We introduce a class of non-commutative curves and study their derived categories of coherent sheaves. In rational case we construct tilting which establishes their derived equivalence with some finite dimensional algebras. In particular, we define the Rouquier dimension of these derived categories.

If G is a group acting on a vector space V by linear transformations, then the invariant polynomial functions on V form a ring. In this talk we will discuss upper bounds for the degrees of generators of this invariant ring. An example of particular interest is the action of the group $SL_n x SL_n$ on the space of m-tuples of n x n matrices by simultaneous left-right multiplication. In this case, Visu Makam and the speaker recently proved that invariants of degree at most $mn^4$ generate the invariant ring. We will explore an interesting connection between this result and the notion of noncommutative rank.

We study the long time behaviour of the heat kernel on Abelian covers of compact Riemannian manifolds. For manifolds without boundary work of Lott and Kotani-Sunada establishes precise long time asymptotics. Extending these results to manifolds with boundary reduces to a 'cute' eigenvalue minimization problem, which we resolve for a Dirichlet and Neumann boundary conditions. We will show how these results can be applied to studying the ``winding'' / ``entanglement'' of Brownian trajectories.

The hook length formula for d-complete posets states that the P-partition generating function for them is given by a product in terms of hook lengths. We give a new proof of the hook length formula using q-integrals. The proof is done by a case-by-case analysis consisting of two steps. First, we express the P-partition generating function for each case as a q-integral and then we evaluate the q-integrals. Several q-integrals are evaluated using partial fraction expansion identities and others are verified by computer. This is joint work with Meesue Yoo.

Schur Q-functions are a family of symmetric functions introduced by Schur in his study of projective representations of symmetric groups. They are obtained by putting t = -1 in the Hall-Littlewood functions associated to the root system of type A. (Schur functions are the t = 0 specialization.) This talk concerns symplectic Q-functions, which are obtained by putting t = -1 in the Hall-Littlewood functions associated to the root system of type C. We present several Pfaffian formulas for symplectic Q-functions similar to those for Schur Q-functions and give a tableau description. Also we discuss some conjectures including the positivity conjecture of structure constants.

We study tensor powers of rank 1 Drinfeld A-modules, where A is
the affine coordinate ring of an elliptic curve. Using the theory of
A-motives, we find explicit formulas for the A-action of these modules.
Then, by developing the theory of vector valued Anderson generating
functions, we give formulas for the coefficients of the logarithm and
exponential functions associated to these A-modules, as well as formulas
for the fundamental period. This allows us to relate function field zeta
values to evaluations of the logarithm function and prove transcendence
facts about these zeta values.

In the 1970's Stark made precise conjectures about the leading
term of the Taylor series expansion at $s=0$ of Artin $L$-functions,
refining Dirichlet's class number formula. Around the same time Barsky,
Cassou-Nogu\`{e}s, and Deligne and Ribet for totally real fields, along
with Katz for CM fields defined $p$-adic $L$-functions of ray class
characters. Since then Stark-type conjectures for these $p$-adic
$L$-functions have been formulated, and progress has been made in some
cases.

The goal of this talk is to discuss a new definition of a $p$-adic
$L$-function and Stark conjecture for a mixed signature character of a real
quadratic field. After stating the definition and conjecture, theoretical
and numerical evidence will be discussed.

When studying supersymmetric equations from physics, Tseng and Yau introduced several Laplacians on symplectic manifolds in 2012. These Laplacians behave different from usual ones in Rimannian case and Complex case. They contain both 2nd and 4th order operators. In this talk, we will discuss these ``symplectic Laplacians'' and their relations with cohomologies on compact symplectic manifolds with boundary. For this purpose, we will introduce new boundary conditions for differential forms on symplectic manifolds. Their properties and importance will be discussed.

