress) with Andre P. Oliveira, Felipe A. Ramírez and Chandrika Sadanand.

Let $X$ be a perfectoid space with tilt $X^\flat$. We build a natural map $\theta:\mathrm{Pic} X^\flat\to\lim\mathrm{Pic} X$ where the (inverse) limit is taken over the $p$-power map, and show that $\theta$ is an isomorphism if $R = \Gamma(X,\mathcal{O}_X)$ is a perfectoid ring.

As a consequence we obtain a characterization of when the Picard groups of $X$ and $X^\flat$ agree in terms of the $p$-divisibility of $\mathrm{Pic} X$. The main technical ingredient is the vanishing of higher derived limits of the unit group $R^*$, whence the main result follows from the Grothendieck spectral sequence.

In this paper, we study supervised learning tasks on the space of probability measures. We approach this problem by embedding the space of probability measures into $L^2$ spaces using the optimal transport framework. In the embedding spaces, regular machine learning techniques are used to achieve linear separability. This idea has proved successful in applications and when the classes to be separated are generated by shifts and scalings of a fixed measure. This paper extends the class of elementary transformations suitable for the framework to families of shearings, describing conditions under which two classes of sheared distributions can be linearly separated. We furthermore give necessary bounds on the transformations to achieve a pre-specified separation level, and show how multiple embeddings can be used to allow for larger families of transformations. We demonstrate our results on image classification tasks.

Based on joint work with Caroline Moosmueller, Harish Kannan, and Alex Cloninger.

In the Wild West of geometry and groups, some familiar objects feel like home: Euclidean spaces, hyperbolic geometry, and more generally all the geometries described by semisimple Lie groups or similar matrix groups over local fields.

Harmonic analysis and operator algebras have their own wild seas, but again with a small safe haven: the ``Type I", home of the commutative world, compact groups, generalizations of Fourier analysis. In a precise sense, objects that can be ``described".

I will present a connection between these two worlds and show how it leads to previously unexpected classification results.

I will explain recent results on modular, geometrically meaningful compactifications of moduli spaces of K3 surfaces, most of which are joint with Philip Engel. A key notion is that of a recognizable divisor: a canonical choice of a divisor in a multiple of the polarization that can be canonically extended to any Kulikov degeneration. For a moduli of lattice-polarized K3s with a recognizable divisor we construct a canonical stable slc pair (KSBA) compactification and prove that it is semi toroidal. We prove that the rational curve divisor is recognizable, and give many other examples.

Various notions of metric approximation for countable groups have been introduced and studied in the last decade, with sofic and hyperlinear approximations being two notable examples among them. The class of linear sofic groups was introduced by Glebsky and Rivera and was subsequently studied by Arzhantseva and Paunescu. This mode of approximation uses the general linear group over, say, the field of complex numbers as model groups, equipped with the distance defined using the normalized rank. Among their other interesting results, Arzhantseva and Paunescu prove that every linear sofic group is 1/4-linear sofic, where the constant 1/4 quantifies how well non-identity elements can be separated from the identity matrix. In this talk, which is based on joint work with Maryam Mohammadi Yekta, we will address the question of optimality of the constant 1/4 and report on some progress in this direction.

In this talk, I will discuss radial and one-dimensional symmetry properties for the stationary incompressible Euler equations in dimension 2 and some related semilinear elliptic equations. I will show that a steady flow of an ideal incompressible fluid with no stagnation point and tangential boundary conditions in an annulus is a circular flow. The same conclusion holds in complements of disks as well as in punctured disks and in the punctured plane, with some suitable conditions at infinity or at the origin. If possible, I will also discuss the case of parallel flows in two-dimensional strips, in the half-plane and in the whole plane. The proofs are based on the study of the geometric properties of the streamlines of the flow and on radial and one-dimensional symmetry results for the solutions of some elliptic equations satisfied by the stream function. The talk is based on joint works with N. Nadirashvili.

For a function $m$ on the real line, its Fourier multiplier $T_m$ is the operator which acts on a function $f$ by first multiplying the Fourier transform of $f$ by $m$, and then taking the inverse Fourier transform of the product. These are well-studied objects in classical harmonic analysis. Of particular interest is when the Fourier multiplier defines a bounded operator on $L_p$. Fourier multipliers can be generalized to arbitrary locally compact groups. If the group is non-abelian, the $L_p$ spaces involved are now the non-commutative $L_p$ spaces associated with the group von Neumann algebra. Fourier multipliers also have a natural extension to the multilinear setting. However, their behaviour can differ markedly from the linear case, and there is much that is unknown even about multilinear Fourier multipliers on the reals.

