The cover time of a graph $G$ is the maximum over vertices $v\in V(G)$ of the expected time for a simple random walk to visit all vertices of $G$, starting at $v$. We will review what we know about this question and then focus on two recent results.

{\bf Dense Graphs:} We consider abritrary graphs $G$ with $n$ vertices and minimum degree at least $\delta n$ where $\delta>0$ is constant. If the conductance of $G$ is sufficiently large then we obtain an asymptotic expression for the cover time $C_G$ of $G$ as the solution to some explicit transcendental equation. Failing this, if the mixing time of a random walk on $G$ is of a lesser magnitude than the cover time, then we can obtain an asymptotic deterministic estimate via a decomposition into a bounded number of dense subgraphs with high conductance. Failing this we give a deterministic asymptotic 2-approximation of $C_G$.

Joint work with Colin Cooper and Wesley Pegden.

{\bf Emerging Giant:} Let $p=\frac{1+\epsilon}{n}$. It is known that if $N=\epsilon^3n\to\infty$ then w.h.p. $G_{n,p}$ has a unique giant largest component. We show that if in additon, $\epsilon=\epsilon(n)\to 0$ then w.h.p. the cover time of $G_{n,p}$ is asymptotic to $n\log^2N$.

Joint work with Wesley Pegden and Tomasz Tkocz.

Classical models of biological populations, for example, Markov branching processes, typically model population size and possibly the distribution of types and/or locations of individuals in the population. The intuition behind these models usually includes ideas about the relationships among the individuals in the population that cannot be directly recovered from the model. This loss of information is even greater if one employs large population approximations such as the diffusion approximations popular in population genetics. ``Lookdown'' constructions provide representations of population models in terms of countable systems of particles in which each particle has a ``type'' which may record both spatial location and genetic type and a ``level'' which incorporates the lookdown structure which in turn captures the genealogy of the population. The original population model can then be viewed as the result of the partial observation of the more complex model. We will exploit ideas from filtering of Markov processes to make the idea of partial observation clear and to justify the lookdown construction.

Resonances are in some sense analogs of discrete eigenvalues for
certain operators with continuous spectrum. Physically they may
correspond to decaying waves.

For Euclidean scattering in odd dimensions, the resonances are points in
the complex plane; in even dimensions, they lie on the logarithmic cover
of the complex plane. For scattering by an obstacle, we consider the
problem of counting the number of resonances in certain regions. In
particular, we show that surprisingly there is a sharp lower bound on a
resonance-counting function in even dimensions for which the analogous
result is not yet known in odd dimensions.

We extend a model for the morphology and dynamics of a crawling eukaryotic cell to describe cells on micro patterned substrates. This model couples cell morphology, adhesion, and cytoskeletal flow in response to active stresses induced by actin and myosin. We propose that protrusive stresses are only generated where the cells adheres, leading to the cell’s effective confinement to the pattern. Simulated cells exhibit a broad range of behaviors, including steady motion, turning, bipedal motion and periodic migration. We further extensively study the turning instability by simplifying the full PDE model into a minimal one. By using the minimal model, we also study the persistent rotational motion (PRM) of small numbers of mammalian cells crawling on micropatterned substrates.

Given an integer vector $a = (a_1,\dots,a_n)$, let $\rho(a)$ be the number of solutions to $a\cdot x=0$, with $x\in \{\pm 1\}^n$. In 1945, Erdos gave a beautiful combinatorial solution to the following problem that was posed by Littlewood and Offord: how large can $\rho(a)$ be if all the entries of $a$ are non-zero?

Following his breakthrough result, several extensions of this problem have been intensively studied by various researchers. In classical works, Erdos-Moser, Sarkozy-Szemeredi, and Halasz obtained better bounds on $\rho(a)$ under additional assumptions on $a$, while Kleitman, Frankl-Furedi, Esseen, Halasz and many others studied generalizations to higher dimensions. In recent years, motivated by deep Freiman-type inverse theorems from additive combinatorics, Tao and Vu brought a new view to this problem by asking for the underlying structural reason for $\rho(a)$ to be large --this is known as the Inverse Littlewood-Offord problem, which is a cornerstone of modern random matrix theory.

In this talk, we will discuss further extensions and improvements for both forward and inverse Littlewood-Offord problems where combinatorial tools and insights have proved to be especially powerful. We also present several applications in (discrete) matrix theory such as: a non-trivial upper bound on the number of Hadamard matrices, an upper bound on the number of $\pm 1$ normal matrices (improving a result of Deneanu and Vu), and a unified approach for counting the number of singular $\pm1$ matrices from various popular models (improving results of Cook, Nguyen and Vershynin).

Let M\_g be the moduli space of smooth curves of genus g. The tautological ring is a subring of the cohomology of M\_g that was introduced by Mumford in the 1980s in analogy with the cohomology of Grassmannians. Work of Faber and Faber-Zagier in the 1990s led to two competing conjectural descriptions of the structure of the tautological ring. After reviewing these conjectures, I will discuss some of the evidence in recent years favoring one conjecture over the other.

Fiber bundles with fiber a surface arise in many areas including hyperbolic geometry, symplectic geometry, and algebraic geometry. Up to isomorphism, a surface bundle is completely determined by its monodromy representation, which is a homomorphism to a mapping class group. This allows one to use algebra to study the topology of surface bundles. Unfortunately, the monodromy representation is typically difficult to 'compute' (e.g. determine its image). In this talk, I will discuss some recent work toward computing monodromy groups for holomorphic surface bundles, including certain examples of Atiyah and Kodaira. This can be applied to the problem of counting the number of ways that certain 4-manifolds fiber
over a surface. This is joint work with Nick Salter.

The Ricci iteration is a discrete analogue of the Ricci flow. Introduced in 2007, it has been studied extensively on Kähler manifolds, providing a new approach to uniformisation. In the talk, we will define the Ricci iteration on compact homogeneous spaces and discuss a number of existence, convergence and relative compactness results. This is largely based on joint work with Timothy Buttsworth (Queensland), Yanir Rubinstein (Maryland) and Wolfgang Ziller (Penn).

Geometric integrator for classic mechanics has provided fruitful results. In this talk, we consider
generalizations to three special settings. One is stiff system which comes from semi-discretization of Hamilton PDE,
traditional exponential integrators are modified to preserve Poisson structure and energy; one is for Lie group,
where configuration space is Lie group, group structure of space is considered to construct variational integrator, in contrast to
constrained mechanics; the final is Control system, we take into account nonobservability analysis of control system, which appears as invariance of special group actions, Kalman Filter is modified based on decomposition of space. Such reduced Filter attains state of
art result.

Complex biological systems involve multiple space and time scales. To get an integrated understanding of these systems involves multiscale modeling, computation and analysis. In this talk, I will discuss two such examples in cell biology and illustrate how to use multiscale methods to explain experimental data. The first example is on chemotaxis of bacterial populations. I will present recent progress on embedding information of single cell dynamics into models of cell population dynamics. I will clarify the scope of validity of the well-known Keller-Segel chemotaxis equation and discuss alternative models when it breaks down. The second example is on the axonal cytoskeleton dynamics in health and disease. I will present a stochastic multiscale model that gave the first mechanistic explanation for the cytoskeleton segregation phenomena observed in many neurodegenerative diseases.

I will present a recent proof of the Multiplicity One Conjecture in Min-max theory. This conjecture was raised by Marques and Neves. It says that in a closed manifold of dimension between 3 and 7 with a bumpy metric, the min-max minimal hypersurfaces associated with the volume spectrum introduced by Gromov, Guth, Marques-Neves are all two-sided and have multiplicity one. As direct corollaries, it implies the generalized Yau's conjecture for such manifolds with positive Ricci curvature, which says that there exist infinitely many pairwise non-isometric minimal hypersurfaces, and the Weighted Morse Index Bound Conjecture by Marques and Neves.

Cathy Pearl is a UC San Diego alumnae and current Head of Conversation Design Outreach at Google. She earned her degrees in Cognitive Science from UC San Diego, and a Computer Science master's from Indiana University. She is the author of the 2016 book, 'Designing Voice User Interfaces,' published by O'Reilly Media and has worked with many innovators in the voice-recognition sphere including Nuance. Her broad background makes her a popular speaker, having done some rocket science, as a software designer for NASA, integrating voice recognition for banks, airlines and healthcare. She's also known for her degital expertise on love and romance, writing a Love Data blog focused on 'data-driven love in our modern world' - even performing analysis of the use of Twitter in dating.

I will describe joint work with Jared Wunsch on propagation of
singularities for some semiclassical Schrodinger equations where the
potential has singularities normal to an interface. Semiclassical
singularities of a given strength propagate across the interface, but only
up to a threshold. This is due to diffracted singularities which are
weaker than the incident singularity by a factor depending on the
regularity of the potential. Time permitting, I will give applications to
logarithmic resonance-free regions in scattering theory.

This talk concerns singular values of M-fold products of i.i.d. right-unitarily invariant N x N random matrix ensembles. As N tends to infinity, the height function of the Lyapunov exponents converges to a deterministic limit by work of Voiculescu and Nica-Speicher for M fixed and by work of Newman and Isopi-Newman for M tending to infinity with N. In this talk, I will show for a variety of ensembles that fluctuations of these height functions about their mean converge to explicit Gaussian fields which are log-correlated for M fixed and have a white noise component for M tending to infinity with N. These ensembles include rectangular Ginibre matrices, truncated Haar-random unitary matrices, and right-unitarily invariant matrices with fixed singular values. I will sketch our technique, which derives a central limit theorem for global fluctuations via certain conditions on the multivariate Bessel generating function, a Laplace-transform-like object associated to the spectral measures of these matrix products. This is joint work with Vadim Gorin.

Classically, general Markov processes were studied through their relationship to operator semigroups. The analytic challenges of operator semigroup theory helped motivate the development of alternative approaches including stochastic equations as introduced by Ito and martingale problems as introduced by Stroock and Varadhan. These approaches have dominated work on Markov processes in the mathematics literature while the Kolmogorov forward equation that characterizes the one dimensional distributions of the process receives much more attention in the physics literature (cf. Fokker-Planck equation, master equation). The talk will include a brief over view of all these approaches paying particular attention to the equivalence of the different approaches in characterizing Markov processes.

Recent developments in numerical optimization show that the augmented Lagrangian method (ALM) is very effective in solving large scale convex semidefinite programming. Due to the possible lack of primal-dual-type error bounds, it was not clear whether the Karush–Kuhn–Tucker (KKT) residuals of the sequence generated by the ALM for solving convex semidefinite programming converge superlinearly. We resolve this issue by establishing the R-superlinear convergence of the KKT residuals generated by the ALM under only a mild dual-type error bound condition, for which neither the primal nor the dual solution is required to be unique.

Deciding nonnegativity of real polynomials is a key question in real algebraic geometry with crucial importance in polynomial optimization. It is well-known that in general this problem is NP-hard, therefore one is interested in finding sufficient conditions (certificates) for nonnegativity, which are easier to check. Since the 19th century, sums of squares (SOS) are a standard certificate for nonnegativity, which can be recognized using semidefinite programming (SDP). This approach, however, has some issues, especially in practice if the optimization problem has many variables or high degree.
In this talk I will introduce sums of nonnegative circuit polynomials (SONC). SONC polynomials are certain sparse polynomials having a special structure in terms of their Newton polytopes and supports and serve as a nonnegativity certificate for real polynomials, which is independent of sums of squares.
Moreover, I will provide an overview about polynomial optimization via SONC polynomials. Similar as SOS correspond to SDP, the new SONC certificates correspond to geometric programming and relative entropy programming. Based on a Positivstellensatz for SONC polynomials we establish a converging hierarchy of efficiently computable lower bounds for constrained optimization problems.
The talk is based on joint work with Sadik Iliman, Adam Kurpisz, and Timo de Wolff.

In this talk, we prove global well-posedness of a system
describing behavior of dilute flexible polymeric fluids. This model is
based on kinetic theory, and a main difficulty for this system is its
multi-scale nature. A new function space, based on moments, is
introduced to address this issue, and this function space allows us to
deal with larger initial data.

This is an introduction to the Birch and Swinnerton-Dyer
Conjecture on L-functions of elliptic curves. The talk is aimed
at graduate and undergraduate students who are strongly encouraged
to attend.

Most of the basic results on martingale problems extend to the setting in which the generator depends on a control. The ``control'' could represent a random environment, or the generator could specify a classical stochastic control problem. The equivalence between the martingale problem and forward equation (obtained by taking expectations of the martingales) provides the tool for extending linear programming methods introduced by Manne in the context of controlled finite Markov chains to general Markov stochastic control problems. The controlled martingale problem can also be applied to the study of constrained Markov processes (e.g., reflecting diffusions), the boundary process being treated as a control. Time permitting: the relationship between the control formulation and viscosity solutions of the corresponding resolvent equation will be discussed. Talk includes joint work with Richard Stockbridge and with Cristina Costantini.

In this talk, we will discuss a particular Hermitian flow on compact or complete non-compact complex manifolds. By using the flow, we will discuss the existence of Kahler-Einstein metric on Hermitian manifolds with quasi-negative bisectional curvature.

MATLAB is generally considered to be the leading software package for scientific computing. In this talk we consider a number of computational examples where MATLAB gives or appears to give wrong answers. These examples are useful to help better understand the inner workings, evolution, limits and tradeoffs of a software package such as MATLAB. The author has collected and developed these examples over the years and has used them in his classes. The examples help students gain greater insight into the fundamentals of matrix computations and also into the basics of finite precision arithmetic and related concepts such as round off error, machine precision, numerical stability, and conditioning.

The Lucas sequences is a sequence of polynomials in $s,t$ defined recursively by $\{0\} = 0, \{1\} = 1,$ and
$\{n\} = s \{n-1\} + t \{n-2\}$ for $n \geq 2$. On specialization of $s$ and $t$ one can recover the Fibonacci numbers, the nonnegative integers, and the $q$-integers $[n]_q$. Given a quantity which is expressed in terms of products and quotients of positive integers, one obtains a Lucas analogue by replacing each factor of $n$ in the expression with $\{n\}$. It is then natural to ask if the resulting rational function is actually a polynomial in $s$ and $t$ and, if so, what it counts. Using lattice paths, we give a combinatorial model for the Lucas analogue of the binomial coefficients. This is joint work with Curtis Bennett, Juan Carrillo, and John Machacek. We then give an algebraic method for proving polynomiality using a connection with cyclotomic polynomials. This part of the talk is joint work with Jordan Tirrell and based on an idea of Richard Stanley. Finally, we also consider Catalan numbers and their relatives, such as those for finite Coxeter groups.

We show how heat kernel bounds can be used to establish lower curvature bounds for the Ricci flow. The method is fairly robust and even can be applied even if initially one only has integral lower curvature bounds.

We use a simple method from harmonic maps theory to investigate singularities
of period mappings from a surface with punctures. More precisely, we derive a harmonic map version
of Schmid’s nilpotent orbit theorem. This is a joint work with J. Jost and K. Zuo.

We consider the problem of minimizing the sum of two convex functions: one is differentiable and relatively smooth with respect to a reference convex function, and the other can be nondifferentiable but simple to optimize. The relatively smooth condition is much weaker than the standard assumption of uniform Lipschitz continuity of the gradients, thus significantly increases the scope of potential applications. We present accelerated Bregman proximal gradient (ABPG) methods that employ the Bregman distance of the reference function as the proximity measure. These methods attain an $O(k^{-\gamma})$ convergence rate in the relatively smooth setting, where $\gamma\in [1, 2]$ is determined by a triangle scaling property of the Bregman distance. We develop adaptive variants of the ABPG method that automatically ensure the best possible rate of convergence and argue that the $O(k^{-2})$ rate is attainable in most cases. We present numerical experiments with three applications: D-optimal experiment design, Poisson linear inverse problem, and relative-entropy nonnegative regression. In all experiments, we obtain numerical certificates showing that these methods do converge with the $O(k^{-2})$ rate. This is joint work with Filip Hanzely and Peter Richtarik.

Nonstandard analysis, a powerful machinery derived from mathematical logic,
has had many applications in probability theory as well as stochastic processes.
Nonstandard analysis allows construction of a single object - a hyperfinite probability
space - which satisfies all the first order logical properties of a finite probability space,
but which can be simultaneously viewed as a measure-theoretical probability space
via the Loeb construction. As a consequence, the hyperfinite/measure duality has
proven to be particularly useful in porting discrete results into their continuous settings.

In this talk, for every general-state-space continuous-time Markov process satisfying appropriate
conditions, we construct a hyperfinite Markov process which has all the basic order logical properties of a finite Markov process to represent it. We show that the mixing time and the hitting time agree with
each other up to some multiplicative constants for discrete-time general-state-space reversible Markov
processes satisfying certain condition. Finally, we show that our result is applicable
to a large class of Gibbs samplers and Metropolis-Hasting algorithms.

In this talk we will discuss a new approach to understanding
eigenfunction concentration. We characterize the features that cause an
eigenfunction to saturate the standard supremum bounds in terms of the
distribution of $L^2$ mass along geodesic tubes emanating from a point. We
also show that the phenomena behind extreme supremum norm growth is
identical to that underlying extreme growth of eigenfunctions when
averaged along submanifolds. Finally, we use these ideas to understand a
variety of measures of concentration including high $L^p$ norms and Weyl
laws; in each case obtaining quantitative improvements over the known
bounds.

Reducing the complexity of system models by averaging fast subsystems has a long history in applied mathematics in general and for stochastic models in particular. The fast components of the model determine an occupation measure, and the averaging argument seeks to replace this occupation measure by a simpler measure. Classically, the simpler measure has been identified as a limit of the occupation measure as some parameter in the model goes to infinity. This averaging argument will be discussed along with a recent approach by Cotter and collaborators that identifies an averaging measure that appears to give a more accurate approximation than the classical limiting argument.

Given an $r$-uniform hypergraph $\mathcal{A} \subset X^{(r)}$, the (lower) shadow of $\mathcal{A}$, denoted $\delta(\mathcal{A})$ is defined as $\delta(\mathcal{A}):= \{ B \in X^{(r-1)} : B \subset A \text{ for some } A \in \mathcal{A} \}$. In this talk, we will explore the classical Kruskal-Katona theorem which gives a lower bound on $|\delta(\mathcal{A})|$ and describe related notions of colex order and compression operators on set families.

Student evaluations of teaching (SET) are widely used in academic personnel decisions as a measure of teaching effectiveness. The way SET are used is statistically unsound--but worse, SET are biased and unreliable. Observational evidence shows that student ratings vary with instructor gender, ethnicity, and attractiveness; with course rigor, mathematical content, and format; and with students' grade expectations. Experiments show that the majority of student responses to some objective questions can be demonstrably false. A recent randomized experiment shows that giving students cookies increases SET scores. Randomized experiments show that SET are negatively associated with objective measures of teaching effectiveness and biased against female instructors by an amount that can cause more effective female instructors to get lower SET than less effective male instructors. Gender bias also affects how students rate objective aspects of teaching. It is not possible to adjust for the bias, because it depends on many factors, including course topic and student gender. Students are uniquely situated to observe some aspects of teaching and students' opinions matter. But for the purposes of evaluating and improving teaching quality, SET are biased, unreliable, and subject to strategic manipulation. Reliance on SET for employment decisions disadvantages protected groups and may violate federal law. For some administrators, risk mitigation might be a more persuasive argument than equity for ending the use of SET in employment decisions: union arbitration and civil litigation over institutional reliance on SET are on the rise. Several major universities in the U.S. and Canada have already de-emphasized, substantially re-worked, or abandoned SET for personnel decisions.

I will consider random matrices in the general linear group GL(N;C) distributed according to a heat kernel measure.
This may also be described as the distribution of Brownian motion in GL(N;C) starting at the identity. Numerically,
the eigenvalues appear to cluster into a certain domain $\Sigma_t$ as $N$ tends to infinity. A natural candidate for
the limiting eigenvalue distribution is the “Brown measure” of the limiting object, which is Biane’s ``free multiplicative Brownian motion.'' I will describe recent work with Driver and Kemp in which we compute this Brown measure. The talk will be self contained and will have lots of pictures.

The coordinate ring $S = \mathbb{C}[x_{i,j}]$ of the space of $m \times n$ matrices carries an action of the group $GL_m \times GL_n$ via row and column operations on the matrix entries. If we consider any $GL_m \times GL_n$-invariant ideal $I$ in $S$, the syzygy modules $\mathrm{Tor}_i(I,\mathbb{C})$ will carry a natural action of $GL_m \times GL_n$. By the BGG correspondence, they also carry an action of $\bigwedge^{\bullet}(\mathbb{C}^m \otimes \mathbb{C}^n)$. It turns out that we can combine these actions together and make them modules over the general linear Lie superalgebra $\mathfrak{gl}(m \mid n)$. We will explain how this works and how it enables us to commute all Betti number of any $GL_m \times GL_n$-invariant ideal $I$. This latter part will involve combinatorics of Dyck paths.