A drawing of a graph on a surface is independently even if every pair of independent edges in the drawing crosses an even number of times. The $\mathbb{Z}_2$-genus of a graph $G$ is the minimum $g$ such that $G$ has an independently even drawing on the orientable surface of genus $g$. An unpublished result by Robertson and Seymour implies that for every $t$, every graph of sufficiently large genus contains as a minor a projective $t\times t$ grid or one of the following so-called $t$-Kuratowski graphs: $K_{3,t}$, or $t$ copies of $K_5$ or $K_{3,3}$ sharing at most $2$ common vertices. We show that the $\mathbb{Z}_2$-genus of graphs in these families is unbounded in $t$; in fact, equal to their genus. Together, this implies that the genus of a graph is bounded from above by a function of its $\mathbb{Z}_2$-genus, solving a problem posed by Schaefer and \v{S}tefankovi\v{c}, and giving an approximate version of the Hanani-Tutte theorem on surfaces.

I'll survey a circle of ideas around the question of measuring ``how irrational'' are various classes of non-rational varieties.

Minors and topological minors are two closely related graph containment relations that have attracted extensive attentions. Though giant breakthroughs on graph minors have been made over decades, several questions about these two relations remain open, especially for topological minors. This talk addresses part of our recent work in this direction, including a proof of Robertson's conjecture on well-quasi-ordering graphs by the topological minor relation, a complete characterization of the graphs having the Erdos-Posa property with respect to topological minors which answers a question of Robertson and Seymour, and a proof of Thomas' conjecture on half-integral packing. More open questions, such as Hadwiger's conjecture on graph coloring and its variations and relaxations, will be discussed in this talk.

Given a tuple $A=(A_1,\ldots,A_g)$ of symmetric matrices of the same size, the affine linear matrix polynomial $L(x):=I-\sum A_j x_j$ is a monic linear pencil. The solution set $S_L$ of the corresponding linear matrix inequality (LMI), consisting of those $x$ in $\mathbb R^g$ for which $L(x)$ is positive semidefinite (PsD), is called a spectrahedron. It is a convex semialgebraic subset of $\mathbb R^g$, and LMIs are ubiquitous in many areas: mathematical optimization, control theory, statistics, etc.

We study the question whether inclusion holds between two spectrahedra. Most of our results concern the case where the included spectrahedron is a hypercube, an NP-hard problem introduced and studied by Ben-Tal and Nemirovskii, who identified a tractable relaxation of the original problem. This relaxation is obtained by considering the inclusion problem for the corresponding ``matricial'' spectrahedra.

To estimate the error inherent in the relaxation we employ probabilistic methods and an old result of Rev. Simmons on flipping biased coins to obtain an elegant scalar optimization formula.

This is based on joint work with Bill Helton, Scott McCullough and Markus Schweighofer.

An intriguing feature of the explicit charged (Reissner-Nordstrom) or spinning (Kerr) black hole spacetimes is the existence of a regular Cauchy horizon, beyond which the Einstein equation loses its predictive power. The strong cosmic censorship conjecture of Penrose is a bold claim that, nevertheless, such a pathological behavior is nongeneric.

In this lecture, I will give a short introduction to general relativity and the strong cosmic censorship conjecture. Then I will describe my recent joint work with J. Luk, where we rigorously establish a version of this conjecture for the Einstein-Maxwell-(real)-scalar-field system in spherical symmetry, which has long been studied by physicists and mathematicians as a useful model for this problem.

The talk concerns (matrices of) noncommutative polynomials $f=f(x)$ from the perspective of free real algebraic geometry. There are several natural notions of a ``zero set'' of $f$. The one we study is the \textbf{free locus} of $f$, $(f)$, which is defined to be the union of hypersurfaces
$$\left\{X\in \operatorname{M}_n(k)^g:\det f(X)=0\right\}$$
over all $n\in\mathbb{N}$. The talk will describe a recent advance on characterizing when $(f_1) \subseteq (f_2)$ holds, which was mainly achieved using linear matrix pencils.

The latter have been for decades an important tool in noncommutative algebra and other areas, e.g. linear systems, automata theory and computational complexity. Given a monic matrix pencil $L=I+\sum_jA_jx_j$, we can evaluate it at an arbitrary tuple of matrices $X$ as
$$L(X)=I\otimes I+\sum_jA_j\otimes X_j.$$
The talk will be mostly concerned with singularity of these evaluations.