One question of interest is this: If $m$ is a function on a group $G$ which defines a bounded $L_p$ multiplier, is the restriction of m to a subgroup $H$ also the symbol of a bounded $L_p$ multiplier on $H$? De Leeuw proved that the answer is yes, when $G$ is $\mathbb{R}^n$. This was extended to the commutative case by Saeki and to the non-commutative case (provided the group $G$ is sufficiently nice) by Caspers, Parcet, Perrin and Ricard. In this talk, I will show how to extend these De Leeuw type theorems to multilinear Fourier multipliers on non-commutative groups. This is part of joint work with Martijn Caspers, Bas Janssens and Lukas Miaskiwskyi.

Motivated by the study of greedy algorithms for graph coloring, we introduce a new graph parameter, which we call weak degeneracy. This notion formalizes a particularly simple way of ``saving" colors while coloring a graph greedily. It turns out that many upper bounds on chromatic numbers follow from corresponding bounds on weak degeneracy. In this talk I will survey some of these bounds as well as state a number of open problems. This is joint work with Eugene Lee.

This is joint work with S. Dinew (Krakow). We investigate the class of compact complex Hermitian-symplectic manifolds $X$. For each Hermitian-symplectic metric $\omega$ on $X$, we introduce a functional acting on the metrics in a certain cohomology class of $\omega$ and prove that its critical points (if any) must be K\"ahler when X is 3-dimensional.

The development of free probability theory has drawn much inspiration from its deep and far reaching analogy with classical probability theory. The same holds for its operator-valued extension, where the fundamental notion of free independence is generalized to free independence with amalgamation as a kind of conditional version of the former. Its development naturally led to operator-valued free analogues of key and fundamental limiting theorems such as the operator-valued free Central Limit Theorem due to Voiculescu and results about the asymptotic behaviour of distributions of matrices with operator-valued entries.

In this talk, we show Berry-Esseen bounds for such limit theorems. The estimates are on the level of operator-valued Cauchy transforms and the L{\'e}vy distance. We address also the multivariate setting for which we consider linear matrix pencils and noncommutative polynomials as test functions. The estimates are in terms of operator-valued moments and yield the first quantitative bounds on the L{\'e}vy distance for the operator-valued free CLT. This also yields quantitative estimates on joint noncommutative distributions of operator-valued matrices having a general covariance profile.

This is a joint work with Tobias Mai.

I will discuss some simplified models for the shape of liquid droplets on rough solid surfaces, especially Bernoulli-type free boundary problems. In these models small scale roughness leads to large scale non-uniqueness, hysteresis, and anisotropies. In technical terms we need to understand laminating/foliating families of plane-like solutions, this is related to ideas of Aubry-Mather theory, but, unlike most results in that area, we need to consider local (but not global) energy minimizers.

In geometry, one often starts with a base space (e.g. a manifold, or a variety, etc) and is interested in constructing global objects over the base. One of the ways to construct these global objects is to glue them over the base from simpler local data.

For example, one builds global vector bundles by first describing them locally as products and then gluing them via isomorphisms satisfying certain cocycle conditions. Said more abstractly, one tries to recover the `category' of vector bundles on the base by looking at the `category' of vector bundles on `small open' sets on the base. The fact that one can do this is succinctly summarised by saying that the `category' of vector bundles satisfies $\emph{descent}$ on open sets over topological spaces. More provocatively, one says that the `category' of vector bundles is a $\emph{stack}$ over the base. This abstraction is not just decorative—the analogous statement fails for isomorphism classes of vector bundles!

In this talk I will discuss this and other ideas which go under the collective name of descent. The first half will be an informal introduction to descent with minimal prerequisites. The second half will discuss counterparts of these ideas in the context of algebraic geometry and commutative algebra, although again with minimal prerequisites.

Rigidity phenomena in homogeneous dynamics have been extensively studied over the past few decades with several striking results and applications.
In this talk, we will give an overview of recent activities related to quantitative aspect of the analysis in this context; we will also highlight some applications.