It is well-known that the Ricci flow will generally develop singularities if one flows an arbitrary initial metric. Ancient solutions arise as limits of suitable blow-ups as the time approaches the singular time and thus play a central role in understanding the formation of singularities. By the work of Hamilton, Perelman, Brendle, and many others, ancient solutions are now well-understood in two and three dimensions. In higher dimensions, only a few classification results were obtained and many examples were constructed. In this talk, we show that for any dimension $n \geq 4$, every noncompact rotationally symmetric ancient $kappa$-solution to the Ricci flow with bounded positive curvature operator must be the Bryant soliton, extending a recent result of Brendle to higher dimensions. This is joint work with Yongjia Zhang.

The starting point for this talk comes from population genetics: how should we estimate evolutionarily relevant parameters from DNA sequence data taken from samples of individuals? I will give a brief overview of what we learned, starting from the Ewens Sampling Formula and touching on Approximate Bayesian Computation as an inference method when likelihoods are intractable. To illustrate ABC, I will give an example concerning inference of the number of distinct DNA sequences in a sample, given only information about the frequency of point mutations in the samples. This example provides an introduction to inference from typical cancer sequencing data, in which individuals are replaced by cells and in which typically we do not know which mutations occur in which cells. I will give a brief overview of what cancer evolution is about, the sort of statistical and computational problems it poses, and where we are in addressing some of them. Time permitting, I will describe some novel experimental methods we are developing to understand the 3D structure of tumors, paving the way for some challenging inferential problems that will require engagement from data scientists and others.

The classical Tannaka reconstruction theorem allows one to recover a compact group $G$ (up to isomorphism) from the monoidal category of finite dimensional representations of $G$ over $\mathbb{C}$, $\text{Rep}_{\mathbb{C}}(G)$, as the tensor preserving automorphisms of the forgetful functor $\text{Rep}_{\mathbb{C}}(G) \longrightarrow \text{Vec}_{\mathbb{C}}$.

Now let $G$ be a profinite group, $K$ a finite extension of $\mathbb{Q}_p$ and $\text{Ban}_G(K)$ the category of $K$-Banach space representations (of $G$). $\text{Ban}_G(K)$ can be equipped with a (completed) tensor product $(-)\hat\otimes_K(-)$ and has a forgetful functor $\omega : \text{Ban}_G(K) \longrightarrow \text{Ban}(K)$.

Using an anti-equivalence of categories between $\text{Ban}_G(K)$ and the category of Iwasawa $G$-modules due to Schneider and Teitelbaum, we prove that a profinite group $G$ can be recovered from $\text{Ban}_G(K)$, in particular $G \cong \text{Aut}^\otimes(\omega)$.

Mathematics can help Art Historians and Art Conservators in studying and understanding art works, their manufacture process and their state of conservation. The presentation will review several instances of such collaborations in the last decade or so. Some of them led (and are still leading) to interesting new challenges in signal and image analysis. In other applications we can virtually rejuvenate art works, bringing a different understanding and experience of the art to museum visitors as well as to experts.

Let G be a countable, discrete, group and f an element of the integral group ring over G. It is well known how to associate to f an action of G on a compact, metrizable, abelian group. It turns out to be particularly interested to consider those f with an $\ell^{2}$-inveres: i.e. a vector $\xi\in \ell^{2}(G)$ so that $f*\xi=\delta_{1}$. Many nice ergodic theoretic properties of the corresponding action have been established in this context. I will give certain examples of f,G for which we can say that this action is a quotient of a Bernoulli shift. When G is amenable, this implies that it *is* a Bernoulli shift.

Standard methods for approximating the solution of time-dependent PDEs typically produce sequential time-stepping algorithms, which are not optimally efficient on today's highly-parallel supercomputers. Spacetime finite element methods have been developed over the last few years as an alternative approach which can harness the massive parallelism of modern computing platforms. In this talk, I will give an overview of what spacetime finite element methods are and how they differ from traditional methods. I will then discuss some a priori error estimates for these methods, as well as strides towards a posteriori error estimators which are reliable and efficient.

For integers $k\geq 1$ and $n\geq 2k+1$, the \emph{Kneser graph} $K(n,k)$ is the graph whose vertices are the $k$-element subsets of $\{1,\ldots,n\}$ and whose edges connect pairs of subsets that are disjoint. The Kneser graphs of the form $K(2k+1,k)$ are also known as the \emph{odd graphs}. We settle an old problem due to Meredith, Lloyd, and Biggs from the 1970s, proving that for every $k\geq 3$, the odd graph $K(2k+1,k)$ has a Hamilton cycle. The proof is based on a reduction of the Hamiltonicity problem in the odd graph to the problem of finding a spanning tree in a suitably defined hypergraph on Dyck words. As a byproduct, we obtain a new proof of the so-called middle levels conjecture. This is joint work with Torsten Mütze and Jerri Nummenpalo.

From Helmholtz to vaping hipsters, the dynamics of vortex filaments, i.e. fluids with vorticity concentrated along a smooth curve, has been a topic of significant interest in fluid dynamics. The global well-posedness of vortex filaments with small circulation follows from the theory of mild solutions of the 3d Navier-Stokes equations at critical regularity. However, for filaments with large circulation these results no longer apply. In this talk we discuss a proof of well-posedness (in a suitable sense) for vortex filaments of arbitrary circulation. Besides their physical interest, these results are the first to give well-posedness in a neighborhood of large self-similar solutions of the 3d Navier-Stokes without additional symmetry assumptions. This is joint work with Jacob Bedrossian and Pierre Germain.

Let $A$ denote a non-constant ordinary abelian surface over a
global function field (of characteristic $p > 2$) with good reduction
everywhere. Suppose that $A$ does not have real multiplication by any real
quadratic field with discriminant a multiple of $p$. Then we prove that
there are infinitely many places modulo which $A$ is isogenous to the
product of two elliptic curves. This is joint work with Davesh Maulik and
Yunqing Tang.

In this talk, we discuss a construction of the discretization of classical field theories, within the Lagrangian and Hamiltonian frameworks, which preserve the various underlying structures inherent to the physical theories. Preservation of structure under discretization is desirable as it ensures similar behavior between the discretized field dynamics and the actual field dynamics, and often provides computational benefits such as long-term stability and reduction of numerical artifacts. We present a Discrete Lagrangian and Discrete Hamiltonian approach to structure-preserving discretization of field theories. As a motivating example, we apply these methods, in conjunction with discretization spaces from the Finite Element Exterior Calculus, to construct a discretization of classical Yang-Mills theories arising in particle physics. As a simple numerical example, we discretize electromagnetism coupled to particle-in-cell plasma dynamics. To conclude, we briefly discuss
future directions for research, including group-equivariant interpolation applied to the discretization of gauge theories, connections with the discretization of quantum field theories, and methods for studying the (generally nonlinear) dynamics of the discretized fields.

In a joint work with D. Hoff and A. Ioana, we have discovered the following product rigidity phenomenon: if $\Gamma$ is an icc group measure equivalent
to a product of non-elementary hyperbolic groups, then any tensor product decomposition of the II$_1$ factor $L(\Gamma)$ arises only from the canonical
direct product decomposition of $\Gamma$. Subsequently, I. Chifan, R. de Santiago and W. Sucpikarnin classified all the tensor product decompositions
for group von Neumann algebras arising from a large class of amalgamated free products. In this talk we will give an overview of these results and discuss
about a similar rigidity phenomenon that appears in the context of von Neumann algebras arising from actions. More precisely, we prove that if $\Gamma$
is a product of certain groups and $\Gamma\curvearrowright (X,\mu)$ is an arbitrary free ergodic measure preserving action, then we show that any tensor
product decomposition of the II$_1$ factor $L^\infty(X)\rtimes\Gamma$ arises only from the canonical direct product decomposition of the underlying
action $\Gamma\curvearrowright X.$

We consider a certain class of finite-dimensional Gorenstein algebras associated to matroids. We show the Lefschetz property in the case where the matroid corresponds to a modular geometric lattice. Our result implies that the modular geometric lattice has the Sperner property. We also discuss the Grobner fan of the defining ideal of our Gorenstein algebra.

We'll discuss an extension of the work of Kudla-Millson on the modularity of special cycles on a non-compact Shimura variety associated to U(n,1) over a split CM field. The volume of their intersections with a diagonally embedded Shimura subvariety is related to Fourier coefficients of a Hilbert modular form coming from the restriction of an Eisenstein series on U(n,n). The main new idea is an application of the regularized Siegel-Weil formula of Gan-Qiu-Takeda.

In the 1970s, C. Fefferman imitated a program of constructing geometric invariants of bounded complex domains by using the canonical Einstein-Kähler metric on it. The program has been generalized to the construction of conformal invariants via complete Einstein metric with prescribed conformal structure on the boundary at infinity.

Later, in 1997, J. Maldacena applied this picture to theoretical physics; it is now known as AdS/CFT correspondence and soon become a very active area of research. Then ideas from physics were imported to Fefferman’s original program on complex domains. In this talk, I will explain some of global invariants of strictly pseudoconvex domains recently obtained in this program, including, renormalized volume and Q-prime curvature.

The algorithmic refinement of triangular meshes is an important component in numerical simulation codes. Newest vertex bisection is one of the most popular methods for geometrically stable local refinement. Its complexity analysis, however, is a fairly intricate recent result
and many combinatorial aspects of this method are not yet fully understood. In this talk, we access newest vertex bisection from the perspective of theoretical computer science. A new result is the amortized complexity analysis over generalized triangulations. An immediate application is the convergence and complexity analysis of adaptive finite element methods over embedded surfaces and singular surfaces. This is joint work with Michael Holst and Zhao Lyu.

The study of group actions on probability measure spaces plays a central role in modern mathematics. The (generalized) Neshveyev-Stormer conjecture states that the group action on a probability measure space can be completely understood by studying the inclusion of the group von Neumann algebra inside the group measure space construction. In my talk I shall show that the Neshveyev-Stormer conjecture is true for a large class of actions. This talk is based on a joint work with Ionut Chifan.

A hypersurface in a compact manifold $M$ with boundary is called a free boundary minimal hypersurface (FBMH) if it is minimal and meets the boundary of $M$ orthogonally. Such hypersurfaces arise naturally as critical points of the area functional in $M$. If we do not assume any boundary convexity of $M$, then the FBMH may be improper, i.e., the interior of the FBMH may touch $\partial M$. We will discuss the compactness and existence results for FBMHs in this general setting.

We propose a novel framework for constructing and computing various stable network observables. Our approach is based on sampling a random homomorphism from a small motif of choice into a given network. Integrals of the law of the random homomorphism induces various network observables, which include well-known quantities such as homomorphism density and average clustering coefficient. We show that these network observables are stable with respect to renormalized cut distance between networks. For their efficient computation, we also propose two Markov chain Monte Carlo algorithms and analyze their convergence and mixing times. We demonstrate how our techniques can be applied to network data analysis, especially for hypothesis testing and hierarchical clustering, through analyzing both synthetic and real world network data.

Semi-supervised classification refers to learning a function that assigns classes to input data using two sets of observations, one where the input and the associated class is recorded and another where only the inputs are observed. This is a very canonical problem in machine learning with strong links to different approaches in Statistics and Mathematics, e.g. probit regression, spectral clustering or the Ginzburg-Landau classifier. In several applications quantifying the uncertainty associated with classification is as important as the classification itself. In semi-supervised learning uncertainty quantification can be used to improve classificationby active-learning, which amounts to manually classifying the subjects for which classification is most uncertain, and then relearning the classification function; this is also known as human-in-the-loop. In the talk I present a recent framework that connects the different approaches to classification and comes automatically with uncertainty quantification. It is based on Gaussian process classification where the covariance operator of the Gaussian process is constructed using information from the graph Laplacian. The plan of the talk is to provide an accessible overview of the key ideas in this paradigm.

In this talk we will discuss an arithmetic analogue of the
gonality of a curve over a number field: the smallest positive integer e
such that the points of residue degree bounded by e are infinite. By work
of Faltings, Harris--Silverman and Abramovich--Harris, it is
well-understood when this invariant is 1, 2, or 3; by work of
Debarre--Fahlaoui these criteria do not generalize to e at least 4. We
will study this invariant using the auxiliary geometry of a surface
containing the curve and devote particular attention to scenarios under
which we can guarantee that this invariant is actually equal to the
gonality. This is joint work with Geoffrey Smith.

Random matrices now play a role in many areas of theoretical, applied, and computational mathematics. Therefore, it is desirable to have tools for studying random matrices that are flexible, easy to use, and powerful. Over the last fifteen years, researchers have developed a remarkable family of results, called matrix concentration inequalities, that balance these criteria. This talk offers an invitation to the field of matrix concentration inequalities and their applications. This talk is designed for a general mathematical audience.

An $(n,k,\ell)$-design is a a family of $k$-sets of $[n]$ such that every $\ell$-set is covered precisely once. The problem of determining whether or not there exists a design for a given set of parameters is a classical and difficult question in combinatorics. We ask a variant of this problem. Namely, given $k,\ell$, can one find a family of $k$-sets of $[n]$ covering every $\ell$-set \textit{at least} once that has ``approximately'' as many sets as an $(n,k,\ell)$-design would have?

In this talk we will solve the above problem using the technique known as the R$\ddot{\text{o}}$
dl nibble. As time permits we will also discuss other problems in design theory, as well as other applications of the R$\ddot{\text{o}}$dl nibble technique.

Polynomial optimization is a special case of dc (Difference of Convex functions) programming, however representing a multivariate polynomial into a dc function is a difficult task. We propose some new results on dc programming formulations for polynomial optimization. We are interested in polynomial decomposition techniques for representing any multivariate polynomial into difference-of-sums-of-squares (DSOS) and difference-of-convex-sums-of-squares (DCSOS) polynomials. We firstly prove that the set of DSOS and DCSOS polynomials are vector spaces and equivalent to the set of real valued polynomials. We also show that the problem of finding DSOS and DCSOS decompositions are equivalent to semidefinite programs (SDPs). Then, we focus on establishing several practical algorithms for DSOS and DCSOS decompositions without solving SDPs. Some examples illustrate how to use our methods.

Following Smirnov’s proof of Cardy’s formula and Schramm’s discovery of SLE, a thorough understanding of the scaling limit of critical percolation on the regular triangular lattice the has been achieved. Smirnorv’s proof in fact gives a discrete approximation of the conformal embedding which we call the Cardy embedding. In this talk I will present a joint project with Nina Holden where we show that the uniform triangulation under the Cardy embedding converges to the Brownian disk under the conformal embedding. Moreover, we prove a quenched scaling limit result for critical percolation on uniform triangulations. Time permitting, I will also explain how this result fits in the the larger picture of random planar maps and Liouville quantum gravity

We consider the family of normal octic fields with Galois group $D_4$, ordered by their discriminant. In forthcoming joint work with Arul Shankar, we verify the strong Malle conjecture for this family of number fields, obtaining the order of growth as well as the constant of proportionality. In this talk, we will discuss and review the combination of techniques from analytic number theory and geometry-of-numbers methods used to prove these results.

We investigate the influence of a stochastically fluctuating step-barrier potential on bimolecular reaction rates by analytical theory and stochastic simulations. We demonstrate that the system exhibits a ``resonant reaction'' behavior with rate enhancement if an appropriately defined fluctuation decay length is of the order of the system size. Importantly, we find that in the proximity of resonance, the standard reciprocal additivity law for diffusion and surface reaction rates is violated due to the dynamical coupling of multiple kinetic processes. Together, these findings may have implications on the correct interpretation of various kinetic reaction problems in complex systems, as, e.g., in biomolecular association or catalysis.

In his seminal work from 1979,
Joseph J. Kohn invented
his theory of multiplier ideal sheaves
connecting a priori estimates for the d-bar problem
with local boundary invariants
constructed in purely algebraic way.

I will explain the origin and motivation of the problem,
and how Kohn's algorithm reduces it
to a problem in local geometry
of the boundary of a domain.

I then present my work with Sung Yeon Kim
based on the technique of jet vanishing orders,
and show how it can be used to
control the effectivity of multipliers in Kohn's algorithm,
subsequently leading to precise a priori estimates.

Buchberger introduced in 1965 the concept of a Groebner basis for a polynomial ideal over a field and gave an algorithm to compute it. Since the 1980s, this concept has been extensively studied and generalized; it has found many applications in diverse areas of mathematics and computer science. The talk will integrate the concepts of a universal Groebner basis which serves as a Groebner basis for all admissible term orderings with a parametric (more popularly called comprehensive) Groebner basis which serves as a Groebner basis for all possible specializations of parameters. This integration defines a mega Groebner basis that works for every admissible ordering as well as for any specialization of parameters. Algorithms for constructing comprehensive Groebner bases, their canonicity, and generalization to universal comprensive Groebner bases will be presented.

In this talk we discuss qualitative behavior of certain solutions to linear and nonlinear dispersive partial differential equations such as Schrodinger and Korteweg-de Vries equations. In particular, we will present results on the fractal dimension of the solution graph and the dependence of solution profile on the algebraic properties of time.

The main problem in phylogenetics is to reconstruct evolutionary relationships between collections of species, typically represented by a phylogenetic tree. In the statistical approach to phylogenetics, a probabilistic model of mutation is used to reconstruct the tree that best explains the data (the data consisting of DNA sequences from homologous genes of the extant species). In algebraic statistics, we interpret these statistical models of evolution as geometric objects in a high-dimensional probability simplex. This connection arises because the functions that parametrize these models are polynomials, and hence we can consider statistical models as algebraic varieties. The goal of the talk is to introduce this connection and explain how the algebraic perspective leads to new theoretical advances in phylogenetics, and also provides new research directions in algebraic geometry. The talk material will be kept at an introductory level, with background on phylogenetics and algebraic geometry.

Bio: Seth Sullivant received his PhD in 2005 from the University of California, Berkeley. After a Junior Fellowship in Harvard's Society of Fellows, he joined the department of mathematics at North Carolina State University in 2008 as an assistant professor. He was promoted to full professor in 2014 and distinguished professor in 2018. Sullivant's work has been honored with a Packard Foundation Fellowship and an NSF CAREER award and he was selected as a Fellow of the American Mathematical Society. He helped to found the SIAM activity group in Algebraic Geometry where he has served as both secretary and chair. Sullivant's current research interests include algebraic statistics, mathematical phylogenetics, applied algebraic geometry, and combinatorics.

Biochemical oscillation is one of the most important way in living systems to track the information of time, or to communicate with population members. A good clock needs to be function accurately in the presence of noise and at the same time respond sensitively to external signals. Low fluctuation and high sensitivity are incompatible in equilibrium systems due to the fluctuation-dissipation theorem (FDT). In biology, biochemical oscillators are fueled by dissipative processes such as ATP hydrolysis, which is inherently nonequilibrium and the FDT is broken. In our recent work, we show that for a single oscillator, the lower bound of oscillation phase fluctuation, and the upper bound of phase sensitivity are determined by the free energy dissipation. Real biological clocks are composed of multiple oscillators and synchronization is necessary to drive their collective dynamics. Inspired by the cyanobacterial circadian clock, we proposed a model of coupled oscillators. We find that synchronization of oscillators cost free energy even though the coupling is conservative. By analytical solving the model, we show that the many-body system goes through a nonequilibrium phase transition driven by energy dissipation.

The Local Langlands program and its variants have lead to the study of smooth, admissible representations of p-adic algebraic groups. The degree to which these are understood depends on the field over which the representations are being taken. Over a field of characteristic p, the usual dual in the category of smooth representations gives less information: in most cases of interest, it is 0! Kohlhaase has defined candidates-the higher duality functors-for a useful replacement. We will go over their properties, and some examples in rank one where they can be computed.

A fundamental tool in graph theory is Szemeredi's regularity lemma which asserts that any dense graph can be partitioned into finitely many parts so that almost all edges are contained in the union of bipartite subgraphs between pairs of the parts and these bipartite subgraphs are random-like under the notion of $\epsilon$-regular.

\medskip

Here, we consider a variation of the regularity lemma for graphs with a nontrivial
clustering coefficient. The clustering coefficient is the ratio of the number triangles and the number of paths of length $2$ in a graph. Note many real-world graphs have large clustering coefficients and such clustering effect is one of the main characteristics of the so-called ``small world phenomenon''.

\medskip

In this talk, We give a regularity lemma for clustering graphs without any restriction on edge density. We also discuss several generalizations of the regularity lemma and mention some related problems.

Given a collection of input rankings provided as permutations $\pi_i : [n] \rightarrow [n]$ for $1 \leq i \leq m$, the rank aggregation problem seeks to find another permutation $\sigma: [n] \rightarrow [n]$ that minimizes $\sum_{i=1}^m K(\sigma, \pi_i)$ where $K$ is the Kendall distance between the two permutations. In this talk, we will discuss motivation for the problem and some existing Markov chain based algorithms along with an investigation of their performance guarantees. Necessary background information will also be provided.