First we will give an algebraic certificate for $(L_1)\subseteq (L_2)$ to hold. Then we will consider a fundamental irreducibility theorem for $(L)$ which is obtained with the aid of invariant theory. Next we will apply the preceding results to factorization in the free algebra. Lastly, smooth points on $(L)$ will be related to one-dimensional kernels of $L(X)$, which leads to the free version of Kippenhahn's conjecture and improves existing Positivstellens\''{a}tze on free semialgebraic sets.

Iwasawa theory is a bridge between algebraic and analytic invariants attached to an arithmetic object,
for a given prime p. When this arithmetic object is an elliptic curve or a modular form, the primes
come in two flavors -- ordinary and supersingular. When p is ordinary, the theory has historically
been relatively well behaved. When p is supersingular, there are several difficulties, and we explain
how to address the difficulties involved in the case of elliptic curves, culminating in the proof of the
Main Conjecture. If time permits, we will sketch joint work in progress with Castella, Ciperiani, and
Skinner concerning main conjecture for weight-two modular forms.

Very often in algebraic geometry, the totality of geometric objects under some natural setting becomes again an algebraic variety, which is one the main object of study in algebraic geometry.
Hilbert scheme is one of such space parametrizing families of projective algebraic varieties having the same fixed Hilbert polynomial, i.e. sharing certain basic extrinsic attributes and intrinsic invariants.
In this talk we will start with the basic construction of the Hilbert scheme of projective algebraic curves due to Alexander Grothendieck.
We then proceed further and discuss about the current state of affairs especially on the irreducibility and the rigidity of the Hilbert scheme of smooth projective curves after reviewing a brief history of the study since the era of Italian school.

The process of converting analog signals, or continuous functions, to digital signals is a classical problem in signal processing. Many analog to digital converters follow the paradigm of taking lots of samples and then compressing afterwards. One might wonder if you could be more prudent and only take as many samples as you'd need to guarantee that you can reconstruct a given signal. That is, can you compress while simultaneously measuring a signal? Compressed sensing's hallmark achievement is proving that for a large class of structured signals this is indeed possible.
In this talk, I will introduce compressed sensing by defining a mathematical model for signal acquisition and outline procedures that guarantee signal reconstruction. There will be lots of pictures, and lots of solving $Ax=b$

We will discuss a 1995 paper of Koll\'ar, which showed via degeneration to characteristic 2 the nonrationality of many low degree hypersurfaces in projective space.

Let $G$ be a finite group. The faithful dimension of $G$ is defined to be the smallest possible dimension for a faithful complex representation of $G$. Aside from its intrinsic interest, the problem of determining the faithful dimension of finite groups is intimately related to the notion of essential dimension, introduced by Buhler and Reichstein.

In this paper, we will address this problem for groups parameterized by a prime parameter $p$ (e.g., Heisenberg groups over finite fields with $p$) and study the question of the dependence of the essential dimension on $p$. As it will be shown, in general, this is always a piecewise polynomial function along certain ``number-theoretically defined'' sets, while, in some specific cases, it is given by a uniform polynomial in $p$.

This talk is based on a joint work with Mohammad Bardestani and Hadi Salmasian.

I will discuss a recent line of research that uses properties of real rooted polynomials to get quantitative estimates in combinatorial linear algebra problems. I will start by discussing the main result that bridges the two areas (the ``method of interlacing polynomials'') and show some examples of where it has been used successfully (e.g. Ramanujan families and the Kadison Singer problem). I will then discuss some more recent work that attempts to make the method more accessible by providing generic tools and also attempts to explain the accuracy of the method by linking it to random matrix theory and (in particular) free probability. I will end by mentioning some current research initiatives as well as possible future directions.