Recently, Taelman defined a ``class module" associated to any Drinfeld module defined over a function field. In the spirit of Iwasawa theory, we will study the structure of these class modules in certain p-adic towers of fields. Using the Equivariant Tamagawa Number Formula for Drinfeld modules, we will propose an Iwasawa main conjecture for these class modules.

In this talk, I will discuss ongoing work to develop a method for proving a shrinking target property on primitive square-tiled surfaces that comes from the action of a subgroup $G$ of its Veech group. Our main tool is the construction of a Fourier-like transform which we can use to relate the $L^2$-norm of the Koopman operator induced by $G$ to the $L^2$-norm of a Markov operator related to a random walk on $G$.

Let $X$ be a smooth, affine, geometrically connected curve
over a finite field of characteristic $p > 2$. Let $\rho:\pi_1(X) \to
\mathbb{C}^\times$ be a character of finite order $p^n$. If $\rho\neq
1$, then the Artin $L$-function $L(\rho,s)$ is a polynomial, and a
theorem of Kramer-Miller states that its $p$-adic Newton polygon
$\mathrm{NP}(\rho)$ is bounded below by a certain Hodge polygon
$\mathrm{HP}(\rho)$ which is defined in terms of local monodromy
invariants. In this talk we discuss the interaction between the polygons
$\mathrm{NP}(\rho)$ and $\mathrm{HP}(\rho)$. Our main result states that
if $X$ is ordinary, then $\mathrm{NP}(\rho)$ and $\mathrm{HP}(\rho)$
share a vertex if and only if there is a corresponding vertex shared by
certain ``local" Newton and Hodge polygons associated to each ramified
point of $\rho$. As an application, we give a local criterion that is
necessary and sufficient for $\mathrm{NP}(\rho)$ and $\mathrm{HP}(\rho)$
to coincide. This is joint work with Joe Kramer-Miller.

The lengths of closed geodesics on a hyperbolic manifold are determined by the traces of its fundamental group. When the latter is a Zariski dense subgroup of an arithmetic group, the trace set is contained in the ring of integers of a number field, and may have some local obstructions. We say that the surface's length set ``saturates" (resp. ``asymptotically saturates") if every (resp. almost every) sufficiently large admissible trace appears. In joint work with Xin Zhang, we prove the first instance of asymptotic length saturation for punctured covers of the modular surface, in the full range of critical exponent exceeding one-half (below which saturation is impossible).

Let $A$ be a (unital) algebra over the complex numbers and $a \in A$. At a very high level, the term \textit{functional calculus} refers to constructions of the form, ``Take some collection $\mathcal{F}$ of scalar functions, and, for all $f \in \mathcal{F}$, define $f(a) \in A$ in a sensible way." One can always take $\mathcal{F} = \mathbb{C}[t]$ with the obvious definition of $p(a) \in A$ for $p \in \mathbb{C}[t]$, but this is pretty much the end of the construction when $A$ has no additional structure. When $A$ has some analytic structure -- as is frequently the case in functional analysis and operator algebras -- one can construct functional calculi for much larger classes of functions. In this slightly experimental talk, it is possible that I will discuss functional calculus in Banach algebras, $C^*$-algebras, and/or von Neumann algebras. The talk will be in a ``choose your own adventure" style, so the audience will decide the exact trajectory of the talk democratically. (I offer my thanks to Max Johnson for the idea to give this kind of talk.) Prerequisites will be minimal: passing familiarity with norms, inner products, bounded/continuous linear maps, completeness, etc. should suffice.

The theory of canonical divisors on curves has witnessed an explosion of interest in recent years, motivated by the recent developments in the study of limits of canonical divisors on nodal curves. Imposing conditions on canonical divisors allows one to construct natural geometric subvarieties of moduli spaces of pointed curves, called strata of canonical divisors. The strata are in fact the projection on moduli spaces of curves of incidence varieties in the projectivized Hodge bundle. I will present a graph formula for the class of the restriction of such incident varieties over the locus of pointed curves with rational tails. The formula is expressed as a linear combination of tautological classes indexed by decorated stable graphs, with coefficients enumerating appropriate weightings of decorated stable graphs. I will conclude with some applications. Joint work with Iulia Gheorghita.

We introduce a new class of permutations, called web permutations. Using these permutations, we provide a combinatorial interpretation for entries of the transition matrix between the Specht and web bases, which answers Rhoades's question. Furthermore, we study enumerative properties of these permutations. This is based on the work with Byung-Hak Hwang and Jihyeug Jang.