Given a family of graphs $\mathcal{F}$, we define a game called the $\mathcal{F}$-saturation game. In this game, two players Mini and Max alternate adding edges to an initially empty graph on $n$ vertices, with the only constraint being that neither player can add an edge that creates a subgraph that lies in $\mathcal{F}$. The game ends when no more edges can be added to the graph. Mini wishes to end the game as quickly as possible, while Max wishes to prolong the game. We let $\textrm{sat}_g(\mathcal{F};n)$ denote the number of edges that are in the final graph when both players play optimally.

The $\{C_3\}$-saturation game was the first saturation game to be considered, but the order of magnitude of $\textrm{sat}_g(\{C_3\},n)$ remains unknown. We consider a variant of this game, the $\{C_3,C_5\}$-saturationgame, and we show that the saturation number $\textrm{sat}_g(\{C_3,C_5\};n)$ is quadratic. As time permits we will discuss other games involving odd cycles, such as the $\{C_3,C_5,\ldots,C_{2k+1}\}$-saturation game and the $\{C_5,C_7,\ldots\}$ saturation game.

The $\partial\overline\partial$-lemma is an important obstruction to K\''ahlerianity on compact complex manifolds. In this talk we will describe the relations between this property and the cohomology groups that one can define on complex manifolds. These are joint works with Daniele Angella, Tatsuo Suwa and Adriano Tomassini.

Szemeredi's regularity lemma and its variants are among the most powerful tools in combinatorics, with myriad applications in combinatorics, number theory, discrete geometry, and theoretical computer science. This talk will survey some of the most exciting recent developments in this area.

A del Pezzo fibration is one of the natural outputs of the Minimal Model Program for threefolds. At the same time, geometry of an arbitrary del Pezzo fibration can be unsatisfying due to the presence of non-integral fibers and terminal singularities of an arbitrarily large index. In 1996, Corti developed a program of constructing `standard models' of del Pezzo fibrations within a fixed birational equivalence class. Standard models enjoy a variety of desired properties, one of which is that all of their fibers are $\mathbb{Q}$-Gorenstein integral del Pezzo surfaces. Corti proved the existence of standard models for del Pezzo fibrations of degree $d \ge 2$, with the case of $d = 2$ being the most difficult. The case of $d = 1$ remained a conjecture. In 1997, Kollár recast and improved the Corti’s result in degree $d = 3$ using ideas from the Geometric Invariant Theory for cubic surfaces. I will present a generalization of Koll\'ar’s approach in which we develop notions of stability for families of low degree ($d \le 2$) del Pezzo fibrations in terms of their Hilbert points (i.e., low degree equations cutting out del Pezzos). A correct choice of stability and a bit of enumerative geometry then leads to (very good) standard models in the sense of Corti. This is a joint work with Hamid Ahmadinezhad and Igor Krylov.

I will describe my recent works, some joint with M. Disconzi and
J. Luk, on the compressible Euler equations and their relativistic analog.
The starting point is new formulations of the equations exhibiting
miraculous geo-analytic structures, including i) a sharp decomposition of
the flow into geometric wave and transport-div-curl parts, ii) null form
source terms, and iii) structures that allow one to propagate one
additional degree of differentiability (compared to standard estimates)
for the entropy and vorticity. I will then describe a main application:
the study of stable shock formation, without symmetry assumptions, in more
than one spatial dimension. I will emphasize the role that nonlinear
geometric optics plays in the analysis and highlight how the new
formulations allow for its implementation. Finally, I will describe some
important open problems, and I will connect the results to the broader
goal of obtaining a rigorous mathematical theory that models the long-time
behavior of solutions in the presence of shock singularities.

We discuss some notoriously hard combinatorial problems for large classes of graphs and hypergraphs arising in geometric, algebraic, and practical applications. These structures escape the “curse of dimensionality”: they can be embedded in a bounded-dimensional space, or have small VC-dimension, or a short algebraic description. What are the advantages of low dimensionality? I will suggest a few possible answers to this question, and illustrate them on classical examples.

After a brief historical introduction, I will present some recent developments in the theory of minimal surfaces in Euclidean spaces which have been obtained by complex analytic methods. The emphasis will be on results pertaining to the global theory of minimal surfaces including Runge and Mergelyan approximation, the conformal Calabi-Yau problem, properly immersed and embedded minimal surfaces, and a new result on the Gauss map of minimal surfaces.

How neuronal variability impacts neural codes is a central question in systems neuroscience, often with complex and model dependent answers. Most population models are parametric, with tacitly assumed structure of neuronal tuning and population variability. While these models provide key insights, they cannot inform how the physiology and circuit wiring of cortical networks impact information flow. In this work, we study information propagation in spatially ordered neuronal networks. We focus on the effects of feedforward and recurrent projection widths relative to columnar width, as well as attentional modulation. We show that narrower feedforward projection width increases the saturation rate of information. In contrast, the recurrent projection width with spatially balanced excitation and inhibition has small effects on information. Further, we show that attention improves information flow by suppressing the internal dynamics of the recurrent network.

We consider point-to-point directed paths in a random environment on the two-dimensional integer lattice. For a general independent environment under mild assumptions we show that the quenched energy of a typical path satisfies a central limit theorem as the mesh of the lattice goes to zero. The proofs rely on concentration of measure techniques and some combinatorial bounds on families of paths. This is joint work with Christian Gromoll and Leonid Petrov.

We will study an obstacle problem coming from consumer search in a product market. I will discuss several properties concerning the geometry of the free boundary. The difficulty is to determine how the geometry depends on the dimension d. This is a joint work with T. Tony Ke, Wenpin Tang and J. Miguel Villas-Boas.

Accurate RNA structural prediction remains challenging, despite its increasing biomedical importance. Sampling secondary structures from the Gibbs distribution yields a strong signal of high probability base pairs. However, identifying higher order substructures requires further analysis. Profiling (Rogers & Heitsch, NAR, 2014) is a novel method which identifies the most probable combinations of base pairs across the Boltzmann ensemble. This combinatorial approach is straightforward, stable, and clearly separates structural signal from thermodynamic noise.

The behavior of the eigenvalues of large random matrices is generally very predictable, on multiple scales. Macroscopically, results like the semi-circle law describe the overall shape of the eigenvalue distributions. Indeed, for many natural ensembles of random matrices, we can describe in great detail the way the distribution of the eigenvalues converges to some limiting deterministic probability measure. On a microscopic scale, we often see the phenomenon of eigenvalue rigidity, in which individual eigenvalues concentrate strongly at predicted locations. I will describe some general approaches to these phenomena, with many examples: Wigner matrices, Wishart matrices, random unitary matrices, truncations of random unitary matrices, Brownian motion on the unitary group, and others.

Neuronal variability is a reflection of recurrent circuitry and cellular physiology, and its modulation is a reliable signature of cognitive and processing state. A pervasive yet puzzling feature of cortical circuits is that despite their complex wiring, population-wide shared spiking variability is low dimensional with all neurons fluctuating en masse. Previous model cortical networks are at loss to explain this variability, and rather produce either uncorrelated activity, high dimensional correlations, or pathologically network behavior. We show that if the spatial and temporal scales of inhibitory coupling match known physiology then model spiking neurons naturally generate low dimensional shared variability that captures in vivo population recordings along the visual pathway. Further, top-down modulation of inhibitory neurons provides a parsimonious mechanism for how attention modulates population-wide variability both within and between neuronal areas, in agreement with our experimental results. Our theory provides a critical and previously missing mechanistic link between cortical circuit structure and realistic population-wide shared neuronal variability.

Stacking a few layers of 2D materials such as graphene and molybdenum disulfide, for example, opens the possibility of tuning the electronic and optical properties of 2D materials. One of the main issues encountered in the mathematical and computational modeling of 2D materials is that lattice mismatch and rotations between the layers destroys the periodic character of the system.
Basic concepts like mechanical relaxation, electronic density of states, and the Kubo-Greenwood formulas for transport properties will be formulated and analyzed in the incommensurate setting. New computational approaches will be presented and the validity and efficiency of these approximations will be examined from mathematical and numerical analysis perspectives.

A bilevel optimization problem is a sequence of two optimization problems where the constraint region of the upper level problem is determined implicitly by the solution set to the lower level problem. It can be used to model a two-level hierarchical system where the two decision makers have different objectives and make their decisions on different levels of hierarchy. Recently more and more applications including those in machine learning have been modelled as bilevel optimization problems. This talk will discuss issues, challenges and discoveries for solving bilevel optimization problems.

A major theme in the study of von Neumann algebras is to investigate which structural aspects of the group extend to its von Neumann algebra. I present recent progress made by Dan Hoff, Ben Hayes, Thomas Sinclair and myself in the case where the group has positive first $L^2$ Betti number. I will also expand on our analysis of s-malleable deformations and their relation to cocylces which forms the foundation of our work.

Ever tried using Google Maps in a parking garage? It sucks. That's because current navigational systems rely heavily on GPS satellites which can't ``see'' you when you're inside a building or multiple stories below ground. I'll be talking about one way of trying to localize an internet connected device using wireless signal strength. If the engineering aspect of this problem didn't scare you off already then look forward to a brief discussion about how we can use wavelets to address this question!

We will discuss the possible closures of geodesic planes in a hyperbolic 3-manifold M. When M has finite volume Shah and Ratner (independently) showed that a strong rigidity phenomenon holds, and in particular such closures are always properly immersed submanifolds of M with finite area. We show that a similar rigidity phenomenon holds for a class of infinite volume manifolds. This is based on joint works with C. McMullen and H. Oh.

Calmness/metric subregularity for set-valued maps is a powerful stability concept in variational analysis. In this talk we first discuss the concept of calmness/metric subregularity and sufficient conditions for verifying it. Then we introduce a perturbation technique for conducting linear convergence analysis of various first-order algorithms for a class of nonsmooth optimization problems which minimizes the sum of a smooth function and a nonsmooth function by using the proximal gradient method as an example. This new perturbation technique enables us to provide some concrete sufficient conditions for checking linear convergence for very general problems where the nonconvexity may appear in each component of the objective function and leads to some improvement for the linear convergence results even for the convex case.

Wave turbulence theory claims that at very long timescales, and in appropriate limiting regimes, the effective behavior of a nonlinear dispersive PDE on a large domain can be described by a kinetic equation called the ``wave kinetic equation''. This is the wave-analog of Boltzmann's equation for particle collisions. We shall consider the nonlinear Schrodinger equation on a large box with periodic boundary conditions, and explore some of its effective long-time behaviors at time scales that are shorter than the conjectured kinetic time scale, but still long enough to exhibit the onset of the kinetic behavior. (This is joint work with Tristan Buckmaster, Pierre Germain, and Jalal Shatah).

I will speak on recent joint work with Brent Nelson, where we introduce a free probabilistic regularity quantity we call the free Stein information. The free Stein information measures in a certain sense how close a system of variables is to admitting conjugate variables in the sense of Voiculescu. I will discuss some properties of the free Stein information and how it relates to other common regularity conditions.

In his monumental book 'On the Functional Equations satisfied by Eisenstein Series' Langlands proved that the K-finite Eisenstein series, initially defined, convergent and holomorphic in appropriate open tube of the parameter space can be meromorphically
continued to the entire parameter space. The K-finiteness was critical to his proof of the theorem. In this lecture I will show how to use Langlands' theorem to prove the meromorphic continuation for smooth Eisenstein series. These results are valid in the full context of Langlands' theorem but I will only talk about arithmetic groups for which the definitions are easier. (Indeed, Langlands' definition of the groups that that would be studied was only completed at the end of the induction in his notorious chapter 7).

Deep learning is a rapidly developing area of machine learning, which uses artificial neural networks to perform learning tasks. Although mathematical description of neural networks is simple, theoretical explanation of spectacular performance of deep learning remains elusive. Even the most basic questions about remain open. For example, how many different functions can a neural network compute? Jointly with Pierre Baldi (UCI CS) we discovered a general capacity formula for all fully connected boolean networks. The formula predicts, counterintuitively, that shallow networks have greater capacity than deep ones. So, mystery remains.

Szemeredi's theorem states that every set of integers $A$ with positive density contains an arithmetic progression of length $k$ for any $k\ge 3$. The case $k=3$ was originally proven by Roth. In this talk we go through the proof of Roth's theorem, as well as other related ideas such as Salem sets and Gower's norms.

Ever since Yao introduced the communication model in 1979, it has played
a pivotal role in our understanding of lower bounds for a wide variety of
problems in Computer Science. In this talk, I will present the lifting method,
whereby communication lower bounds are obtained by ``lifting'' much simpler
lower bounds. I will present several lifting theorems that we have obtained and
explain what makes the exciting/useful but also difficult to prove. Finally I will
highlight how they have been used to solve several open problems in
circuit complexity, proof complexity, optimization, cryptography, game theory
and privacy.

High-order numerical methods promise higher fidelity and more predictive power when compared with traditional low-order methods. Furthermore, many properties of these methods make them well-suited for modern computer architectures. However, the use of these methods also introduces several new challenges. For example, in the time-dependent setting, high-order spatial discretizations can result in severe time step restrictions, motivating the use of implicit solvers. The resulting systems are large and often ill-conditioned, posing a challenge for traditional solvers. In this talk, I will discuss the development of efficient solvers and preconditioners designed specifically for high-order finite element and discontinuous Galerkin methods. An entropy-stable sparse line-based discretization will be developed to make these methods suitable for use on GPU- and accelerator-based architectures. These methods will then be applied to relevant problems in compressible flow.

The shrinking target problem for a dynamical system tries to answer the question of how fast can a sequence of targets shrink so that a typical orbit will keep hitting them indefinitely. I will describe some new and old results on this problem for flows on homogenous spaces, with various applications to problems in Diophantine approximations.

Monotone systems are of great interest for their numerous applications and close connections to many physical and biological systems. In linear spaces, a local characterisation of monotonicity is provided by differential positivity with respect to a constant cone field, which combines positivity theory with a local analysis of nonlinear dynamics. Since many dynamical systems are naturally defined on nonlinear spaces, it is important to develop the concept on such spaces. The question of how to define monotonicity on a nonlinear manifold is complicated by the absence of a general and well-defined notion of order in such settings. Fortunately, for Lie groups and important examples of homogeneous spaces that are ubiquitous in many problems of engineering and applied mathematics, symmetry provides a way forward. Specifically, the existence of a notion of geometric invariance on such spaces allows for the generation of invariant cone fields, which in turn induce conal orders. We propose differential positivity with respect to invariant cone fields as a natural and powerful generalisation of monotonicity to nonlinear spaces. We illustrate the key concepts with examples from consensus theory on Lie groups and operator theory on the set of positive definite matrices.

Consider a kind of generalized Nash equilibrium problems (GNEPs) whose objective functions are polynomials, and the constraints can be represented by polynomial equalities and inequalities. Gauss-Seidel typed Approach is one kind of natural easy implemented method to solve GNEP. We study some properties for this approach, and give out a computable criterion for Generalized Potential game (GPGs), the condition under which the convergence of this approach could be guaranteed.

The concept of disjointness of dynamical systems (both topological and
measure-theoretic) was introduced by Furstenberg in the 60s and has
since then become a fundamental tool in dynamics. In this talk, I will
discuss disjointness of topological systems of discrete groups. More
precisely, generalizing a theorem of Furstenberg (who proved the result
for the group of integers), we show that for any discrete group $G$, the
Bernoulli shift $2^G$ is disjoint from any minimal dynamical system. This
result, together with techniques of Furstenberg, some tools from the
theory of strongly irreducible subshifts, and Baire category methods,
allows us to answer several open questions in topological dynamics: we
solve the so-called ``Ellis problem'' for discrete groups and characterize
the underlying topological space for the universal minimal flow of
discrete groups. This is joint work with Eli Glasner, Benjamin Weiss,
and Andy Zucker.

The Kronecker coefficients are the notoriously elusive structure constants for the decomposition into irreducibles of the tensor product of irreducible representations of the symmetric group. In recent joint work with Marni Mishna and Mercedes Rosas, we study the piecewise quasipolynomial nature of the Kronecker function using tools from polyhedral geometry. In this talk I will describe this approach, which begins with Jacobi's definition of the Schur function as a quotient of alternants, an idea originally exploited by Rosas in her thesis. I will then illustrate its power by focusing on the first nontrivial case, showing how we derive, for this case, new exact formulas and an upper bound for the Kronecker coefficients (the {\em atomic} Kronecker coefficient) as well as other properties. The polyhedral geometry gives a curious digraph whose nodes are certain monomials in the alternant. An additional advantage of this approach is that asymptotic estimates for dilations can be computed using techniques of analytic combinatorics in several variables.

In 1908, Paul Koebe conjectured that every open connected set in the plane is conformally diffeomorphic to an open connected set whose boundary components are either round circles or points. The Weyl problem, in the hyperbolic setting, asks for isometric embedding of surfaces of curvature at least -1 in to the hyperbolic 3-space. We show that there are close relationships among the Koebe conjecture, the Weyl problem and the work of Alexandrov and Thurston on convex surfaces. This is a joint work with Tianqi Wu.

In this talk, I will focus on Principal Component Analysis (PCA) in the presence of missing data. Under a homogeneous and independent missingness mechanism, we showed that the leading eigenspaces of a Hadamard-reweighted sample covariance matrix achieves the (nearly) minimax optimal rate with a phase transition. If the true leading eigenspaces satisfy an incoherence assumption, we can embrace much more flexible missingness mechanisms: we derived the statistical rate of this reweighted-covariance-based estimator under arbitrary deterministic observation regime. Then we proposed to use this estimator to initialize a tuning-free iterative algorithm called primePCA to further enhance the statistical accuracy. We showed that under the noiseless setting, primePCA achieves exact recovery of the true leading eigenspaces with geometric convergence, provided that the initializer is close to the truth. Simulation study shows that primePCA performs similarly as softImpute with oracle tuning within a wide range of heterogeneity levels of observation probabilities and signal-to-noise ratios.

I will review and present some new results on construction, long time behaviour and coercive inequalities for Markov semigroups on nilpotent Lie groups.

The Hall-magnetohydrodynamics (MHD) system is obtained from the ideal MHD system by incorporating a quadratic second-order correction, called the Hall current term, that takes into account the motion of electrons relative to positive ions. In recent work with Sung-Jin Oh, we investigated the Cauchy problem in the irresistive case. We first study the linearized systems around a special class of stationary magnetic fields with certain symmetries, and obtain ill- and well-posedness results, depending on the profile of the magnetic field. We then pass from linear to nonlinear results: near a non-zero constant magnetic field, the system is well-posed but it is ill-posed (in the strongest sense of Hadamard) near the trivial magnetic field. We are mainly guided by the behavior of bicharacteristics for the principal symbol. The key tools are: dispersive smoothing in the well-posedness case and construction of degenerating wave packets together with a systematic use of a generalization of the energy identity in the ill-posedness case.

Information Geometry is the differential geometric study of the manifold of probability models, and promises to be a unifying geometric framework for investigating statistical inference, information theory, machine learning, etc. Instead of using metric for measuring distances on such manifolds, these applications often use “divergence functions” for measuring proximity of two points (that do not impose symmetry and triangular inequality), for instance Kullback-Leibler divergence, Bregman divergence, f-divergence, etc. Divergence functions are tied to generalized entropy (for instance, Tsallis entropy, Renyi entropy, phi-entropy) and cross-entropy functions widely used in machine learning and information sciences. It turns out that divergence functions enjoy pleasant geometric properties – they induce what is called “statistical structure” on a manifold M: a Riemannian metric g together with a pair of affine connections D, D*, such that D and D* are both Codazzi coupled to g while being conjugate to each other. Divergence functions also induce a natural symplectic structure on the product manifold MxM for which M with statistical structure is a Lagrange submanifold. In joint work with M. Leok, we shown how divergence functions allow us to decouple Hamiltonian and Lagrangian dynamics in geometric mechanics. We recently characterize (para-) holomorphicity of D, D* in the (para-)Hermitian setting, and show that statistical structures can be enhanced to (para-)Hermitian and (para-)Kahler manifolds. The surprisingly rich geometric structures and properties of a statistical manifold open up the intriguing possibility of geometrizing statistical inference, information, and machine learning in string-theoretic languages.

There are several major obstructions that arise in the attempt
to formulate an analog of the Maeda Conjecture in the setting of Drinfeld
modular forms. I will report on an undergoing joint work with G. Boeckle
and P. Graef wherein we identify various Hecke stable filtrations on the
spaces of cuspidal Drinfeld modular forms of full level and a given weight
by identifying them with spaces of positive characteristic valued,
$SL_2(\mathbb{F}_q[\theta])$-invariant, harmonic cocycles on the oriented
edges of the Bruhat-Tits tree for $PGL_2 / \mathbb{F}_q((1/\theta))$ made
possible by an isomorphism of Tietelbaum. I will compare with the
classical situation along the way and provide some of the relevant
background on Drinfeld modular forms in a pre-talk aimed at graduate
students and postdocs.

We discuss some of the recent work on discrete conformal geometry of polyhedral surfaces. The following topics will be addressed: the relationship among discrete conformal geometry, convex surfaces in hyperbolic 3-space, and the Koebe circle domain conjecture, and the convergence of the discrete uniformization metrics to the Ponicare metric. This is a joint work with D. Gu, J. Sun, and T. Wu.