We study the 2D Euler equations linearized around smooth, radially symmetric vortices with strictly decreasing vorticity profiles. Under a trivial orthogonality condition, we prove that the perturbation vorticity winds up around the vortex and weakly converges to a radially symmetric configuration, as time goes to infinity. This process is known as ``vortex axisymmetrization'' in the physics literature and is thought to stabilize vortex structures such as hurricanes and cyclones. Additionally, the velocity field converges strongly in L2 to the corresponding equilibrium (as time goes to infinity) and we give optimal decay rates in weighted L2 spaces. Interestingly, the rate of decay is faster for the linearized 2D Euler equations than for the passive scalar equation. The passive scalar rate is degraded by the slow mixing at the vortex core, but the linearized 2D Euler equations expel vorticity from the origin leading to a faster decay rate.

What is a random graph and what do random graphs look like? We will discuss some basic properties, mathematical methods and applications of random graphs.

In this talk, we introduce the LRHB algorithm, which is an extension of the reduced-Hessian method of Gill and Leonard for unconstrained problems to problems with simple bound constraints. Numerical results for LRHB will be presented. We will also consider computational and practical issues with methods for nonlinear optimization and present results on a large test collection of problems indicating the reliability and efficiency of sequential quadratic programming methods and interior-point methods on certain classes of problems. This is joint work with Michael Ferry and Philip E. Gill.

Gene conversion is a ubiquitous phenomenon that leads to the exchange of genetic information between homologous DNA regions and maintains co-evolving multi-gene families in most pro- and eukaryotic organisms. In this talk, I will consider its implications for the evolution of a single functional gene with a silenced duplicate, using two different mathematical models of evolution on rugged fitness landscapes. Our analytical and numerical results show that, by helping to circumvent valleys of low fitness, gene conversion with an inactive duplicate gene can cause a significant speedup of adaptation which depends non-trivially on the frequency of gene conversion and the structure of the landscape. Stochastic effects due to finite population sizes further increase the likeliness of exploiting this evolutionary pathway. Our results reveal the potential for duplicate genes to act as a ``scratch paper'' that frees evolution from being limited to strictly beneficial mutations in strongly selective environments.

Deninger and Werner developed an analogue for $p$-adic curves of the
classical correspondence of Narasimhan and Seshadri between stable
bundles of degree zero and unitary representations of the topological
fundamental group for a complex smooth proper curve. Using parallel
transport, they associated functorially to every vector bundle on a
$p$-adic curve whose reduction is strongly semi-stable of degree 0 a
$p$-adic representation of the \'etale fundamental group of the curve. They
asked several questions: whether their functor is fully faithful and
what is its essential image; whether the cohomology of the local systems
produced by this functor admits a Hodge-Tate filtration; and whether
their construction is compatible with the $p$-adic Simpson correspondence
developed by Faltings. We will answer these questions in this talk.

Do you love factoring numbers? Me too. Whether you're helping your little cousin with her algebra homework or trying to crack an RSA cryptosystem, there's nothing quite like the thrill of taking a big number and writing it as a product of smaller numbers. In this talk we present a brief historical overview of some primality tests and factorization methods; we will also develop in more detail the quadratic sieve method developed by Pomerance in the 1980's, which for many years held the record for fastest implementation of a factorization algorithm. We leave the converse question of how to create a large number from several smaller numbers as an exercise to the interested listener.

The Ricci iteration is a discrete analogue of the Ricci flow. According to Perelman, the Kahler-Ricci flow converges to a Kahler-Einstein metric whenever one exists, and it has been conjectured that the Ricci iteration should behave similarly. In joint work with Y. Rubinstein we are able to confirm this conjecture for general Fano manifolds.

n this talk we will characterize the irreducible quasifinite highest weight modules of some subalgebras of the Lie algebra of matrix quantum pseudodifferential operators $N \times N$.

In order to do this, we will first give a complete description of the anti-involutions that preserve the principal gradation of the algebra of $N\times N$ matrix quantum pseudodifferential operators and we will describe the Lie subalgebras of its minus fixed points. We will obtain, up to conjugation, two families of anti-involutions that show quite different results when $n=N$ and $n< N$. We will then focus on the study of the ``orthogonal'' and ``symplectic'' type subalgebras found for case $n=N$, specifically the classification and realization of the quasifinite highest weight modules.

Pearson's Chi-squared test is very widely used in practice and has a long history. The validity with a large number of cells or small expected frequencies has been open for a long time. We provide a solution to this open problem by rigorously establishing a distributional approximation of Pearson's Chi-squared test statistic by using a high-dimensional central limit theorem for quadratic forms of random vectors. We also propose a modified chi-squared statistic with a faster convergence rate and propose the concept of adjusted degrees of freedom. Our procedure is applied to goodness-of-fit test for the social life feeling data and the Rochdale data.

Sen attached to each $p$-adic Galois representation of a $p$-adic field a multiset of numbers
called generalized Hodge-Tate weights. In this talk, we regard a $p$-adic local system on a
rigid analytic variety as a geometric family of Galois representations and show that the
multiset of generalized Hodge-Tate weights of the local system is constant.

Communicating Mathematics in Movies: Join award-winning documentary filmmaker Ekaterina Eremenko for a screening of The Discrete Charm of Geometry, followed by a Q&A led by Fields Medalist Efim Zelmanov about Eremenko's work portraying mathematics and mathematicians on film. At the UCSD Price Center Theatre from 5:00-7:00pm on Monday, February 5, 2018.

Equivalence relations are among the most basic concepts in mathematics. Nevertheless, it turns out that there are some pretty complicated equivalence relations out there. In this talk we will discuss different ways of classifying them - whatever that means. We will look at some (more or less) concrete examples. Finally, we will construct some fairly natural equivalence relations for which we can nevertheless prove a strong nonclassifiability result.

In this talk we will show a scattering - type result for Schrodinger maps with small Besov norm. The proof extends an earlier result of Bejenaru, Ionescu, and Kenig.

One of the fundamental problems in combinatorics involves deciding whether some given linear equation has solutions with all its variables lying in some restricted set, and if so, estimating how many such solutions there are. In this talk, we will describe some of the old and new results in this area, as well as discuss a number of unsolved problems.

The generalized Nash equilibrium problem (GNEP for short)is an extension of Nash equilibrium problem where the feasible set of each player may depend on the rivals strategies. It has many applications in areas such as economics, engineering, transportation and management sciences. In this talk, we present an exact penalty function method to reduce the GNEP into a Nash equilibrium problem. Here the penalty function is smooth, which is different from the most existing function. We also report numerical results so as to illustrate the efficiency of the proposed method.

Independent sets in hypergraphs can be encoded as 0-1 configurations on the set of vertices such that each hyperedge is adjacent to at least one 0. This model has been studied in the CS community for its large gap between efficient MCMC algorithms (previously $d <(k-1)/2$) and the conjectured onset of computational hardness ($d > O(2^{k/2})$ ), where $d$ is the largest degree of vertices and $k$ is the minimum size of hyperedges. In this talk we use a percolation approach to show that the Glauber dynamics is rapid mixing for $d < O(2^{k/2}$), closing the gap up to a multiplicative constant.

The three-dimensional spatiotemporal organization of genetic material inside the cell nucleus remains an open question in cellular biology. During the time between two cell divisions, the functional form of DNA in cells, known as chromatin, fills the cell nucleus in its uncondensed polymeric form, which allows the transcriptional machinery to access DNA. Recent in vivo imaging experiments have cast light on the existence of coherent chromatin motions inside the nucleus, in the form of large-scale correlated displacements on the scale of microns and lasting for seconds. To elucidate the mechanisms for such motions, we have developed a coarse-grained active polymer model where chromatin is represented as a confined flexible chain acted upon by active molecular motors, which perform work and thus exert dipolar forces on the system. Numerical simulations of this model that account for steric and hydrodynamic interactions as well as internal chain mechanics demonstrate the emergence of coherent motions in systems involving extensile dipoles, which are accompanied by large-scale chain reconfigurations and local nematic ordering. Comparisons with experiments show good qualitative agreement and support the hypothesis that long-ranged hydrodynamic couplings between chromatin-associated active motors are responsible for the observed coherent dynamics. The connection between our model and mechanisms proposed for self-organization of other biological systems including bacterial suspensions will also be discussed.