Vinogradov's three prime theorem states that every sufficiently large odd number $N$ can be written in terms of the sum
of three odd primes. We will give a very light sketch of the proof of Vinogradov's three prime theorem and give a comparison to
the previous talk on Roth's theorem to arithmetic progressions.

Finite element methods are numerical methods that approximate solutions to PDEs using functions on a mesh representing the problem domain. Discontinuous-Petrov Galerkin Methods are a class of finite element methods that are aimed at achieving stability of the Petrov-Galerkin finite element approximation through a careful selection of the associated trial and test spaces. In this talk, I will present DPG theorems as they apply to linear problems, and then approaches for those theorems in the case of semi-linear problems. In particular, I will explore a particular case of semilinear problems, that allows for results in the linear case to hold.

A central theme in the theory of von Neumann algebras is to determine all possible tensor product decompositions of a given factor. I will present a recent joint work with Yusuke Isono where we use the rigidity of full factors and a new flip automorphism approach in order to study this problem. Among other things, we show that a separable full factor admits at most countably many tensor product decompositions (up to stable unitary conjugacy). We also establish new primeness and Unique Prime Factorization results for crossed products coming from compact actions of irreducible higher rank lattices (e.g. $SL_n(\mathbb{Z})$ for $n>2$) as well as noncommutative Bernoulli shifts with arbitrary base (not necessarily amenable).

We will discuss some important classical results from Glimm and Thoma about the existence of ``big'' irreducible representations. No prerequisites required beyond curiosity.

In this talk, we explore explicit cross sections to the horocycle and geodesic flows on $\operatorname{SL}(2, \mathbb{R})/G_q$, with $q \geq 3$. Our approach relies on extending properties of the primitive integers $\mathbb{Z}_\text{prim}^2 := \{(a, b) \in \mathbb{Z}^2 \mid \gcd(a, b) = 1\}$ to the discrete orbits $\Lambda_q := G_q (1, 0)^T$ of the linear action of $G_q$ on the plane $\mathbb{R}^2$. We present an algorithm for generating the elements of $\Lambda_q$ that extends the classical Stern-Brocot process, and from that derive another algorithm for generating the elements of $\Lambda_q$ in planar strips in increasing order of slope. We parametrize those two algorithm using what we refer to as the \emph{symmetric $G_q$-Farey map}, and \emph{$G_q$-BCZ map}, and demonstrate that they are the first return maps of the geodesic and horocycle flows resp. on $\operatorname{SL}(2, \mathbb{R})/G_q$ to particular cross sections. Using homogeneous dynamics, we then show how to extend several classical results on the statistics of the Farey fractions, and the symbolic dynamics of the geodesic flow on the modular surface to our setting using the $G_q$-BCZ and symmetric $G_q$-Farey maps. This talk is self-contained, and does not assume any prior knowledge of Hecke triangle groups or homogeneous dynamics.

An important result of X.-J. Wang states that a convex ancient solution to mean curvature flow either sweeps out all of space or lies in a stationary slab (the region between two fixed parallel hyperplanes). We will describe recent results on the construction and classification of convex ancient solutions and convex translating solutions to mean curvature flow which lie in slab regions, highlighting the connection between the two. Work is joint with Theodora Bourni and Giuseppe Tinaglia.

Tverberg's Theorem says that a set with sufficiently many points in $\mathbb{R}^d$ can always be partitioned into $m$ parts so that the $(m-1)$-simplex is the (nerve) intersection pattern of the convex hulls of the parts. In this talk we will talk about intersection patterns and how Tverberg's Theorem is but a special case of a more general situation where other simplicial complexes arise as nerves.

In this talk, we shall discuss recent progress in the global
uniqueness issues for inverse boundary problems for second order elliptic
equations, such as the conductivity and magnetic Schrodinger equations,
with low regularity coefficients. Generally speaking, in an inverse
boundary problem, one wishes to determine the coefficients of a PDE inside
a domain from the knowledge of its solutions along the boundary of the
domain. While ubiquitous in practice, the mathematical analysis of such
problems is quite challenging, and the consideration of the low regularity
setting, motivated by applications, brings additional substantial
difficulties. In this talk, we shall discuss the case of full, as well as
partial, measurements, both for domains in the Euclidean space, as well as
in the more general setting of transversally anisotropic compact
Riemannian manifolds with boundary. Some of the important ingredients in
our approach are semiclassical Carleman estimates with limiting Carleman
weights with an optimal gain of derivatives, precise smoothing estimates,
as well as a construction of Gaussian beam quasimodes in a low regularity
setting. This is joint work with Gunther Uhlmann.

Biomolecules are inherently flexible and their dynamics affects ligand binding. To investigate the internal dynamics of proteins and nucleic acids, we apply all-atom molecular dynamics
simulations. However, atomistic level of detail makes simulations too computationally demanding to describe folding and global motions so reduced representations of molecules are often
applied. These so-called coarse-grained models are sufficient to capture global collective motions on biologically relevant spatial and temporal scales. However, due to reduction of the
degrees of freedom, coarse-grained models require parameterizations of the potential energy function (force field). Moreover, coarse-grained force field parameters are typically not transferable between different molecules and problems.

I will present our efforts to design an automatic parameterization procedure to obtain force fields for reduced models of biomolecules. The procedure for the optimization of potential
energy parameters is based on metaheuristic methods. I will also show examples of applications to dynamics of proteins and nucleic acids.

In this talk, I'll give a brief introduction on the study of the Nevanlinna theory (theory of holomorphic curves) using the stochastic calculus, following the works fo B. Davis, T. K. Carne and A. Atsuji etc.. In particular, I will outline the ideas and compare its method with the classical approach, and outline its advantage on studying the maps on the (general) Kahler manifolds (or even the Rieamnnian manifolds).

This is a joint work with Wanke Yin. We discuss the connection between the type defined by the Lie-Bracket of vector fields, the finite type condition in terms of the trace of Levi form and the order of contact with smooth complex submanifolds. We discuss a partial solution to an old conjecture of Bloom asked about 40 years ago.

Formulated in 1984, Green's Conjecture predicts that one can recognize
the intrinsic complexity of an algebraic curve from the syzygies of its
canonical embedding. Green's Conjecture for a general curve has been
resolved using geometric methods in two landmark papers by Voisin in the
early 00s. I will explain how the theory of Koszul modules provides an
alternative solution to this problem, by relating it via Hermite
reciprocity to the study of the syzygies of the tangent developable
surface to a rational normal curve. Joint work with M. Aprodu, G.
Farkas, S. Papadima, and J. Weyman.

Recent advances in electron microscopy have, for the first time enabled imaging of single cells in 3D at a nanometer length scale resolution. An uncharted frontier for in silico biology is the ability to simulate cellular processes using these observed geometries. However, this will require a system for going from EM images to 3D volume meshes which can be used in finite element simulations. In this paper, we develop an end-to-end pipeline for this task by adapting and extending computer graphics mesh processing and smoothing algorithms. Our workflow makes use of our recently rewritten mesh processing software, GAMer 2, which implements several mesh conditioning algorithms and serves as a platform to connect different pipeline steps. We apply this pipeline to a series of electron micrographs of dendrite morphology explored at three different length scales and show that the resultant meshes are suitable for finite element simulations. Our pipeline, which consists of free
and open-source community driven tools, is a step towards routine physical simulations of biological processes in realistic geometries. We posit that a new frontier at the intersection of computational technologies and single cell biology is now open. Innovations in algorithms to reconstruct and simulate cellular length scale phenomena based on emerging structural data will enable realistic physical models and advance discovery.

A locally compact group $G$ is \emph{Hermitian} if the spectrum $\sigma_{L^1(G)}(f)$ is contained in $\mathbb R$ for every $f=f^*\in L^1(G)$. Examples of Hermitian groups include all abelian locally compact groups. A question from the 1960s asks whether every Hermitian group is amenable. I will speak on the history and recent affirmative solution to this problem.

Given a finite set $X$ of size $n$, we can form a metric space on the power set $\mathcal{P}(X)$ by the metric $d(A,B) = |A \triangle B|$ where $A \triangle B := (A \cap B^c) \cup (A^c \cap B)$. An $\alpha$-well separated set system is a subset $\mathcal{F} \subset \mathcal{P}(X)$ so that for all distinct $A, B \in \mathcal{F}$, we have that $d(A,B) \geq \alpha n$. In this talk, we will focus on the case where $\alpha= \frac{1}{2}$ and use linear algebra techniques to explore bounding the size of an $\alpha$-well separated family. We will also discuss the construction of these large $\alpha$-well separated set systems via Hadamard matrices.

The c-projective metrizability equation is an invariant overdetermined linear geometric PDE on an almost c-projective manifold governing the existence of quasi-Kahler metrics compatible with the c-projective structure. I will show that the degeneracy locus of a solution to the c-projective metrizability equation satisfying a generic condition on its prolonged system is a smoothly embedded submanifold of codimension 1 which inherits a partially-integrable nondegenerate almost CR structure. Phrased differently, this result explicitly links the Levi-form of the boundary CR structure of a c-projectively compact quasi-Kahler manifold satisfying a non-vanishing 'generalized scalar curvature' condition to the interior metric.

For a (connected, finite) graph $G$, we define its genus to be the quantity $g := E - V + 1$, where $E$ is the number of edges of $G$ and $V$ is the number of vertices. While it is not the case that graph homomorphisms preserve this invariant, it is the case that contractions between graphs do. In this talk we will consider the category of all genus $g$ graphs and contractions. More specifically, we consider integral representations of the opposite category, i.e. functors from the opposite category to Abelian groups. Using the combinatorics of graph minors, we will show that representations of this kind satisfy a Noetherian property. As applications of this technical result, we show that configuration spaces of graphs as well as Kazhdan-Lusztig polynomials of graphical matroids must satisfy strong finiteness conditions. This is joint work with Nick Proudfoot.

We prove a rigidity result for solutions to the normalized Ricci flow whose rate of convergence is faster than exponential and discuss a connection to the classification problem for noncompact shrinking solitons.

We provide a new modeling framework and adopt modern optimization tools to attack an important open problem in statistics. In particular, we consider the optimal adaptive trial design problem in personalized medicine. Adaptive enrichment designs involve preplanned rules for modifying enrollment criteria based on accruing data in a randomized trial. We focus on designs where the overall population is partitioned into two predefined subpopulations, e.g., based on a biomarker or risk score measured at baseline for personalized medicine. The goal is to learn which populations benefit from an experimental treatment. Two critical components of adaptive enrichment designs are the decision rule for modifying enrollment, and the multiple testing procedure. We provide a general framework for simultaneously optimizing these components for two-stage, adaptive enrichment designs through Bayesian optimization. We minimize the expected sample size under constraints on power and the familywise Type I error rate. It is computationally infeasible to directly solve this optimization problem due to its nonconvexity and infinite dimensionality. The key to our approach is a novel, discrete representation of this optimization problem as a sparse linear program, which is large-scale but computationally feasible to solve using modern optimization techniques. Applications of our approach produce new, approximately optimal designs. In addition, we shall further discuss several extensions to solve other related statistical problems.

The Patlak-Keller-Segel equations (PKS) are widely applied to model
the chemotaxis phenomena in biology. It is well-known that if the
total mass of the initial cell density is large enough, the PKS
equations exhibit finite time blow-up. In this talk, I present some
recent results on applying additional fluid flows to suppress
chemotactic blow-up in the PKS equations. These are joint works with
Jacob Bedrossian and Eitan Tadmor.

I will present two examples of dynamic cell-membrane processes we have been working on, highly inspired by recent experimental results, and described by combining scaling, mathematical modelling and numerical simulations. i) \underline{Immunological synapse}: The cellular basis for the adaptive immune response during antigen recognition relies on a specialized protein interface known as the immunological synapse. We propose a minimal mathematical model for the dynamics of the immunological synapse that encompass membrane mechanics, hydrodynamics and protein kinetics. Simple scaling laws describe the time and length scales of the self-organizing protein clusters as a function of membrane stiffness, rigidity of the adhesive proteins, and the fluid flow in the synaptic cleft. ii) \underline{Formation of Intraluminal Vesicles}: The endosome is a membrane-bound compartment, which encapsulates cargo as it matures into a multi-vescular body that regulate cell activity as well as enabling communication with surrounding cells. The cargo encapsulation process take place as Intraluminal Vesicles form at the endosome membrane, a process in part regulated by the Endosomal Sorting Complex Required for Transport (ESCRTs). We develop a membrane model including membrane elasticity, protein crowding (steric repulsions) and gaussian bending rigidity, which suggests that the vesicles form passively only needing to overcome a small energy barrier.

Escobar proved a sharp Sobolev inequality for the embedding of $W^{1,2}(X^{n+1})$ into $L^{2n/(n-1)}(\partial X)$ by exploiting the conformal properties of the Laplacian in X and the normal derivative along the boundary. More recently, an alternative proof was given by using a Dirichlet-to-Neumann operator along the boundary and its close relationship to the 1/2-power of the Laplacian. In this talk, I describe a new relationship between the conformally covariant fractional powers of the Laplacian due to Graham--Zworski and higher-order Dirichlet-to-Neumann operators in the interior, and use it to prove sharp Sobolev inequalities for embeddings of $W^{k,2}$. Other consequences of this relationship, such as a surprising maximum principle for the conformal 3/2-power of the Laplacian, will also be discussed.

This thesis is devoted to incorporating censoring and truncation to state-of-art Statistical
methodology and theory, to promote the evolution of survival analysis and support Medical
research with up-to-date tools. In Chapter 1, I study the mixture cure-rate model with left
truncation and right-censoring. We propose a Nonparametric Maximum Likelihood Estimation
(NPMLE) approach to effectively handle the truncation issue. We adopt an efficient and stable
EM algorithm. We are able to give a closed form variance estimator giving rise to valid
inference. In Chapter 2, I study the estimation and inference for the Fine-Gray competing
risks model with high-dimensional covariates. We develop confidence intervals based on a
one-step bias-correction to an initial regularized estimator. We lay down a methodological
and theoretical framework for the one-step bias-corrected estimator with the partial
likelihood. In Chapter 3, I study the inference on treatment effect with censored
time-to-event outcome while adjusting for high-dimensional covariates. We propose an
orthogonal score method to construct honest confidence intervals for the treatment effect.
With a slight modification, we obtain a doubly robust estimator extremely tolerant to both
estimation inconsistency and volatility. All the methods in aforementioned chapters are tested
through extensive numerical experiments and applied on real data with authentic medical
interests.

An outstanding challenge in large scale computing is to enable the casual
application programmer to realize performance obtained by an expert.
The challenge has grown in recent years due to disruptive technological
changes, which are expected to continue. In HPC, performance programming
generally relies on a priori knowledge about the application. However,
it is important to avoid entangling application software with knowledge
about the hardware.

The HPC community relies heavily on libraries, which have helped insulate
application software against technological change. However, not all
change can be accommodated via libraries, and an alternative approach
is to restructure the source using a custom translator that incorporates
the required a priori knowledge.

I will describe custom source-to-source translators targeting different
performance programming problems arising in large scale computation.
The first translator, Saaz, reduces the overheads of abstraction by up
to an order of magnitude in application libraries used to construct tools
for data discovery in turbulent flow simulation. The second translator,
MATE, restructures MPI applications to tolerate significant amounts
of communication on distributed memory computers. The third translator,
Mint, transforms annotated C++ stencil codes into highly optimized CUDA
that comes close (80%) to the performance of carefully hand coded CUDA
running on GPUs.

Each translator incorporates application semantics into the optimization
process, which are unavailable through a traditional compiler working
with conventional language constructs. In effect, the translators treat
idiomatic constructs or library APIs as a domain specific language
embedded within a conventional programming language--in our case C or C++.

Domain specific translation is an effective means of managing development
costs, enabling the domain scientist to remain focused on the domain
science, while realizing performance usually attributed to expert coders.

I will conclude the talk with earlier work on run times, that led to the
research in domain specific translation.

In 1995, Gessel introduced a multivariate formal power series $G$ tracking the distribution of ascents and descents in labeled binary trees. In addition to showing the $G$ is a symmetric function, he conjectured that $G$ is Schur-positive. In this talk, we'll see how to expand $G$ positively in terms of ribbon Schur functions. Moreover, we'll see how certain specializations of $G$ relate to actions on hyperplane arrangements. As an application of our work, we get a proof of gamma-positivity of the distribution of right edges over the set of local binary search trees.

Classical multidimensional scaling is an important tool for data reduction in many applications. It takes in a distance matrix and outputs low-dimensional embedded samples such that the pairwise distances between the original data points can be preserved, when treating them as deterministic points. However, data are often noisy in practice. In such case, the quality of embedded samples produced by classical multidimensional scaling starts to break down, when either the ambient dimensionality or the noise variance gets larger. This motivates us to propose the modified multidimensional scaling procedure which applies a nonlinear shrinkage to the sample eigenvalues. The nonlinear transformation is determined by sample size, the ambient dimensionality, and moment of noise. As an application, we consider the problem of clustering high-dimensional noisy data. We show that modified multidimensional scaling followed by various clustering algorithms can achieve exact recovery, i.e., all the cluster labels can be recovered correctly with probability tending to one. Numerical studies lend strong support to our proposed methodology.

Dr. Qiang Sun is currently an Assistant Professor of Statistics at the University of Toronto within the Department of Statistical Sciences and Department of Computer and Mathematical Sciences. Previously, he worked at Princeton University as an associate research scholar. He earned his Ph.D. from the University of North Carolina Chapel Hill in 2014 and his B.S. from University of Science and Technology of China in 2010. His research interests span a broad spectrum, including hypothesis-driven imaging genetics, clustering, manifold learning, nonconvex optimization and robust statistics.

A semitoric system is a type of 4-dimensional integrable system which possesses a circular symmetry; semitoric systems are classified in terms of five invariants by a result of Pelayo-Vu Ngoc. In this talk we will introduce semitoric systems, discuss their classification, and discuss several recent results related to explicitly constructing such systems. The general strategy of such constructions is via a one-parameter family of systems, known as a semitoric family, which passes through certain degeneracies to transition into the desired system. Using these families, we find several new explicit semitoric systems which display various behavior and are of importance in the semitoric minimal models program. The work presented is joint with Y. Le Floch and S. Hohloch (and work joint with D. Kane and A. Pelayo will also be mentioned).

In this talk, I will discuss some recent work on radial symmetry property for stationary or uniformly-rotating solutions for 2D Euler and SQG equation, where we aim to answer the question whether every stationary/uniformly-rotating solution must be radially symmetric, if the vorticity is compactly supported. This is a joint work with Javier Gómez-Serrano, Jaemin Park and Jia Shi.

We use Erman and Wood's semiample extension of Poonen's closed
point sieve to compute the probability that a semiample complete
intersection over a finite field is smooth. This generalizes work of Bucur
and Kedlaya, who provided the analogous calculation in the ample setting.
We further extend the result by allowing the requirement that the complete
intersection meet a closed subscheme transversely, so long as the subscheme
satisfies a mild Altman Kleiman type condition. In both cases the
probability stabilizes to a product of local factors determined by the
semiample divisor in question.

We prove a gap theorem for asymptotic entropy on ancient Ricci flows. We also prove an assertion made by Perelman in his paper ``The entropy formula for the Ricci flow and its geometric applications'', saying that for an ancient solution with bounded nonnegative curvature operator, bounded entropy is equivalent to noncollapsing on all scales.

The Shuffle Theorem of Carlsson and Mellit gives a well-studied combinatorial expression for the bigraded Frobenius characteristic of the ring of diagonal harmonics. The Rational Shuffle Theorem of Mellit and the Delta Conjecture proposed by Haglund, Remmel and Wilson are two natural generalizations of the Shuffle Theorem. The Primary goal of this dissertation is to prove some special cases of the conjectures, and compute the Schur function expansion of the corresponding symmetric function expressions.

It has been a puzzling question why several organisms reproduce sexually. Fisher and Muller hypothesized that reproducing by sex can speed up the evolution. They explained that in the sexual reproduction, recombination can combine beneficial alleles that lie on different chromosomes, which speeds up the time that those beneficial alleles spread to the entire population. We consider a population model of fixed size $N$, in which we will focus on two loci on a chromosome. Each allele at each locus can mutate into a beneficial allele at rate $\mu_N$. The individuals with 0, 1, and 2 beneficial alleles die at rates $1, 1-s_N$ and $1-2s_N$ respectively. When an individual dies, with probability $1-r_N$, the new individual inherits both alleles from one parent, chosen at random from the population, while with probability $r_N$, recombination occurs, and the new individual receives its two alleles from different parents. Under certain assumptions on the parameters $N, \mu_N, s_N$ and $r_N$, we obtain an asymptotic approximation for the time that both beneficial alleles spread to the entire population. When the recombination probability is small, we show that recombination does not speed up the time that the two beneficial alleles spread to the entire population, while when the recombination probability is large, we show that recombination decreases the time, which agrees with Fisher-Muller hypothesis, and confirms the advantage of reproducing by sex.

A new primal-dual path-following shifted penalty-barrier method will be described for solving nonlinear inequality constrained optimization problems (NIP). The proposed method has a bi-level structure in which a trajectory parameterized by the penalty and barrier parameters and Lagrangian multipliers estimates is closely followed towards a constrained local minimizer of NIP. This method inherits some features of the primal-dual augmented Lagrangian method for solving nonlinear equality constraint problems (NEP) but has been extended to handle inequality constraints. Global and local convergence results will be presented. Finally, numerical results from the CUTEst test collection will be provided to support the robustness of the proposed algorithm.

A simple character with delusions of grad school applies his abstraction-through-incomprehension style of independent squalorship to banalities which appear, from an obtuse angle, to concern foundational questions in mathematics, providing an abject lesson in how common misconceptions may be avowed.

Let $k$ be a global field, that is, a number field of finite
degree
over $\Bbb Q$ or the function field of a smooth projective curve $C$
over a finite field $F$. Let $X$ be a smooth projective variety over $k$,
and let $K$
be the maximal cyclotomic extension of $k$, obtained by adjoining
all roots of unity. If $X$ is an abelian variety, a famous theorem,
due to Ribet in the number field case and Lang-Neron in the
function field case when $X$ has trace zero over the constant subfield
of $K$, asserts that the torsion subgroup of the Mordell-Weil group of $X$
over $K$ is finite. Denoting by $k^{sep}$ a separable closure of $k$,
this result is equivalent to finiteness of the fixed part
by $G=Gal(k^{sep}/K)$ of the etale cohomology group
$H^1(X_{k^{sep}},\Bbb Q/\Bbb Z)$,
where we ignore the $p$-part in positive characteristic $p$. In a recent
paper, Roessler-Szamuely generalize this result to all odd cohomology groups.
The trace zero assumption in the function field case is replaced by
a ''large variation'' assumption on the characteristic polynomials of
Frobenius acting on the cohomology of the fibres of a
morphism $f: \mathcal{X}\to C$ from a smooth projective variety
$\mathcal{X}$
over a finite field to $C$ with generic fibre $X$.
In this talk, I will discuss the case of even degree, proving some
positive results in the number field case and negative results in the
function field case.

Non-abelian $C^\ast$-algebras can be understood better from the examination of their maximal abelian subalgebras. In particular, Renault showed that in the presence of a Cartan subalgebra, a $C^\ast$-algebra can be associated in a canonical way with a topological twisted groupoid.

In joint work with Jon Brown, Adam Fuller, and David Pitts, we extend Renault's result by identifying Cartan pairs revealed by gradings by a group.

The shape of a degree $n$ number field is a $n-1$-variable real quadratic form
(up to equivalence and scaling) which keeps track of the lattice shape of its ring
of integers relative to $\mathbb{Z}$. For number fields of small degree, in previous
joint work with Bhargava, we showed that shapes of $S_n$-number fields are
equidistributed, when ordered by absolute discriminant. The proof relies heavily
on Bhargava's parametrizations which introduces but ultimately ignores the notion
of resolvent rings. This talk discusses work in progress, joint with Christelle Vincent,
in which we define the joint shape of a ring and its resolvent ring in order to prove
equidistribution of joint shapes of quartic fields and their cubic resolvent fields.

Almost everything we encounter in our 3-dimensional world is a surface - the outside of a solid object. Comparing the shapes of surfaces is, not surprisingly, a fundamental problem in both theoretical and applied mathematics. Deep mathematical results are now being used to study objects such as bones, brain cortices, proteins and biomolecules. This talk will discuss recent joint work with Patrice Koehl that introduces a new metric on the space of genus-zero surfaces and applies it in this context.

A landmark result by Dykema in 1993 classified free products of finite-dimensional von Neumann algebras equipped with tracial states. In 1997, Shlyakhtenko constructed the almost periodic free Araki-Woods factors, a natural non-tracial analogue to free group factors. He asked whether free products of finite-dimensional von Neumann algebras with respect to non-tracial states can be described in terms of free Araki-Woods factors. In this talk, I will answer Shlyakhtenko's question in the affirmative, therefore providing a complete classification of free products of finite dimensional von Neumann algebras. This is joint work with Brent Nelson.

The category {\bf FI} and its variants have been of great interest recently. Being a finitely generated {\bf FI}-module implies many desirable properties about sequences of symmetric group representations, in particular representation stability. We define a new combinatorial category analogous to {\bf FI} for the 0-Hecke algebra, denoted by $\mathcal{H}$, indexing sequences of representations of $H_n(0)$ as $n$ varies under suitable compatibility conditions. We then provide examples of $\mathcal{H}$-modules and use these to discuss some properties finitely generated $\mathcal{H}$-modules possess, including a new form of representation stability and eventually polynomial growth.

We prove the conjecture that any Grothendieck $(\infty,1)$-topos can be presented by a Quillen model category that interprets homotopy type theory with strict univalent universes. Thus, homotopy type theory can be used as a formal language for reasoning internally to $(\infty,1)$-toposes, just as higher-order logic is used for 1-toposes. As part of the proof, we give a new, more explicit, characterization of the fibrations in injective model structures on presheaf categories. In particular, we show that they generalize the coflexible algebras of 2-monad theory.

The interesting part of the cohomology of the theta divisor $D$ of an abelian fivefold $A$ shares numerical properties with the Lie algebra $E_6$. We define 27 surfaces inside $D$, one for each realisation of $A$ as a Prym variety, and explain how they generate a sublattice of $H^4(D, \mathbb{Z})$ isomorphic to the root lattice of $E_6$. This gives an effective proof of the Hodge conjecture for the theta divisor.

An alternative to selecting an integer uniformly from $1$ to $N$ and letting $N$ go to infinity is to select an
integer according to the Riemann zeta distribution: the probability of selecting $n$ is $1/\zeta(s)n^s$, and
letting $s$ go to $1$. We will explain several results that arise naturally due to the multiplicative property of this distribution.

We study the effective behavior of a Brownian motion in both one and two dimensional comb like domains. This problem arises in a variety of physical situations such as transport in tissues, and linear porous media. We show convergence to a limiting process when when both the spacing between the teeth, and the probability of entering a tooth vanish at the same rate. This limiting process exhibits an anomalous diffusive behavior, and can be described as a Brownian motion time-changed by the local time of an independent sticky Brownian motion. At the PDE level, this leads to equations that have fractional time derivatives and are similar to the Bassett differential equation.

Fix an integer $N$ and a prime $(p,N)=1$ where $p>3$. We show that the number of newforms $f$ (up to a scalar multiple) of level $N$ and even weight $k$ such that $T_p(f) = 0$ is bounded independently of $k$, where $T_p$ is the Hecke operator.

Consider the Bergman kernel associated to the tensor power of a positive line bundle on a compact Kähler manifold. We will present our work on its near-diagonal asymptotic and off-diagonal decay properties. This is joint work with H. Hezari and Z. Lu.

Epitaxial growth is an important physical process for forming solid films or other nano-structures. It occurs as atoms, deposited from above, adsorb and diffuse on a crystal surface. Modeling the rates that atoms hop and break bonds leads in the continuum limit to degenerate 4th-order PDE that involve exponential nonlinearity and the p-Laplacian with p=1, for example. We discuss a number of analytical results for such models, some of which involve subgradient dynamics for Radon measure solutions and a new notion of weak solutions.

Malle's conjecture can be thought of as a
generalization of the inverse Galois problem, which asks for every
finite group $G$, is there a number field $K$ such that their Galois
group over $\mathbb{Q}$ is isomorphic to $G$? Although open, this
question is widely believed to be true, and Malle went further to
predict the asymptotics of how many number fields there are with a
given Galois group that only depended on the group structure of $G$
and the degree of the number field. In this talk, we will discuss the
history as well as recent results and techniques surrounding these
conjectures.

(joint with B.Bakker and Y.Brunebarbe) One
very fruitful way of studying complex algebraic varieties is by
forgetting the underlying algebraic structure, and just thinking of
them as complex analytic spaces. To this end, it is a natural and
fruitful question to ask how much the complex analytic structure
remembers. One very prominent result is Chows theorem, stating that
any closed analytic subspace of projective space is in fact
algebraic. A notable consequence of this result is that a compact
complex analytic space admits at most one algebraic structure - a
result which is false in the non-compact case. This was generalized
and extended by Serre in his famous GAGA paper using the language of
cohomology.

We explain how we can extend Chows theorem and in fact all of GAGA to
the non-compact case by working with complex analytic structures that
are 'tame' in the precise sense defined by o-minimality. This leads to
some very general 'algebraization' theorems, which can be used to
obtain new results in Hodge Theory. In particular, we use this
technology to prove a conjecture of Griffiths on the algebraicity and
quasi-projectivity of images of period maps. As prerequisities for
this talk, it would be helpful to have gone through a first year
course in Algebraic Geometry, covering in particular the theory of
sheaves.

I will present several results (that have been obtained jointly with Stefanos Aretakis and Dejan Gajic) from the analysis of solutions of linear and nonlinear wave equations on extremal Reissner-Nordstrom spacetimes, including sharp asymptotics on the horizon and at infinity for linear waves, and instability phenomena for nonlinear waves. These results can be seen as stepping stones to the fully nonlinear problem of stability/instability of extremal black holes.

There is a rich connection between homogeneous dynamics and number theory, especially when dynamical results are effective (i.e. when rates of convergence for dynamical phenomena are known). In this final defense, I describe my research on the asymptotic distribution of almost-prime times in horospherical flows on the space of lattices, as well as on compact quotients of SL(n,R). In the compact setting, I obtain a result that implies density for almost-primes in horospherical flows, where the number of prime factors is independent of the basepoint, and in the space of lattices I show the density of almost-primes in abelian horospherical orbits of points satisfying a certain Diophantine condition. To prove this, I first give an effective equidistribution result for arbitrary horospherical flows on the space of lattices, which I then use to prove an effective rate for the equidistribution of arithmetic progressions in abelian horospherical flows, to which I then apply a combinatorial sieve.

Motivated by the study of mirror symmetry, Strominger-Yau-Zaslow (SYZ) conjectured that Calabi-Yau manifolds admit certain minimal Lagrangian fibrations. These minimal Lagrangians are the special Lagrangian submanifolds studied earlier by Harvey-Lawson. Many of the implication of the SYZ conjecture is proved and it has been the guiding principle for studying mirror symmetry for a long time. However, not many special Lagrangians are known in the literature. In this talk, I will prove the existence of special Lagrangian fibration on the complement of a smooth anti-canonical divisor in a (weak) Del Pezzo surface. If the time allows, I will explain its impact to mirror symmetry. This is joint work with Tristan Collins and Adam Jacob.

The talk concerns inequalities on functions of matrix variables, in particular a convex set C of matrices defined by them. The functions are typically (noncommutative) polynomials or rational functions and the sets include matrices of all sizes, hence are dimension free
convex sets.

Extreme points are getting to be understood. But optimizing a linear functional leads in our experiments to surprising properties which are unexplained. The talk describes the sets, the extreme points and the experimental findings.

Ratner's celebrated equidistribution theorem states that the trajectory of any point in a homogeneous space under a unipotent flow is getting equidistributed with respect to some algebraic measure. In the case where the action is horospherical, one can deduce an effective equidistribution result by mixing methods, an idea that goes back to Margulis' thesis. When the homogeneous space is non-compact, one needs to impose further
diophantine conditions'' over the base point, quantifying some recurrence rates, in order to get a quantified equidistribution result. In the talk I will discuss certain diophantine conditions, and in particular I will show how a new Margulis' type inequality for translates of horospherical orbits helps verify such conditions, leading to a quantified equidistribution result for a large class of points, akin to the results of A. Strombergsson regarding the SL2 case. In particular we deduce a fully effective quantitative equidistribution statement for horospherical trajectories of lattices defined over number fields.

A challenging problem in the brain imaging research is a principled incorporation of information from different imaging modalities in regression models. Frequently, data from each modality is analyzed separately using, for instance, dimensionality reduction techniques, which result in a loss of information. We propose a novel regularization method, griPEER (generalized ridgified Partially Empirical Eigenvectors for Regression) to estimate the association between the brain structure features and a scalar outcome within the generalized linear regression framework. griPEER provides a principled approach to use external information from the structural brain connectivity to improve the regression coefficient estimation. Our proposal incorporates a penalty term, derived from the structural connectivity Laplacian matrix, in the penalized generalized linear regression. We address both theoretical and computational issues and show that our method is robust to the incomplete structural brain connectivity information. griPEER is evaluated via extensive simulation studies and it is applied in classification of the HIV+ and HIV- individuals.

The generalized Dickman distribution ${\cal D}_\theta$ with parameter $\theta>0$ is the unique solution to the distributional equality
$W=_d W^*$, where
\begin{align*}
W^*=_d U^{1/\theta}(W+1),
\end{align*}
with $W$ non-negative with probability one, $U \sim {\cal U}[0,1]$ independent of $W$, and $=_d$ denoting equality in distribution. Members of this family appear
in the study of algorithms, number theory, stochastic geometry, and perpetuities.

The Wasserstein distance $d(\cdot,\cdot)$ between such a $W$ with finite mean, and $D \sim {\cal D}_\theta$ obeys
\begin{align*} d(W,D) \le (1+\theta)d(W^*,W).
\end{align*}
The specialization of this bound to the case $\theta=1$ and coupling constructions yield for $n \ge 1$ that
\begin{align*}
d_1(W_n,D) \le \frac{8\log (n/2)+10}{n} \quad \mbox{where } \quad W_n=\frac{1}{n}C_n-1,
\end{align*}
and $C_n$ is the number of comparisons made by the Quickselect algorithm to find the smallest element of a list of $n$ distinct numbers.

Joint with Bhattacharjee, using Stein's method, bounds for Wasserstein type distances can also be computed between ${\cal D}_\theta$ and weighted sums arising
in probabilistic number theory of the form
\begin{align*}
S_n=\frac{1}{\log(p_n)} \sum_{k=1}^n X_k \log(p_k)
\end{align*}
where $(p_k)_{k \ge 1}$ is an enumeration of the prime numbers in increasing order and $X_k$ is, for instance, Geometric with parameter $1-1/p_k$.

I will describe a new set of estimates for the 2d water waves problem, in which the free surface has an angled crest (or corner) with a time-dependent angle that changes with the evolution of the water wave, and with a corner vertex that can move in all directions. There are no symmetry constraints on the crest, and the fluid can have bulk vorticity. This is joint work with D. Coutand.

Queueing systems operating under the processor sharing discipline are relevant for studying time-sharing in computer and communication systems. Measure-valued processes, which track the residual service times of all jobs in the system, have been used to describe the dynamics of such systems. However, exact analysis of these infinite-dimensional stochastic processes is rarely possible. As a tool for approximate analysis of such systems, it has been proved that a fluid model arises as a functional law of large numbers limit of a multi-class processor sharing queue. This talk will focus on the asymptotic behavior of such a fluid model in the interesting regime of critical loading, where the average inflow of work to the system is equal to the capacity of the system to process that load.

Using an approach involving a certain relative entropy functional, we show that critical fluid model solutions converge to a set of invariant states as time goes to infinity, uniformly for all initial conditions lying in certain relatively compact sets. This generalizes an earlier single-class result of Puha and Williams to the more complex multiclass setting. In particular, several new challenges are overcome, including formulation of a suitable relative entropy functional and identifying a convenient form of the time derivative of the relative entropy applied to trajectories of critical fluid model solutions.

This is joint work with Justin A. Mulvany (USC) and Ruth J. Williams (UCSD)

This talk is concerned with the inverse problem of recovering a discrete measure on the torus consisting of S atoms, given M consecutive noisy Fourier coefficients. Super-resolution is sensitive to noise when the distance between two atoms is less than 1/M. We connect this problem to the minimum singular value of non-harmonic Fourier matrices. New results for the latter are presented, and as consequences, we derive results regarding the information theoretic limit of super-resolution and the resolution limit of subspace methods (namely, MUSIC and ESPRIT). These results rigorously establish the super-resolution phenomena of these algorithms that were empirically discovered long ago, and numerical results indicate that our bounds are sharp or nearly sharp. We also discuss how to take advantage of redundant measurements for the purpose of reducing quantization error. Interesting connections to trigonometric interpolation and uncertainty principles are also presented. Joint work with John Benedetto, Albert Fannjiang, Sinan Gunturk, and Wenjing Liao.

For environmental microbial communities, environment is destiny in the sense that, frequently, microbial community form and function are strongly linked to chemical and physical conditions. Moreover, most environments outside of the lab are physically and chemically heterogeneous, further shaping and complicating the metabolisms of their resident microbial communities: spatial variation introduce physics such as diffusive and advective transport of nutrients and byproducts for example. Conversely, microbial metabolic activity can strongly effect the environment in which the community must function. Hence it is important to link metabolism at the cellular level to physics and chemistry at the community level.

In order to introduce metabolism to community-scale population dynamics, many modeling methods rely on large numbers of reaction kinetics parameters that are unmeasured and likely effectively unmeasurable (because they are themselves coupled to environmental conditions), also making detailed metabolic information mostly unusable. The bioengineering community has, in response to these difficulties, moved to kinetics-free formulations at the cellular level, termed flux balance analysis. These cellular level models should respond to system level environmental conditions. To combine and connect the two scales, we propose to replace classical kinetics functions (almost) entirely in community scale models and instead use cell-level metabolic models to predict metabolism and how it is influenced and influenced by the environment. Further, our methodology permits assimilation of many types of measurement data.

Lattices of minimal covolume have been studied fairly
intensively in real Lie groups, particularly in the hyperbolic
isometry groups. On the other hand, their $p$-adic analogues only
have been determined in (some) lower rank groups. In this talk,
we will discuss the higher rank behavior of lattices of minimal
covolume in $\mathrm{SL}_n(\mathbb{Q}_p)$. We will briefly introduce
their general structure, then use Prasad's volume formula and
Borel-Prasad techniques to compute their covolume. This quantity involves
a variety of number-theoretical objets, and its understanding gives rise
to some number-theoretical questions. As this is work in progress, joint
with Alireza Salehi Golsefidy, the scope of the talk will be to give a
general overview of the techniques and problems involved, rather than
stating precise results.

This talk will include an introduction to the topic of V(D)J rearrangements of particular subsets of T cells and B cells of the adaptive human immune system, in particular of IgG heavy chains. There are many statistical problems that arise in understanding these cells. This presentation will be my attempt to provide some mathematical and computational details that arise in trying to understand the data.

In his influential disjointness paper, H. Furstenberg proved that weakly-mixing systems are disjoint from irrational rotations (and in general, Kronecker systems), a result that inspired much of the modern research in dynamics. Recently, A. Venkatesh managed to prove a quantitative version of this disjointness theorem for the case of the horocyclic flow on a compact Riemann surface. I will discuss Venkatesh's disjointness result and present a generalization of this result to more general actions of nilpotent groups, utilizing structural results about nilflows proven by Green-Tao-Ziegler. If time permits, I will discuss applications of such theorems in sparse equidistribution problems and number theory.

Reaction-diffusion equations have been widely used to describe biological pattern formation. Nonuniform steady states of reaction-diffusion models correspond to stationary spatial patterns supported by these models. Frequently these steady states are not unique, which correspond to various spatial patterns observed in biology. Traditionally, time-marching methods or steady state solvers based on Newton’s method were used to compute such solutions. However, the solution that any of these methods leads to highly depends on the initial condition/guess. In this talk, I present a systematic method to compute multiple nonuniform steady states for reaction-diffusion models and determine the dependence on model parameters. The method is based on homotopy continuation techniques and multigrid methods. We apply the method to two classic reaction-diffusion models and compare our results with available theoretical analysis in the literature. The first is the Schnakenberg model that has been used to describe biological pattern formation due to diffusion-driven instability. The second is the Gray-Scott model which was proposed in 1980’s to describe autocatalytic glycolysis reactions. In each case, our method uncovers many, if not all, nonuniform steady states and their stabilities. We also compared our computational results with analytical results in the literature and the comparison suggests some errors in prior results obtained using asymptotic analysis.

Bacterial quorum sensing (QS) is a form of intercellular communication that relies on the production and detection of diffusive signaling molecules called autoinducers. Such a mechanism allows the bacteria to track their cell density in order to regulate group behavior, such as biofilm formation and bioluminescence. In a number of bacterial QS
systems, including V. harveyi, multiple signaling pathways are integrated into a single phosphorylation-dephosphorylation cycle. In this talk, we will explore how QS uses feedback loops to 'decode' the integrated signals by actively changing the sensitivity in different pathways.

Using a new method “coefficient shuffling” we are apple to sharpen the comparison theorems of Bishop and Gromov. The theorem derives from work with physics L. Susskind and A. Brown on black hole dynamics, or viewed another way the problem of optimal compilation of a quantum algorithm.

Motivated by biological models of solvation, this defense consists of analysis of models of electrostatic free energy of charged systems that incorporate both continuum and discrete idealizations of charges.