We consider the eigenvalue problem for octonionic $3\times3$ Hermitian
matrices (the exceptional Jordan algebra, also known as the Albert
algebra). For real eigenvalues, most of the properties expected by
analogy with the complex case still hold, provided they are
reinterpreted to take into account of the lack of commutativity and
associativity. There are nevertheless some interesting surprises along
the way, among them the existence of nonreal eigenvalues, and the fact
that the components of primitive idempotents (elements of $OP^2$, the
Cayley--Moufang plane) always associate.

Applications of these results will be briefly discussed, both to the
study of exceptional Lie groups (the Albert algebra is the minimal
representation of $e_6$) and to physics ($OP^2$ can be interpreted as
the solution space of the Dirac equation in 10 spacetime dimensions).

Geometric Recursion is a very general machinery for constructing mapping class group invariants objects associated to two dimensional surfaces. After presenting the general abstract definition we shall see how a number of constructions in low dimensional geometry and topology fits into this setting. These will include the Mirzakhani-McShane identies, mapping class group invariant closed forms on Teichmuller space (including the Weil-Petterson symplectic form) and the Goldman symplectic form on moduli spaces of flat connections for general compact simple Lie groups. If time permits we shall also discuss the process which establishes that any application of Topological Recursion can be lifted to a Geometric Recursion setting involving continuous functions on Teichmuller space, where the connection back to Topological Recursion is obtained by integration over the moduli space of curve. The work presented is joint with G. Borot and N. Orantin.

Asked by some graduate students what my current research is about, I will try to introduce the very basic ideas and definition of Segal--Bargmann transform, which intertwines the Heisenberg and Fock pictures.

A \textbf{repetition} in an edge-colored graph is a path in which the sequence of colors in the first half of the path is identical to that in the second half. In 2002 Alon, Grytczuk, Haluszczak, and Riordan showed that every $k$-ary tree has a repetition-free edge-coloring with at most $4k$ colors. We present a simple procedure for obtaining repetition-free edge-colorings of $k$-ary trees with at most $3k+1$ colors. We will also discuss some related vertex-coloring questions.

Arising frequently in sciences and engineering, ill-posed problems remain a formidable challenge and a frontier in scientific computing because their solutions appear to be infinitely sensitive to data perturbations. Many common algebraic problems are ill-posed, such as matrix rank, singular linear and nonlinear systems, polynomial factorizations and Jordan Canonical Forms. On the other hand, the instability of such problems may be a ``misconception'', as argued by W. Kahan 40 years ago, since the solutions are well behaved when certain structures of the problems are preserved. Furthermore, the hypersensitivity is not random but one-directional: Tiny perturbations can only decrease the singularity of the problem and never increase it. From a geometric perspective, ill-posed problems with a specific structure form a smooth manifold that is embedded in similar manifolds of lower codimensions. Based on this property and the Tubular Neighborhood Theorem, ill-posed algebraic problems can be regularized to remove the instability through two optimization problems: Maximizing the codimension of nearby manifolds and minimizing the distance from the manifold to the data point. In this talk we shall elaborate the computing strategy based on the regularization approach along with algorithms/software for finding accurate solutions of many ill-posed algebraic problems from empirical data.

In this talk, we discuss the Diederich--Forn\ae ss index in several complex variables. A domain $\Omega\subset\mathbb{C}^n$ is said to be pseudoconvex if $-\log(-\delta(z))$ is plurisubharmonic in $\Omega$, where $\delta$ is a signed distance function of $\Omega$. The Diederich--Forn\ae ss index has been introduced since 1977 as an index to refine the notion of pseudoconvexity. After a brief review of pseudoconvexity, we discuss this index from the point of view of geometric analysis. We will find an equivalent index associated to the boundary of domains and with it, we are able to obtain accurate values of the Diederich--Forn\ae ss index for many types of domains.