Topics: This is an introductory course in mathematical logic at the graduate level. Topics to be covered during the Fall quarter to include first order logic, soundness, completeness, cut-elimination, Herbrand's theorem, decidability, undecidability, Robinson resolution, Lowerheim-Skolem, Craig interpolation, quantifier elimination, elementary embeddings, model completeness, preservation theorems.

There are no particular prerequisites beyond sufficient mathematical maturity. Suitable for graduate students in mathematics, computer science, philosophy. Please email me if you are interested in attending and cannot make the first lecture. (Thurs, Sep 26).

There is no textbook. Supplemental reading includes:

For Proof Theory:
Handbook of Proof Theory, chapters 1 and 2 by the instructor.(Available freely online.)
Proof Theory, by Gaisi Takeuti. (Low-priced Dover edition available.)
For Model Theory:
A Shorter Model Theory by Wilfrid Hodges. (Dover edition available)
Model Theory (Chang and Keisler) is a bit more advanced.

In this talk we apply Fourier analysis of Boolean functions to solve problems in social choice theory and property testing. In particular we examine Arrow's impossibility theorem and the BLR test.

In 1968 V. Strassen discovered the usual way we multiply matrices is not the most efficient one. This raised the question as to just how efficiently matrices can be multiplied, and led to the astounding conjecture that for large matrices, it is almost as easy to multiply them asto add them. After giving a brief history of the problem, I will explain how algebraic geometry and representation theory gives insight into this central question in computer science.

While much is known about existence of algorithms in the study of 3-manifolds and knot theory, much less is known about lower bounds on their complexity (hardness results). In this talk we will discuss hardness of several problems. We prove:

Theorem 1: given a 2- or 3-dimensional complex $X$, deciding if $X$ embed in $R^3$ in NP-hard.

We also prove that certain link invariants that are defined using 4-dimensional topology give rise to NP-hard problems; for example:

Theorem 2: deciding if a link in the 3-sphere bounds a smooth surface of non-negative Euler characteristic in the 4-ball is NP-hard.

For the main event we turn our attention to knots. The unknot recognition problem (solved by Haken in the 60's) is known to be in NP and co-NP, and as such, is not expected to be hard. Lackenby proved a polynomial bound on the number of Reidemeister moves needed to untangle an unknot diagram. In light of these two facts, one might hope for an efficient algorithm that find this optimal untangling. Unfortunately this is unlikely to happen since we prove:

Theorem 3: given an unknot diagram D and a positive integer n, deciding if D can be untangled using n Reidemeister moves is NP-hard.

This is joint work with Arnaud de Mesmay, Eric Sedgwick, and Martin Tancer.

We consider a class of parametric regression problems where the signal is observed through random nonlinear functions with a difference of convex (DC) form. This model describes a broad subset of nonlinear regression problems that includes familiar special cases such as phase retrieval/quadratic regression and blind deconvolution/bilinear regression. Given the DC decomposition of the observation functions as well as an approximate solution, we formulate a convex program as an estimator that operates in the natural space of the signal. Our approach is computationally superior to the methods based on semidefinite/sum-of-squares relaxation---tailored for polynomial observation functions---and can compete with the non-convex methods studied in special regression problems. Furthermore, under mild moment assumptions, we derive the sample complexity of the proposed convex estimator using a PAC-Bayesian argument. We instantiate our results with bilinear regression with Gaussian factors and provide a method for constructing the required initial approximate solution.

In random constraint satisfaction problems, first and second moment methods are used to yield upper and lower bounds for the threshold of edge density for existence of solutions. In random hypergraph 2-coloring, Achlioptas and Moore obtain a lower bound using a standard second moment method based on the Paley-Zygmund inequality. Coja-Oghlan and Zdeborova later use an enhanced second moment method involving the Hamming geometry of the set of colorings to improve the lower bound up to a so-called condensation transition.

We adapt their methods and setup to a subshift of finite type over a sofic group and show results analogous to the aforementioned, exploiting the combination of phenomena that occur at densities between the standard and enhanced second moment thresholds to conclude that there exists an interesting example of a topological dynamical system having two different positive sofic entropies relative to two different sofic approximations. This is joint work with Dylan Airey and Lewis Bowen.

Quantum computing can be effected by a sequence of projective measurements. Such strategies are called, ``measurement only''. The key geometric concept which ensures that system information does not leak to the environment during measurement is ``equiangularity''. I will explain what the Octonions are and how they provide the largest families of equiangular subspaces of a Hilbert space.

Morse theory can be generalized to the study of maps to $R^k$. I will discuss where this leads, focusing on ambient Morse functions of embedded 3-manifolds. The relevance to the question of finite generation of the Goeritz groups $G_g$ will be explained.

In our fast paced world, it's important to take things slow every once in a while. In this talk, we'll take a leisurely stroll through some of the research I've done recently on slow Fibonacci walks, in which we try and generate numbers in a ``Fibonacci-like way'' as slowly as possible. This talk is part of the graduate student seminar ``Food for Thought,'' and in particular (1) no prior knowledge of anything is assumed, and more importantly (2) snacks will be provided.

Web geometry deals with foliations in general position. In the planar case and the complex setting, a $d$-web is given by the generic family of integral curves of an analytic or an algebraic differential equation \textit{F(x,y,y')=0} with y'-degree $d$. Invariants of these configurations as abelian relations (related to Abel's addition theorem), Lie symmetries or Godbillon-Vey sequences are investigated. This viewpoint enlarges the qualitative study of differential equations and their moduli. In the nonsingular case and through the singularities, Cartan-Spencer and meromorphic connections methods will be used. Basic examples will be given from different domains including classic algebraic geometry and WDVV-equations. Standard results and open problems will be mentioned. Illustration of the interplay between differential and algebraic geometry, new results will be presented.

Based on the review paper in the area of stochastic network: Stochastic Processing Networks by Ruth J. Williams, I will recall some cornerstones in queueing networks. In addition, I will introduce some recent progress and open problems in this area if time allows.

We develop a simple semi-algebraic 2-level solver built
on traditional multigrid ideas. It is designed to be easily
incorporated into existing simulation software. It exhibits good
convergence for many classes of challenging problems including
discontinuous diffusion, convection-diffusion, and Helmholtz
equations. It has built-in structure that makes it simple to
generalize to a hierarchical basis multigrid solver.

Every area-minimizing hypercone having only an isolated singularity fits into a foliation by smooth, area-minimizing hypersurfaces asymptotic to the cone itself. In this talk I will present the following epsilon-regularity result: every minimal surfaces lying sufficiently close to a minimizing quadratic cone (for example, the Simons' cone), is a perturbation of either the cone itself, or some leaf of its associated foliation. This result also implies the Bernstein-type result of Simon-Solomon, which characterizes area-minimizing hypersurfaces asymptotic to a quadratic cone as either the cone itself, or some leaf of the foliation, and it also allows to study convergence to singular minimal hyper surfaces. This is a joint result with N. Edelen.

Let $\beta\equiv \beta^{(m)} = \{\beta_{i}\}_{i\in \mathbb{Z}_{+}^{n},
|i|\le m}$, $\beta_{0}>0$, denote a real $n$-dimensional multisequence of degree $m$.
The \textit{Truncated Moment Problem} for $\beta$ (TMP) concerns the existence
of a positive Borel measure $\mu$, supported in $\mathbb{R}^{n}$, such that
$
\beta_{i} = \int_{\mathbb{R}^{n}} x^{i}d\mu ~~~~~~~~( i\in \mathbb{Z}_{+}^{n},~~|i|\le m).
$
(Here, for $x\equiv (x_{1},\ldots,x_{n})\in \mathbb{R}^{n}$
and $i\equiv (i_{1},\ldots,i_{n})\in \mathbb{Z}_{+}^{n}$,
we set
$|i| = i_{1}+\cdots + i_{n}$ and
$x^{i} = x_{1}^{i_{1}}\cdots x_{n}^{i_{n}}$.)
A measure $\mu$ as above is a {\it{representing measure}} for $\beta$.
We discuss three equivalent ``solutions" to TMP, based on: 1) flat extensions
of moment matrices, 2) positive extensions of Riesz functionals, and 3) the
\textit{core variety} of a multisequence. In work with G. Blekherman
[J. Operator Theory, to appear]
We proved that $\beta$ has a representing measure if and only if the core variety
is nonempty, in which case the core variety is the union of supports of all
finitely atomic representing measures. We discuss open questions concerning
difficulties in applying of any of the above solutions to TMP in special cases or in
numerical examples.

We consider general self-adjoint polynomials and rational expressions in several independent random matrices whose entries are centered and have constant variance. Under some numerically checkable conditions, we establish for these models the optimal local law, i.e., we show that the empirical spectral distribution on scales just above the eigenvalue spacing follows the global density of states which is determined by free probability theory. We show that the above results can be applied to prove the optimal bulk local law for two concrete families of polynomials: general quadratic forms in Wigner matrices and symmetrized products of independent matrices with i.i.d. entries. Moreover, in the framework of the developed theory for rational expressions in random matrices, we study the density of transmission eigenvalues in the random matrix model for transport in quantum dots coupled to a chaotic environment.
This is a joint work with Laszlo Erd$\ddot{\text{o}}$s and Torben Kr$\ddot{\text{u}}$ger.

Faltings's isogeny theorem states that two abelian varieties
over a number field are isogenous precisely when the characteristic
polynomials associated to the reductions of the abelian varieties at all
prime ideals are equal. This implies that two abelian varieties defined
over the rational numbers with the same L-function are necessarily
isogenous, but this is false over a general number field.

In order to still use the L-function to determine the underlying field,
we extract more information from the L-function by "twisting": a twist
of an L-function is the L-function of the tensor of the underlying
representation with a character. We discuss a theorem stating that
abelian varieties over a general number field are characterized by their
L-functions twisted by Dirichlet characters of the underlying number field.

Calculus in normed vector spaces is the basis for several areas of
mathematics and physics, but it is not a topic that is often covered with
very much care or detail. This talk's focus is the theory of
differentiation in normed vector spaces, more specifically the Gateaux and
Fr\'{e}chet derivatives. Towards the end, we shall cover an
infinite-dimensional Taylor's Theorem, and we shall likely get to discuss
some applications. Plus, there will be plenty of examples throughout! This
talk is Part I of a (likely) three- or four-part series, with future
topics including integration and complex analysis.

Last week, Yingjia gave an excellent overview of results in the theory of Stochastic Processing networks. I will be taking this opportunity to
demonstrate some of the concepts she mentioned in depth using a specific example. In particular, I will be discussing the fluid limit of a
processor sharing queue, how its properties can be used to prove a dimension reduction (state space collapse), and how this can be used to ultimately prove the diffusion approximation.

We study tuples $(X_1,\dots,X_d)$ of self-adjoint operators in a
tracial $W^*$-algebra whose non-commutative distribution is the free Gibbs
law for a (sufficiently regular) convex potential $V$. Such tuples model
the large $N$ behavior of random matrices $(X_1^{(N)}, \dots, X_d^{(N)})$
chosen according to the measure $e^{-N^2 V(x)}\,dx$ on
$M_N(\mathbb{C})_{sa}^d$. Previous work showed that
$W^*(X_1,\dots,X_d)$ is isomorphic to the free group factor
$L(\mathbb{F}_d)$. In a recent preprint, we showed that an isomorphism
$\phi: W^*(X_1,\dots,X_d)$ can be chosen so that $W^*(X_1,\dots,X_k)$ is
mapped to the canonical copy of $L(\mathbb{F}_k)$ inside $L(\mathbb{F}_d)$
for each $k$. The idea behind the proof is to apply PDE methods for
constructing transport to Gaussian to the conditional density of
$X_j^{(N)}$ given $X_1^{(N)}, \dots, X_{j-1}^{(N)}$. Then we analyze the
asymptotic behavior of these transport maps as $N \to \infty$ using a new
type of functional calculus, which applies certain
$\|\cdot\|_2$-continuous functions to tuples of self-adjoint operators
to self-adjoint tuples in (Connes-embeddable) tracial $W^*$-algebras.

Line search methods for unconstrained optimization based on satisfying the Wolfe conditions impose a restriction on the value of the directional derivative of the objective function at the new iterate. Projected search methods for bound-constrained optimization involve a line search along a continuous piecewise-linear path, which makes it impossible to apply the conventional Wolfe conditions. We propose a new quasi-Wolfe line search for piecewise differentiable functions. The behavior of the line search is similar to that of a conventional Wolfe line search, except that a step is accepted under a wider range of conditions. These conditions take into consideration steps at which the line search function is not differentiable. Some basic results associated with a conventional Wolfe line search are established for the quasi-Wolfe case. After identifying the practical considerations needed for converting a Wolfe line search into a quasi-Wolfe line search, details of the imp
lementation along with some numerical results will be presented.

We show that any set of $n$ points in general position in the plane
determines $n^{1-o(1)}$ pairwise crossing segments. The best
previously known lower bound, $\Omega(\sqrt{n})$ was proved more than
25 years ago by Aronov, Erd\H os, Goddard, Kleitman, Klugerman, Pach,
and Schulman. Our proof is fully constructive, and extends to dense
geometric graphs. This is joint work with J\'anos Pach and G\'abor
Tardos.

I will present the recent tools I have developed to prove existence and
regularity properties of the critical points of anisotropic functionals.
In particular, I will provide the anisotropic extension of Allard's
celebrated rectifiability theorem and its applications to the
anisotropic Plateau problem. Three corollaries are the solutions to the
formulations of the Plateau problem introduced by Reifenberg, by
Harrison-Pugh and by Almgren-David. Furthermore, I will present the
anisotropic counterpart of Allard's compactness theorem for integral
varifolds. To conclude, I will focus on the anisotropic isoperimetric
problem: I will provide the anisotropic counterpart of Alexandrov's
characterization of volume-constrained critical points among finite
perimeter sets. Moreover I will derive stability inequalities associated
to this rigidity theorem.
Some of the presented results are joint works with De Lellis, De
Philippis, Ghiraldin, Gioffr\'e, Kolasinski and Santilli.

Finding roots of a function is one of the most fundamental tasks in mathematics. What if
the function is random ?

We are going to survey some of the main developments in the theory of random functions in the last 80 years or so, from the works of Polya, Erdos, Littlewood, Offord in the early 1900s to this very sunny day.

Gerrymandering is the (currently very relevant) issue in
representative democracies of drawing electoral districts to give a
political advantage to one party or group. Political events within the
last few years have sparked a lot of research activity from
mathematicians, computer scientists, and statisticians related to
detecting and quantifying gerrymandered electoral plans. In this talk I
will give an introduction to the problem of gerrymandering, discuss some
historical attempted methods of quantifying gerrymanders and their
shortcomings, and then talk about a promising new method called ``metagraph
Markov Chain Monte Carlo'' currently being researched and implemented.

We examine the relationship between having an algebraic Hecke character attached to the cohomology of a smooth projective variety $X$ equipped with a finite-order automorphism and the algebraicity of some Hodge/Tate classes on the product $X^n$. As a consequence, we deduce the Hodge and Tate conjectures for some self-products of varieties, including some self-products of $K3$ surfaces.

I will report some recent progress on p-adic Simpson
correspondence.

The stochastic block model (SBM) is a generative model for random graphs
with a community structure, which has been one of the most fruitful
research topics in community detection and clustering. A phase
transition behavior for detection was conjecured by Decelle el al.
(2011), and was confirmed by Mossel et al. (2012,2013) and Massouli\'e
(2013). We consider the community detection problem in random
hypergraphs. Angelini et al. (2015) conjectured a phase transition for
community detection in sparse hypergraphs generated by a hypergraph
stochastic block model (HSBM). We confirmed the positive part of the
phase transition for the 2-block case by a generalization of the method
developed in Massouli\'e (2013). We introduced a matrix which counts
self-avoiding walks on hypergraphs, whose leading eigenvectors give us a
correlated reconstruction. This is joint work with Soumik Pal.

We work out an explicit theory for the shtuka
function of rank 1 sign-normalized Drinfeld modules over the function
field of an elliptic curve. Using these explicit formulas, we obtain a
product formula for the fundamental period of the exponential function
associated to the Drinfeld module. We also find identities for
deformations of reciprocal sums and as a result prove special value
formulas for L-series over the function field.

A main goal of algebraic geometry is the
classification of algebraic varieties and a central tool in this
endeavor is the study of moduli spaces. I will discuss the moduli
space of plane curves of degree d: a parameter space where each point
corresponds to an isomorphism class of a certain curve. There are many
techniques to compactify this space, including GIT, the minimal model
program, and a differential geometric approach called K stability. In
joint work with K. Ascher and Y. Liu, we interpolate between these
different compactifications and study the problem via wall crossings.

Given $s$ finite sets $A_1, \ldots, A_s$, determining the size of the
union of the $s$ sets is an easy problem. Determining the maximimum number
of size $k$ subsets of an $n$ element set for which there does not exist
$s$ sets which union has size $q$ is a very hard problem in general. Many
problems in extremal set theory can be restated in this language for
particular choices of $s,k,q$. For instance, the case where $s=2$ is
equivalent to the complete intersection theorem, and when $sk=q$, this is
equivalent to the Erd{\H o}s matching conjecture; one of the biggest open
problems in the field. This talk is based off a recent paper of Peter
Frankl and Andrey Kupavskii.

Hacon and McKernan have proved that there exist integers $r_n$ such that if $X$ is a smooth variety of general type and dimension $n$, then the pluricanonical maps $|rK_X|$ are birational for all $r\geq r_n$. These values are typically very large: for example $r_3\geq 27$ and $r_4\geq 94$. In this talk we will show that the $r^{\textup{th}}$ canonical maps of smooth threefolds and fourfolds of general type have birationally bounded fibers for $r\geq 2$ and $r\geq 4$ respectively. Furthermore, we will generalize these results to higher dimensions in terms of the constants $r_n$ and we will discuss recent progress on a conjecture of Chen and Jiang.

A famous theorem of Y. T. Siu states that plurigenera of projective complex manifolds are invariant under deformation. The only known proof of this result uses deep techniques from complex analysis, which are not available in the algebraic category. In this talk, we will illustrate some recent progress toward an algebraic proof of Siu's result, and explain how these methods can be used to prove analogous results in positive and mixed characteristic

Bootstrap method, being an alternative of
statistical inference based on normal distribution, is popular in
modern time statistics. In this talk, I will introduce the theoretical
background of bootstrap algorithm and demonstrate how to use bootstrap
methods to deal with high dimensional inference problem, like
confidence interval for high dimensional mean and high dimensional
Lasso.

For a fixed set of positive integers $R$, we say $\mathcal{H}$ is an $R$-uniform hypergraph, or $R$-graph, if the cardinality of each edge belongs to $R$. For a graph $G=(V,E)$, a hypergraph $\mathcal{H}$ is called a \textit{Berge}-$G$, denoted by $BG$, if there is an injection $i\colon V(G)\to V(\mathcal{H})$ and a bijection $f\colon E(G) \to E(\mathcal{H})$ such that for all $e=uv \in E(G)$, we have $\{i(u), i(v)\} \subseteq f(e)$. We present some recent results about extremal problems on Berge hypergraphs from the perspectives of the shadow graph. In particular, we define variants of the Ramsey number and Tur\'an number in Berge hypergraphs, namely the \emph{cover Ramsey number} and \emph{cover Tur\'an number}, and show some general lower and upper bounds on these variants. We also determine the cover Tur\'an density of all graphs when the uniformity of the host hypergraph equals to $3$. These results are joint work with Linyuan Lu.

Free probability provides a unifying framework for studying random multi-matrix models in the large $N$ limit. Typically, the purview of these techniques is limited to invariant or mean-field ensembles. Nevertheless, we show that random band matrices fit quite naturally into this framework. Our considerations extend to the infinitesimal level, where finer results can be stated for the $\frac{1}{N}$ correction. As an application, we consider the question of outliers for finite-rank perturbations of our model. In particular, we find outliers at the classical positions from the deformed Wigner ensemble. No background in free probability is assumed.

The theory of perfectoid spaces was initially developed by Scholze to
prove new cases of the weight-monodromy conjecture. He constructed
perfectoid covers of toric varieties that allowed him to translate
results from characteristic p to characteristic 0. We will give an
overview of Scholze's method, then explain how to use an analogous
construction for abelian varieties to prove the weight-monodromy
conjecture for complete intersections in abelian varieties.

This talk is based on joint work with Slim Ibrahim, Nader
Masmoudi and Federica Sani. The Trudinger-Moser inequality gives
uniform exponential integrability in place of the (failed) critical
Sobolev embedding. In this talk, we consider existence of maximizers
for general nonlinearity of the optimal growth on the disk and on the
whole plane, respectively. The problem is delicate because
concentration of energy may or may not happen depending on lower order
nonlinearity. We derive a very sharp threshold between existence and
non-existence cases for the nonlinearity in an explicit asymptotic
expansion.