High dimensional time series data arise in a wide range of disciplines, including finance, signal processing, neuroscience, meteorology, seismology and many other areas. For low dimensional time series there is a well-developed estimation and inference theory. Inference theory in the high dimensional setting has been rarely studied. In this talk, I will give an overview of the work that is proposed to develop and advance statistical inference theory for high dimensional time series data analysis including parameter estimation, construction of simultaneous confidence intervals, prediction, model selection, Granger causality test, hypothesis testing and spectral domain estimation.

We explain a recent work with Huang where we combine various results in complex analysis including Fefferman's invariant theory, the Chern-Moser theory, the Cheng-Yau solution of Kahler-Einstein metrics, to provide an affirmative solution of a conjecture posed by Cheng-Yau. The conjecture stated that the Bergman metric of a bounded strongly pseudoconvex domain is Einstein if and only if the domain is holomorphically equivalent to the ball.

Regularization techniques are widely employed in the solution of inverse problems in data analysis and scientific computing due to their effectiveness in addressing difficulties due to ill-posedness. In their most common manifestation, these methods take the form of penalty functions added to the objective in optimization-based approaches for solving inverse problems. The purpose of the penalty function is to induce a desired structure in the solution, and these functions are specified based on prior domain-specific expertise. We consider the problem of learning suitable regularization functions from data in settings in which precise domain knowledge is not directly available; the objective is to identify a regularizer to promote the type of structure contained in the data. The regularizers obtained using our framework are specified as convex functions that can be computed efficiently via semidefinite programming. Our approach for learning such semidefinite regularizers combines recent techniques for rank minimization problems along with the Operator Sinkhorn iteration. (Joint work with Yong Sheng Soh).

I will discuss stochastic interacting particle systems in the KPZ universality class evolving in one-dimensional inhomogeneous space. The inhomogeneity means that the speed of a particle depends on its location. I will focus on integrable examples of such systems, i.e., for which certain observables can be written in exact form suitable for asymptotic analysis. Examples include a continuous-space version of TASEP (totally asymmetric simple exclusion process), and the pushTASEP (=long-range TASEP). For integrable systems, density limit shapes can be described in an explicit way. We also obtain asymptotics of fluctuations, in particular, around slow bonds and infinite traffic jams caused by slowdowns.

I will present work that aim at understanding the failure of analytic hypoellipticity of special differential operators, namely sums of squares of (analytic) vector fields. According to Hörmander, given such an operator P that satisfies the ``bracket condition'' and given a smooth function f, if u is distribution solution to Pu=f then u is also a smooth function. P is then said to be hypoelliptic. But if f is real analytic, then u need not to be analytic but merely smooth Gevrey for some indices, usually to be guessed. Examples were built by Metivier, Matsuzawa, Bove, Baouendi-Goulaouic..., using real variables methods. In this joint work with Paulo Cordaro, by using methods of complex analysis, we show that this failure of analytic hypoellipticity due to the presence of irregular singularity of some holomorphic ODEs the analysis of which defines the best Gevrey indices to be expected. The theory of summability of formal solutions of holomorphic ODEs as developped by Ramis, Malgrange, Braaksma is a fundemental tool here.

In this talk, I will describe a new algebraic characterization of sectional curvature bounds that only involves curvature terms in the Weitzenboeck formulae for symmetric tensors. This characterization is further clarified by means of a symmetric analogue of the Kulkarni-Nomizu product, which renders it computationally amenable. Furthermore, a related application of the Bochner technique to closed 4-manifolds with indefinite intersection form and positive or nonnegative sectional curvature will be discussed, yielding some new insight about the Hopf Conjecture. This is based on joint work with R. Mendes (Univ. zu K$\ddot{\text{o}}$ln, Germany).

The Gaussian entropy, introduced by Colding and Minicozzi, is a rigid motion and scaling invariant functional which measures the complexity of hypersurfaces of the Euclidean space. It is defined to be the supremal Gaussian area of all dilations and translations of the hypeprsurface, and as such, is well adapted to be studied by mean curvature flow. In the case of the n-th sphere in $R^{n+1}$, the entropy can be computed explicitly, and is decreasing as a function of the dimension n. A few years ago, Colding Ilmanen Minicozzi and White proved that all closed, smooth self-shrinking solutions of the MCF have larger entropy than the entropy of the n-th sphere. In this talk, I will describe a generalization of this result, which derives better (sharp) entropy bounds under topological constraints. More precisely, we show that if M is any closed self-shrinker in $R^{n+1}$ with a non-vanishing k-th homotopy group (with $k\leq n$), then its entropy is higher than the entropy of the k-th sphere in $R^{k+1}$. This is a joint work with Brian White.