Not only is queueing theory important for understanding congestion in
the modern world, but examples that arise in queueing theory motivate
interesting mathematical problems. I'm here to talk with you about these
problems: why randomness is so important in making an accurate model, the
use of measure-valued random variables, and the mysteries behind the term
'scaling limit.'

In this talk we will introduce the basic definitions and tools for understanding One-dimensional Stochastic Differential Equations. We will discuss well-posedness, the scale function, the speed measure, hitting times, and Feller's Test: the definitive characterization of explosion for these equations. Relevant examples will be shown to motivate. If time permits, we will introduce Feller's boundary classification.

We investigate infinite highly arc transitive digraphs with two additional properties, descendant-homogeneity and Property $Z$.
A digraph $D$ is {\itshape {highly arc transitive}} if for each $s \geq 0$ the automorphism group of $D$ is transitive on the set of directed paths of length $s$; and $D$ is {\itshape {descendant-homogeneous}} if any isomorphism between finitely generated subdigraphs of $D$ extends to an automorphism of $D$. A digraph is said to have {\itshape {Property $Z$}} if it has a homomorphism onto a directed line.
We show that if $D$ is a highly arc transitive descendant-homogeneous digraph with Property $Z$ and $F$ is the subdigraph spanned by the descendant set of a directed line in $D$, then $F$ is a locally finite 2-ended digraph with equal in- and out-valencies. If, moreover, $D$ has prime out-valency then $F$ is isomorphic to the digraph $\Delta_p$. This knowledge is then used to classify the highly arc transitive descendant-homogeneous digraph of prime out-valency which have Property $Z$.

The asymptotics of the second-largest eigenvalue in random regular graphs (also referred to as the ``Alon conjecture'') have been computed by Joel Friedman in his celebrated 2004 paper. Recently, a new proof of this result has been given by Charles Bordenave, using the non-backtracking operator and the Ihara-Bass formula. In the same spirit, we have been able to translate Bordenave's ideas to bipartite biregular graphs in order to calculate the asymptotical value of the second-largest pair of eigenvalues, and obtained a similar spectral gap result. Applications include community detection in equitable graphs or frames, matrix completion, and the construction of channels for efficient and tractable error-correcting codes (Tanner codes). This work is joint with Gerandy Brito and Kameron Harris.

A 2-ton interactive sculpture came to life at Burning Man 2018, the world's most influential large-scale sculpture showcase. Rising 12 feet tall with an 18-foot wingspan in the Nevada desert, the unfolding dodecahedron was illuminated by 16,000 LEDs, requiring 6,500 person hours and \$50,000 in funds.

Its interior, large enough to hold 15 people, was fully lined with massive mirrors, alluding to a possible shape of our universe. The unfolding exterior points to the 500-year-old work of Albrecht D$\ddot{\text{u}}$rer, and the tantalizing open problem of discovering a geometric unfolding for every convex polyhedron. We discuss the state-of-the-art, and consider higher-dimensional unfolding analogs, with elegant geometric and combinatorial relationships.

In this talk, we prove sharp lower bound of the first nonzero eigenvalue of the weighted $p$-Laplacian on compact Bakry-Emery manifolds, without boundary or with convex boundary and Neuman boundary condition. This is joint work with Kui Wang.

Sampling Gibbs measures at low temperature is a very important task but computationally very challenging. Numeric evidence suggest that the infinite-swapping algorithm (isa) is a promising method. The isa can be seen as an improvement of replica methods which are very popular. We rigorously analyze the ergodic properties of the isa in the low temperature regime deducing Eyring-Kramers formulas for the spectral gap (or Poincar\'e constant) and the log-Sobolev constant. Our main result shows that the effective energy barrier can be reduced drastically using the isa compared to the classical over-damped Langevin dynamics. As a corollary we derive a deviation inequality showing that sampling is also improved by an exponential factor. Furthermore, we analyze simulated annealing for the isa and show that isa is again superior to the over-damped Langevin dynamics. This is joint work with Georg Menz, Andr\'e Schlichting and Tianqi Wu.

In this series of lectures we will describe the construction
of a $G$-equivariant $L$-function $Theta^E_{K/F}(s)$ associated to an abelian
extension $K/F$ of characteristic $p$ global fields of Galois group $G$ and a
suitable Drinfeld module $E$ defined over $F$, as well as state and sketch the
proof of a theorem linking the special value
$Theta^E_{K/F}(0)$ to a quotient of volumes of certain compact topological
spaces canonically associated to the pair $(K/F, E)$.

In lecture I (1-2pm), Cristian will define the $L$--function, give an
arithmetic interpretation of its special value at $s=0$ and state the
main theorem.

In lecture II (2-3pm), Joe will introduce the main ingredients involved
in the proof of the main theorem and sketch the main ideas of proof.

These lectures describe joint work of J. Ferrara, N. Green, Z. Higgins and C.
Popescu. The results within generalize to the Galois equivariant setting
earlier work of L. Taelman on special values of Goss zeta functions associated
to Drinfeld modules (Taelman, Annals of Math. 2010).

Let $(X, T^{1, 0}X)$ be a compact CR manifold and $(L, h)$ be a Hermitian CR line bundle over $X$. When $X$ is Levi-flat and $L$ is positive, Ohsawa and Sibony constructed for every $\kappa\in\mathbb N$ a CR projective embedding of $C^\kappa$-smooth
of the Levi-flat CR manifold. Adachi constructed a counterexample to show that the $C^k$-smooth can not be generalized to $C^\infty$-smooth. The difficulty comes from the fact that the Kohn Laplacian is not hypoelliptic on Levi flat manifolds.

In this talk, we will consider CR manifold $X$ with a transversal CR $G$-action where $G$ is a compact Lie group and $G$ can be lifted to a CR line bundle $L$ over $X$.
The talk will be divided into two parts. In the first part, we will talk about the Morse inequalities for the Fourier components of Kohn-Rossi cohomology on CR manifolds with transversal CR $S^1$-action.
By studying the partial Szeg\"o kernel on $(0

A sunflower with $r$ petals is a collection of $r$ sets so
that the intersection of each pair is equal to the intersection of
all. Erdos and Rado in 1960 proved the sunflower lemma: for any fixed
$r$, any family of sets of size $w$, with at least about $w^w$ sets,
must contain a sunflower. The famous sunflower conjecture is that the
bound on the number of sets can be improved to $c^w$ for some constant
$c$. Despite much research, the best bounds until recently were all of
the order of $w^{cw}$ for some constant c. In this work, we improve
the bounds to about $(log w)^w$.

There are two main ideas that underlie our result. The first is a
structure vs pseudo-randomness paradigm, a commonly used paradigm in
combinatorics. This allows us to either exploit structure in the given
family of sets, or otherwise to assume that it is pseudo-random in a
certain way. The second is a duality between families of sets and DNFs
(Disjunctive Normal Forms). DNFs are widely studied in theoretical
computer science. One of the central results about them is the
switching lemma, which shows that DNFs simplify under random
restriction. We show that when restricted to pseudo-random DNFs, much
milder random restrictions are sufficient to simplify their structure.

Joint work with Ryan Alweiss, Kewen Wu and Jiapeng Zhang.

Suppose that an infinite subset $A$ of the natural numbers is
partitioned into finitely many subsets. A property of $A$ that is always
inherited by at least one of the elements of the partition is known as a
partition regular property. Suppose we have a method of measuring the size
of an infinite subset of the natural numbers. A property $P$ is said to be
a density property if every infinite subset of the natural numbers which
is large enough, according to our yardstick, satisfies $P$. In the
qualitative sense, partition regular properties guarantee that order is
conserved and density properties guarantee that order is always achieved
at a threshold size. We will provide some interesting examples and
applications of partition regular properties, density properties, and
related ideas.

We establish non-asymptotic concentration bound and Bahadur representation for quantile regression estimator in the random design setting. Smoothed quantile regression is then proposed with fast computation and high estimation accuracy. Finally, we provide rigorous theoretical guarantees for the validity of inference via multiplier bootstrap.

Consider a rational curve, described by a map f :$P^1 \to P^n$. The
Shapiro--Shapiro conjecture says the following: if all the inflection
points of the curve (the roots of the Wronskian of f) are real, then the
curve itself is defined by real polynomials (up to change of
coordinates). An equivalent statement is that certain real Schubert
varieties in the Grassmannian intersect transversely -- a fact with
broad combinatorial and topological consequences. The conjecture, made
in the 90s, was proven by Mukhin--Tarasov--Varchenko in '05/'09 using
methods from quantum mechanics.

I will present a generalization of the Shapiro--Shapiro conjecture,
joint with Kevin Purbhoo, where we allow the Wronskian to have complex
conjugate pairs of roots. We decompose the real Schubert cell according
to the number of such roots and define an orientation of each connected
component. For each part of this decomposition, we prove that the
topological degree of the restricted Wronski map is given by a symmetric
group character. In the case where all the roots are real, this implies
that the restricted Wronski map is a topologically trivial covering map;
in particular, this gives a new proof of the Shapiro-Shapiro conjecture.

Consider a rational curve, defined by a map $f:\mathbf{P}^1\rightarrow\mathbf{P}^n$. The Shapiro-Shapiro conjecture says the following: if all the inflection points of the curve (the roots of the Wronskian of $f$) are real, then the curve itself is defined by real polynomials (up to change of coordinates). An equivalent statement is that certain real Schubert varieties in the Grassmannian intersect transversely - a fact with broad combinatorial and topological consequences. The conjecture, made in the 90s, was proven by Mukhin-Tarasov-Varchenko in '05/'09 using methods from quantum mechanics. I will present a generalization of the Shapiro-Shapiro conjecture, joint with Kevin Purbhoo, where we allow the Wronskian to have complex conjugate pairs of roots. We decompose the real Schubert cell according to the number of such roots and define an orientation of each connected component. For each part of this decomposition, we prove that the topological degree of the restricted Wronski map is given by a symmetric group character. In the case where all the roots are real, this implies that the restricted Wronski map is a topologically trivial covering map; in particular, this gives a new proof of the Shapiro-Shapiro conjecture.

The cohomology groups of complex algebraic
varieties come equipped with a powerful but intrinsically analytic
invariant called a Hodge structure. The fact that Hodge structures of
certain very special algebraic varieties are nonetheless parametrized
by algebraic varieties has led to many important applications in
algebraic and arithmetic geometry. While this fails in general,
recent joint work with Y. Brunebarbe, B. Klingler, and J. Tsimerman
shows that parameter spaces of Hodge structures always admit a "tame"
analytic structure in a sense made precise using ideas from model
theory. A salient feature of the tame analytic category is that it
allows for the local flexibility of the full analytic category while
preserving the global behavior of the algebraic category.

In this talk I will explain this perspective as well as some important
applications, including an easy proof of a celebrated theorem of
Cattani--Deligne--Kaplan on the algebraicity of Hodge loci and the
resolution of a longstanding conjecture of Griffiths on the
quasiprojectivity of the images of period maps.

A unitary representation $\pi\colon G\to B(H)$ of a locally compact group $G$ is an \emph{$L^p$-representation} if $H$ admits a dense subspace $H_0$ so that the matrix coefficient

$ G\ni s\mapsto \langle \pi(s)\xi,\xi\rangle$

belongs to $L^p(G)$ for all $\xi\in H_0$. The \emph{$L^p$-C*-algebra} $C^*_{L^p}(G)$ is the C*-completion $L^1(G)$ with respect to the C*-norm

$ \|f\|_{C^*_{L^p}}:=\sup\{\|\pi(f)\| : \pi\textnormal{ is an }L^p\textnormal{-representation of $G$}\}\qquad (f\in L^1(G)).$

Surprisingly, the C*-algebra $C^*_{L^p}(G)$ is intimately related to the enveloping C*-algebra of the Banach $*$-algebra $PF^*_p(G)$ ($2\leq p\leq \infty$). Consequently, we characterize the states of $C^*_{L^p}(G)$ as corresponding to positive definite functions that ``almost'' belong to $L^p(G)$ in some suitable sense for ``many'' $G$ possessing the Haagerup property, and either the rapid decay property or Kunze-Stein phenomenon. It follows that the canonical map

$$ C^*_{L^p}(G)\to C^*_{L^{p'}}(G)$$

is not injective for $2\leq p' \leq p \leq \infty$ when $G$ is non-amenable and belongs to the class of groups mentioned above. As a byproduct of our techniques, we give a near solution to a 1978 conjecture of Cowling.

This is primarily based on joint work with E. Samei.

Adaptive cubic regularization methods have several favorable properties for nonconvex optimization. In particular, under mild assumptions, they are globally convergent to a second-order stationary point. In this talk, I will introduce an adaptive cubic regularization method for unconstrained optimization. Methods analogous to those used to solve the trust-region subproblem will be discussed for solving the local cubic model. Some numerical results will be presented that compare a cubic regularized Newton's method, a standard trust-region method and a trust-search method.

For every fixed $s$ and large $n$, we describe all values of $n_1,\ldots,n_s$ such that for every $2$-edge-coloring of the complete $s$-partite graph $K_{n_1,\ldots,n_s}$ there exists a monochromatic (i) cycle $C_{2n}$ with $2n$ vertices, (ii) cycle $C_{\geq 2n}$ with at least $2n$ vertices, (iii) path $P_{2n}$ with $2n$ vertices, and (iv) path $P_{2n+1}$ with $2n+1$ vertices. This implies a generalization of the conjecture by Gy\' arf\' as, Ruszink\' o, S\' ark\H ozy and Szemer\' edi that for every $2$-edge-coloring of the complete $3$-partite graph $K_{n,n,n}$ there is a monochromatic path $P_{2n+1}$.

An important tool is our recent stability theorem on monochromatic connected matchings (A matching $M$ in $G$ is connected if all the edges of $M$ are in the same component of $G$). We will also talk about exact Ramsey-type bounds on the sizes of monochromatic connected matchings in $2$-colored multipartite graphs. Joint work with J\' ozsef Balogh, Alexandr Kostochka and Mikhail Lavrov.

For each regular coadjoint orbit of a compact group, we construct an exhaustion by symplectic embeddings of toric domains. As a by-product we arrive at a conjectured formula for the Gromov width of coadjoint orbits. Our method combines ideas from Poisson-Lie groups and from the geometric crystals of Berenstein-Kazhdan. We also prove similar results for multiplicity-free spaces. This is joint work with A. Alekseev, J. Lane, and Y. Li.

The classical uniformization theorem transforms the study of moduli spaces of marked Riemann surfaces into the study of constant curvature metrics with singularities. I will give a survey on constant curvature metrics with cusp and conical singularities, including works joint with Richard Melrose, Rafe Mazzeo and Bin Xu, where new analytic tools have been developed to understand the uniformization. ``Resolution of singularities'' is the key idea in the analysis, which can be seen as an analogue of the Deligne--Mumford compactification of Riemann moduli spaces.

In this talk, we will discuss recent results on stable self-similar singularity formation for the 2D Boussinesq and singularity formation for the 3D Euler equations in the presence of the boundary with $C^{1,alpha}$ initial data for the velocity field that has finite energy. The blowup mechanism is based on the Hou-Luo scenario of a potential 3D Euler singularity. We will also discuss some 1D models for the 3D Euler equations that develop stable self-similar singularity in finite time. For these models, the regularity of the initial data can be improved to $C_c^{infty}$. Some of the results are joint work with Thomas Hou and De Huang.

Contact manifolds, which arise naturally in mechanics, dynamics, and geometry, carry natural Riemannian and sub-Riemannian structures and it was shown by Gromov that the latter can be obtained as a limit of the former. Subsequently, Rumin found a complex of differential forms reflecting the contact structure that computes the singular cohomology of the manifold. He used this complex to describe the behavior of individual eigenvalues of the Riemannian Hodge Lapacians in the sub-Riemannian limit but was unable to determine the behavior of global spectral invariants. I will report on joint work with Hadrian Quan in which we determine the global behavior of the spectrum by explaining the structure of the heat kernel along this limit in a uniform way.

In this talk we shall look at an overview of discrete Morse theory in the context of simplicial complexes. Discrete Morse theory, based on the work by R. Forman, provides a framework to study the ``shape'' (i.e. the topology) of a simplicial complex via discrete Morse functions which are real valued functions defined on the simplices of the complex. The critical simplices, which are determined by the respective discrete Morse function, reveal key topological features of the simplicial complex. This is, in essence, a discrete adaptation of Morse theory in differential topology which allows us to study the topology of a manifold by looking at the differentiable functions on the manifold. The talk will cover the basics of discrete Morse theory with multiple examples and will also discuss possible applications in the context of persistent homology.

In this talk, we will explore various $q$ analogs of previous results from the seminar. The main
result will be the vector space analog of a Theorem of Frankl and Wilson. We will also discuss
some applications to constructive lower bounds on Ramsey numbers. The talk will be based off a
paper with the same title written by Peter Frankl and Ronald Graham.

An ordinary Gushel-Mukai fourfold $X$ is a smooth quadric section of a linear section of the Grassmannian $G(2,5)$. Kuznetsov and Perry proved that the bounded derived category of $X$ admits a semiorthogonal decomposition whose non-trivial component is a subcategory of K3 type. In this talk I will report on a joint work in progress with Alex Perry and Laura Pertusi, in which we construct Bridgeland stability conditions on the K3 subcategory of $X$. Then I will explain some applications concerning the existence of a homological associated K3 surface, and related algebraic constructions in hyperkaehler geometry.

Ideas from theoretical physics have had a
profound impact on geometry, topology, and representation theory over
the last several decades. An early high point of this interaction was
Witten's quantum field theoretic interpretation of the celebrated
Donaldson invariants, which in turn opened the door to his discovery
of the even-more-celebrated Seiberg-Witten invariants. In this talk,
we'll explain how more recently this interaction has made possible
dramatic advances in geometric representation theory, with a focus on
joint work with Sabin Cautis revealing the structure of the coherent
Satake category of a complex Lie group. This is an intricate cousin of
the constructible Satake category appearing in the geometric Satake
equivalence, a cornerstone of the geometric Langlands program. The
coherent Satake category turns out to have rich connections to the
Fomin-Zelevinsky theory of cluster algebras, as well as to the
representation theory of quantum groups and quiver Hecke
algebras. However, while these connections can be stated in purely
mathematical terms, their discovery hinged crucially on first
understanding how to interpret the coherent Satake category in terms
of physics --- in fact, the very same physics (4d N=2 supersymmetric
Yang-Mills theory) behind the Donaldson and Seiberg-Witten invariants.

In this talk, we will introduce the basic concept of the neural network
and discuss one specific type of neural network: Recurrent Neural
Network (RNN). We will discuss the backpropagation algorithm and analyze
the structure of RNN. Furthermore, we will introduce LSTM neural
network. If time permits, an example of time series forecasting by this
method will be provided.

It is a classical fact that free (discrete) groups can be embedded
in $GL_{2}(\mathbb{Z})$. In 1987, Zubkov showed that for a free pro-$p$
group $F_{\hat{p}}$, the situation changes, and when $p>2$, $F_{\hat{p}}$
cannot be embedded in $GL_{2}(\Delta)$ when $\Delta$ is a profinite
ring. In 2005, inspired by Kemer's solution to the Specht problem,
Zelmanov sketched a proof for the following generalization: For every
$d\in\mathbb{N}$ and large enough prime $p\gg d$, $F_{\hat{p}}$
cannot be embedded in $GL_{d}(\Delta)$.

The natural question then is: What can be said when $p$ is not large
enough? What can be said in the case $d=p=2$ ? In the talk I am going
to describe the proof of the following theorem: $F_{\hat{2}}$ cannot
be embedded in $GL_{2}(\Delta)$ when $char(\Delta)=2$. The main
idea of the proof is the use of trace identities in order to apply
finiteness properties of a Noetherian trace ring through the Artin-Rees
Lemma (Joint with E. Zelmanov).

Schubert calculus studies the algebraic geometry and combinatorics of matrix factorizations. I will discuss recent developments in $K$-theoretic Schubert calculus, and their connections to problems in combinatorics and representation theory.

From a probabilistic perspective, 2D quantum gravity is the study of natural probability measures on the space of all possible geometries on a topological surface. One natural approach is to take scaling limits of discrete random surfaces. Another approach, known as Liouville quantum gravity (LQG), is via a direct description of the random metric under its conformal coordinate. In this talk, we review both approaches, featuring a joint work with N. Holden proving that uniformly sampled triangulations converge to the so called pure LQG under a certain discrete conformal embedding.

Since the ground breaking work of Donaldson in the 1980s, topologists has achieved huge success in using gauge theory to study smooth 4-manifolds with nonzero second homology. The case of 4-manifolds with trivial second homology is relatively less known. In particular, when the 4-manifold have the same homology as S1 cross S3, there are several gauge theoretic invariants. The first one is the Casson-Seiberg-Witten invariant LSW(X) defined by Mrowka-Ruberman-Saveliev; the second one is the Fruta-Ohta invariant LFO(X). It is conjecture that these two invariants are equal to each other (This is an analogue of Witten's conjecture relating Donaldson and Seiberg-Witten invariants.) In this talk, I will recall the definition of these two invariants, give some applications of them (including a new obstruction for metric with positive scalar curvature), and sketch a proof of this conjecture for finite-order mapping tori. This is based on a joint work with Danny Ruberman and Nikolai Saveliev.

Since the ground breaking work of Donaldson in the 1980s, topologists has achieved huge success in using gauge theory to study smooth 4-manifolds with nonzero second homology. The case of 4-manifolds with trivial second homology is relatively less known. In particular, when the 4-manifold have the same homology as S1 cross S3, there are several gauge theoretic invariants. The first one is the Casson-Seiberg-Witten invariant LSW(X) defined by Mrowka-Ruberman-Saveliev; the second one is the Fruta-Ohta invariant LFO(X). It is conjecture that these two invariants are equal to each other (This is an analogue of Witten's conjecture relating Donaldson and Seiberg-Witten invariants.) In this talk, I will recall the definition of these two invariants, give some applications of them (including a new obstruction for metric with positive scalar curvature), and sketch a proof of this conjecture for finite-order mapping tori. This is based on a joint work with Danny Ruberman and Nikolai Saveliev.

Motivated by problems in distributed optimization and computation, we discuss a generalization of the Perron-Frobenius Theorem to products of random stochastic matrices. To do so, we introduce several objects such as infinite flow graph, infinite flow property, and show the connection of these concepts to ergodicity of chains of random stochastic matrices.

Fractal uncertainty principle states that no function can be localized to a fractal set simultaneously in position and in frequency. The strongest version so far has been obtained in one dimension by Bourgain and the speaker with recent higher dimensional advances by Han and Schlag.

I will present two applications of the fractal uncertainty principle. The first one (joint with Jin and Nonnenmacher) is a frequency-independent lower bound on mass of eigenfunctions on compact negatively curved surfaces, which in particular implies control for the Schr$\ddot{\text{o}}$dinger equation by any nonempty open set. The second application (joint with Zahl) is an essential spectral gap for convex co-compact hyperbolic surfaces, which implies exponential energy decay of high frequency waves.

We show that the average size of the automorphism group over
$\mathbb{F}_q$ of a smooth degree $d$ hypersurface in
$\mathbb{P}^{n}_{\mathbb{F}_q}$ is equal to $1$ as $d\rightarrow
\infty$. We also discuss some consequences of this result for the moduli
space of smooth degree $d$ hypersurfaces in $\mathbb{P}^n$.

It is a classical theorem of Roth that every dense subset of $\left\{1,\ldots,N\right\}$ contains a nontrivial three-term arithmetic progression. Quantitatively, results of Sanders, Bloom, and Bloom-Sisask tell us that subsets of relative density at least $1/(\log N)^{1-\epsilon}$ already have this property. In this talk, we will discuss about some sets of $N$ integers which unlike $\left\{1,\ldots,N\right\}$ do not contain nontrivial four-term arithmetic progressions, but which still have the property that all of their subsets of density at least $1/(\log N)^{1-\epsilon}$ must contain a three-term arithmetic progression. Perhaps a bit surprisingly, these sets turn out not to have as many three-term progressions as one might be inclined to guess, so we will also address the question of how many three-term progressions can a four-term progression free set may have. Finally, we will also discuss about some related results over $\mathbb{F}_{q}^n$. Based on joint works with Jacob Fox and Oliver Roche-Newton.

Modular forms are holomorphic functions invariant under a
certain group action. They have a surprising amount of number theoretic
information. We introduce their basic theory and explain how their
connection to Galois theory can be used to study congruences between
modular forms.

In this talk, we present recent results on the geometry of centrally-symmetric random polytopes, generated by $N$ independent copies of a random vector $X$ taking values in ${\mathbb{R}}^n$. We show that under minimal assumptions on $X$, for $N \gtrsim n$ and with high probability, the polytope contains a deterministic set that is naturally associated with the random vector -- namely, the polar of a certain floating body. This solves the long-standing question on whether such a random polytope contains a canonical body.

Moreover, by identifying the floating bodies associated with various random vectors we recover the estimates that have been obtained previously, and thanks to the minimal assumptions on $X$ we derive estimates in cases that had been out of reach, involving random polytopes generated by heavy-tailed random vectors (e.g., when $X$ is $q$-stable or when $X$ has an unconditional structure). Finally, the structural results are used for the study of a fundamental question in compressive sensing -- noise blind sparse recovery.

This is joint work with the speaker's PhD student Christian K$\ddot{\text{u}}$mmerle (now at Johns Hopkins University) as well as Olivier Gu{\'e}don (University of Paris-Est Marne La Vall{\'e}e), Shahar Mendelson (Sorbonne University Paris), and Holger Rauhut (RWTH Aachen).

The 2-party political system defines a natural partition of a network of individuals into 2 teams. One can view these individuals as players in a network-sized game, and the utility (or equivalently cost) functions for each player can be realized as wanting to be connected to individuals on the same political party and distanced from those in the opposing political party. When considered from a game theoretic point of view, the ``greedy'' (or myopic/optimal) strategies can be examined. Thus, the game turns into a dynamical system, which can be investigated to understand when each political party will become totally connected. When the original network is sampled from an Erdos-Renyi graph G(n, q), we find a one-sided threshold of when a political party will become completely connected.

Until recently, it was unclear if there was a natural way to construct (compactified) moduli spaces of Fano varieties. One approach to solving this problem is the K-moduli Conjecture, which predicts that K-polystable Fano varieties of fixed dimension and volume are parametrized by a projective good moduli space. In this talk, I will survey recent progress on this conjecture and discuss a result with Yuchen Liu and Chenyang Xu proving the openness of K-stability (a step in constructing K-moduli spaces).

Invariant random subgroups (IRS) are conjugacy invariant probability measures on the space of subgroups of a given locally compact group G. They arise naturally as point stabilizers of probability measure preserving actions. Invariant random subgroups can be regarded as a generalization both of normal subgroups and of lattices in topological groups. As such, it is interesting to extend results from the theories of normal subgroups and of lattices to the IRS setting.
Stuck-Zimmer proved that for higher rank simple Lie groups, any nontrivial IRS comes from a lattice. In rank 1 however the situation is far more complex. Indeed, the space of invariant random subgroups of $SL_{2}R$ contains all moduli spaces of Riemann surfaces, and can be used to obtain an interesting compactification thereof related to the Deligne-Mumford compactification.

Nevertheless, jointly with Arie Levit, we prove a different type of rigidity result valid in the rank 1 setting. We show that the critical exponent (exponential growth rate) of an infinite IRS in an isometry group of a Gromov hyperbolic space (such as a rank 1 Lie group, or a hyperbolic group) is almost surely greater than half the Hausdorff dimension of the boundary. This can be reinterpreted by saying that for any probability measure preserving action of such a group, stabilizers are almost surely either trivial or ``very big''.
This generalizes an analogous result of Matsuzaki-Yabuki-Jaerisch for normal subgroups.
As a corollary, we obtain that if $\Gamma$ is a typical subgroup and $X$ a rank 1 symmetric space then $\lambda_{0}(X/\Gamma)\< \lambda_{0}(X)$ where $\lambda_0$ is the smallest eigenvalue of the Laplacian. The proof uses ergodic theorems for actions of hyperbolic groups.

We summarize symmetrization, contraction principles and Talagrand's concentration inequality with several refined versions for empirical process. These results serve as useful tools in statistical learning theory. Proof sketch with basic ideas will be discussed.

The computation of stable homotopy groups of
spheres is one of the most fundamental problems in topology.
Despite its simple definition, it is notoriously hard to compute.
It has connections to many areas of mathematics. In this talk, I
will discuss a recent breakthrough on this problem, which depends
on motivic homotopy theory in a critical way. I will also talk
about applications to smooth structures on spheres, and towards
the open problem of Kervaire invariant one in dimension 126. This
talk is based on several joint work with Bogdan Gheorghe, Daniel
Isaksen, and Guozhen Wang.

In this talk we will present several classical results on
finite-dimensional Leibniz algebras. We give main examples of Leibniz
algebras and show nilpotency of Leibniz algebras in terms of special
kinds of derivations. Also, we present the structure of solvable Lie
algebras with a given nilradical and with the maximality condition for
the complementary subspace to the nilradical. Moreover, among such
solvable Lie algebras we shall indicate a subclass of Lie algebras whose
cohomology group is trivial. Finally, we provide some examples of
infinite-dimensional Lie algebras with a similar structure.

Trefftz methods rely, in broad terms, on the idea of approximating solutions to PDEs using basis functions which are exact solutions of the Partial Differential Equation (PDE), making explicit use of information about the ambient medium. But wave propagation problems in inhomogeneous media is modeled by PDEs with variable coefficients, and in general no exact solutions are available. Generalized Plane Waves (GPWs) are functions that have been introduced, in the case of the Helmholtz equation with variable coefficients, to address this problem: they are not exact solutions to the PDE but are instead constructed locally as high order approximate solutions. We will discuss the origin, the construction, and the properties of GPWs. The construction process introduces a consistency error, requiring a specific analysis.

The Erd\H{o}s--Szekeres theorem can be interpreted as saying that in any red-blue edge-coloring of an ordered complete graph on $rs+1$ vertices, there is a red ordered path of length $r$ or a blue ordered path of length $s$. We consider the size Ramsey version of this problem and show that $\tilde{r}(P_r, P_s)$, the least number of edges in an ordered graph with this Ramsey property, satisfies
\[
\frac18 r^2 s \le \tilde{r}(P_r, P_s) \le C r^2 s (\log s)^3
\]
for any $2 \le r \le s$, where $C>0$ is a constant. This is joint work with J\'ozsef Balogh, Felix Clemen, and Emily Heath.

Differential graded algebra techniques have played a crucial role in the
development of homological algebra, especially in the study of
homological properties of commutative rings carried out by Serre, Tate,
Gulliksen, Avramov, and others. In our work, we extend the construction
of the Koszul complex and acyclic closure to a more general setting. As
an application of our constructions, we show that the Ext algebra of
quotients of skew polynomial rings by ideals generated by normal
elements is the universal enveloping algebra of a color Lie algebra, and
therefore a color Hopf algebra. As a consequence, we give a presentation
of the Ext algebra when the elements generating the ideal form a regular
sequence, this generalizes a theorem of Bergh and Oppermann. It follows
that in this case the Ext algebra is noetherian, providing a partial
answer to a question of Kirkman, Kuzmanovich and Zhang.

Minimal surfaces (critical points of the area functional) have a rich and successful history in the study of the interaction between geometry and topology that goes back to the 1960s. In practice, the presence and properties of minimal surfaces inside a Riemannian manifold profoundly influences the ambient geometry. In this talk, we will discuss how one can use the Allen--Cahn equation to guarantee the existence of a rich class of geometrically and topologically distinct minimal surfaces inside a generic Riemannian 3-manifold. As a byproduct, one obtains a pure PDE resolution of a number of previously unapproachable questions in minimal surface theory, which parallels recent simultaneous advances that instead use geometric measure theory.

In order to approximate the solution of a PDE by the finite element method, the problem domain is generally first subdivided into a collection of cells, like quadrilaterals or triangles. This collection of cells, called a mesh, has a direct impact on the accuracy of the numerical solution, and the properties of 2D and 3D meshes are very well studied. However, some new schemes for solving numerical PDEs require 4D meshes. These schemes, called Space-Time Finite Element Methods (STFEMs), treat time and the spatial variables in the same way when approximating PDEs. As a result, time-dependent problems in three spatial variables are considered in a four-dimensional space-time domain. This talk introduces techniques for creating and manipulating four-dimensional conforming simplicial meshes for use with STFEMs. The approach relies on the theory of combinatorial maps, which will be introduced and considered from the computational perspective. After a discussion of the
challenges and benefits of this approach, we present some initial results in creating space-time meshes.

n 1974, Erd\H{o}s and Rothchild conjectured that the complete bipartite graph has the maximum number of two-edge-colorings without monochromatic triangles over all n-vertex graphs. Since then, a new class of colored extremal problems has been extensively studied by many researchers on various discrete structures, such as graphs, hypergraphs, Boolean lattices and sets. In this talk, I will first give an overview of some previous results on this topic. The second half of this talk is to explore the rainbow variants of the Erd\H{o}s-Rothschild problem. With Jozsef Balogh, we confirm conjectures of Benevides, Hoppen and Sampaio, and Hoppen, Lefmann, and Odermann, and complete the characterization of the extremal graphs for the edge-colorings without rainbow triangles. Next, we study a similar question on sum-free sets, where we describe the extremal configurations for integer colorings with forbidden rainbow sums. The latter is joint work with Yangyang Cheng, Yifan Jing, Wenling Zhou and Guanghui Wang.

Modern machine learning algorithms have achieved remarkable performance in a myriad of applications, and are increasingly used to make impactful decisions in the hiring process, criminal sentencing, healthcare diagnostics and even to make new scientific discoveries. The use of data-driven algorithms in high-stakes applications is exciting yet alarming: these methods are extremely complex, often brittle, notoriously hard to analyze and interpret. Naturally, concerns have raised about the reliability, fairness, and reproducibility of the output of such algorithms. This talk introduces statistical tools that can be wrapped around any ``black-box'' algorithm to provide valid inferential results while taking advantage of their impressive performance. We present novel developments in conformal prediction and quantile regression, which rigorously guarantee the reliability of complex predictive models, and show how these methodologies can be used to treat individuals equitably. Next, we focus on reproducibility and introduce an operational selective inference tool that builds upon the knockoff framework and leverages recent progress in deep generative models. This methodology allows for reliable identification of a subset of important features that is likely to explain a phenomenon under-study in a challenging setting where the data distribution is unknown, e.g., mutations that are truly linked to changes in drug resistance.

The cellular cytoskeleton is essential in proper cell function as well as in organism development. These filaments represent the roads along which most protein transport occurs inside cells. I will discuss several examples where questions about filament-motor protein interactions require the development of novel mathematical modeling, analysis, and simulation.

In the development of egg cells into embryos, RNA molecules bind to and unbind from cellular roads called microtubules, switching between bidirectional transport, diffusion, and stationary states. Since models of intracellular transport can be analytically intractable, asymptotic methods are useful in understanding effective cargo transport properties as well as their dependence on model parameters. We consider these models in the framework of partial differential equations as well as stochastic processes and derive large time properties of cargo movement for a general class of problems. The proposed methods have applications to macroscopic models of protein localization and microscopic models of cargo movement by teams of motor proteins. I will also discuss an agent-based modeling and data analysis framework for understanding how actin filaments and myosin motors interact to form contractile ring channels essential in development. In particular, we propose tools drawing from topological data analysis to analyze time-series data of filament network interactions and illustrate the impact of key parameters on significant ring emergence, thus giving insight into formation and maintenance of these biological channels.

The homology cobordism group of integer homology three-spheres is a natural invariant of interest to four-dimensional topologists. In this talk, we recall its definition and give a short introduction to involutive Floer homology, As an application, we see that there is an infinite-rank summand of the homology cobordism group. This includes joint work with Irving Dai, Jen Hom, and Linh Truong.

The homology cobordism group of integer homology three-spheres is a natural invariant of interest to four-dimensional topologists. In this talk, we recall its definition and give a short introduction to involutive Floer homology, As an application, we see that there is an infinite-rank summand of the homology cobordism group. This includes joint work with Irving Dai, Jen Hom, and Linh Truong.

The Stiefel manifold is the set of orthonormal bases for $k$-planes in an $n$-dimensional space. We compute its degree as an algebraic variety in the set of $k$-by-$n$ matrices using techniques from classical algebraic geometry, representation theory, and combinatorics. We give an interpretation of this degree in terms of non-intersecting lattice paths. This is joint work with Fulvio Gesmundo.

The Landau equation is a mesoscopic model in plasma physics that describes the evolution in phase-space of the density of colliding particles. Due to the non-local, non-linear terms in the equation, an understanding of the existence, uniqueness, and qualitative behavior of solutions has remained elusive except in some simplified settings (e.g., homogeneous or perturbative). In this talk, I will report on recent progress in the application of ideas of parabolic regularity theory to this kinetic equation. Using these ideas we can, in contrast to previous results requiring boundedness of fourth derivatives of the initial data, construct solutions with low initial regularity (just $L^\infty$) and show they are smooth and bounded for all time as long as the mass and energy densities remain bounded. This is a joint work with S. Snelson and A. Tarfulea.

Let $C$ be a curve over a finite field and let $\rho$ be a
nontrivial representation of $\pi_1(C)$. By the Weil conjectures, the
Artin $L$-function associated to $\rho$ is a polynomial with algebraic
coefficients. Furthermore, the roots of this polynomial are
$\ell$-adic units for $\ell \neq p$ and have Archemedian absolute
value $\sqrt{q}$. Much less is known about the $p$-adic properties of
these roots, except in the case where the image of $\rho$ has order
$p$. We prove a lower bound on the $p$-adic Newton polygon of the
Artin $L$-function for any representation in terms of local monodromy
decompositions. If time permits, we will discuss how this result
suggests the existence of a category of wild Hodge modules on Riemann
surfaces, whose cohomology is naturally endowed with an irregular
Hodge filtration.

The presentation will start by reviewing data on the current status of gender equity in science, technology, engineering and mathematics (STEM) disciplines and summarizing social science research that illuminates some causes of gender disparities in STEM. With this context established, the focus will shift to how women enter into leadership roles in academic settings, what they experience and how gender impacts the way they exercise their authority. The final part of the talk will discuss how we can all contribute to changing the face of leadership for the future, to the benefit of all of us in STEM.

The Fontaine-Wintenberger theorem relates the absolute Galois group
of $\mathbb{Q}_p(\mu_{p^{\infty}})$ to the absolute Galois group of a
certain local field over $\mathbb{F}_p$. Such an equivalence enables us to
understand the Galois representation of $\mathbb{Q}_p$ with some extra
work. In this talk I will explain several key ideas behind its classical
proof and its generalization via Faltings's almost purity theorem if time
permits.

The McKay correspondence takes many guises but at its core connects the
geometry of minimal resolutions for quotient singularities $C^n / G$ to
the representation theory of the group $G$. When $G$ is an abelian subgroup
of $SL(3)$, Craw-Ishii showed that every minimal resolution can be
realised as a moduli space of stable quiver representations naturally
associated to $G$, although the chamber structure for the stability
parameter and associated wall-crossing behaviour is poorly understood. I
will describe my recent work giving explicit representation-theoretic
descriptions of the walls and wall-crossing behaviour for the chamber
corresponding to a particular minimal resolution called the G-Hilbert
scheme. Time permitting, I will also discuss ongoing work with Yukari
Ito (IPMU) and Tom Ducat (Bristol) to better understand the geometry,
chambers, and corresponding representation theory for other minimal
resolutions.

Causal inference from observational data is a vital problem, but it
comes with strong assumptions. Most methods assume that we observe all
confounders, variables that affect both the causal variables and the
outcome variables. But whether we have observed all confounders is a
famously untestable assumption. We describe the deconfounder, a way to
do causal inference from observational data allowing for unobserved
confounding.

How does the deconfounder work? The deconfounder is designed for
problems of multiple causal inferences: scientific studies that
involve many causes whose effects are simultaneously of interest. The
deconfounder uses the correlation among causes as evidence for
unobserved confounders, combining unsupervised machine learning and
predictive model checking to perform causal inference. We study the
theoretical requirements for the deconfounder to provide unbiased
causal estimates, along with its limitations and tradeoffs. We
demonstrate the deconfounder on real-world data and simulation
studies.

In a perforated domain, the asymptotic behavior of the fluid motion depends on the rate (inter-hole distance)/(size of the holes). We will present the standard framework and explain how to find the critical rate where "strange terms" appear for the Laplace and Navier-Stokes equations. Next, we will study Euler equations where the critical rate is totally different than for parabolic equations. These works are in collaboration with V.Bonnaillie-Noel, M.Hillairet, N.Masmoudi, C.Wang and D.Wu.

The planarity testing problem is the algorithmic problem of testing whether
a given input graph is planar, that is, whether it can be drawn in the
plane without edge crossings. Clustered planarity (c-planarity, for short)
was introduced in 1995 by Feng, Cohen, and Eades as a
generalization of graph planarity, in which the vertex set of the
input graph is endowed with a
hierarchical clustering and we seek an embedding (edge crossing-free
drawing) of the graph in the
plane that respects the clustering in a certain natural sense.

A seemingly unrelated problem of thickenability for simplicial complexes
emerged in the topology of manifolds in the 1960s. A 2-dimensional
simplicial complex is thickenable if it embeds in some orientable
3-dimensional manifold.

We study the atomic embeddability testing problem, which is a common
generalization of clustered planarity and thickenability
testing, and present a polynomial time algorithm for this problem,
thereby giving the first polynomial time algorithm for c-planarity.

Until now, it has been an open problem whether c-planarity can be
tested efficiently, despite relentless efforts. Recently, Carmesin announced
that thickenability can be tested in polynomial time.
Our algorithm for atomic embeddability combines ideas from Carmesin's
work with algorithmic tools previously developed for so-called weak
embeddability testing.

Joint work with Csaba Toth.