Many scientific datasets are of high dimension, and the analysis usually requires retaining the most important structures of data. Many existing methods work only for data with structures that are mathematically formulated by curves, which is quite restrictive for real applications. To get more general graph structures, we develop a novel graph structure learning framework that captures the local information of the underlying graph structure based on reversed graph embedding. A new learning algorithm is developed that learns a set of principal points and a graph structure from data, simultaneously. Experimental results on various synthetic and real world datasets show that the proposed method can uncover the underlying structure correctly.

Study of algebraic equations is one of the oldest and most celebrated themes in mathematics. It was understood that finite systems of equations are decidable in the fields of complex and real numbers. The celebrated Hilbert tenth problem stated in 1900 asks for a procedure which, in a finite number of steps, can determine whether a polynomial equation (in several variables) with integer coefficients has or does not have integer solutions. In 1970 Matiyasevich, following the work of Davis, Putnam and Robinson, solved this problem in the negative. Similar questions can be asked for arbitrary rings. We will give a survey of results on the Diophantine problem in different rings and prove the undecidability of equations (the undecidability of the Diophantine problem) for free Lie algebras of rank at least 3 over an arbitrary field. These are joint results with A. Miasnikov.

The class number formula connects the residue of the Dedekind zeta
function at s=1 to the regulator, which measures the covolume of the
lattice generated by logarithms of units. Beilinson defined a vast
generalization of the regulator morphism and conjectured a class number
formula associated to the cohomology of any smooth proper variety over a
number field. His formula provides arithmetic meaning for the orders of
the so-called trivial zeros of L-functions at integer points as well as
the value of the first nonzero derivative at these points.

We study this conjecture for the middle degree cohomology of
compactified Shimura varieties associated to unitary groups of signature
(2,1) and (2,2) over Q. We construct explicit Beilinson-Flach elements
in the motivic cohomology of these varieties and compute their
regulator. This is joint work with Aaron Pollack.

Modular forms are ubiquitous mathematical objects. Roughly speaking, they are functions in the complex upper half plane satisfying certain symmetry properties. More importantly, they occur naturally in connection with many problems in different areas of mathematics, e.g., number theory, geometry, mathematical physics, etc.

In this talk, I will give a brief introduction to the theory of classical modular forms while providing examples of various applications.

Wave maps are natural hyperbolic analogue of harmonic maps. The study of the wave maps over the last twenty plus years has led to many beautiful and deep ideas, culminating in the proof of the ``ground state conjecture'' by Sterbenz and Tataru (with independent proof by Tao, and by Krieger&Schlag when the map has hyperbolic plane as target). The understanding of wave maps is now quite satisfactory. There are however still some remaining interesting problems, involving more detailed dynamics of wave maps. In this talk, we shall look at the problem of ruling out the so called ``null concentration of energy'' for wave maps. We will briefly review the history of wave maps, and explain why the null concentration of energy is relevant, and why the channel of energy inequality seems to be uniquely good in ruling out such possible energy concentration in the presence of solitons.

In this talk, I will describe my results on superconvergence estimates of mixed methods using Raviart--Thomas finite elements. First I prove the canonical interpolant and finite element solution approximating the vector variable are superclose in $L^2$ norm. The main tool is a triangular integral identity in Bank and Xu SIAM J. Numer. Anal 41 (2003) 2294-2312, and a discrete Helmholtz decomposition. Comparing to previous supercloseness results (eg. Brandts Numer. Math. 68 (1994) 311--324), my proof is constructive and works on irregular triangular meshes. Even on a special uniform grid, my result shows that the previous supercloseness result is suboptimal. Next I will describe several postprocessing operators based on simple edge averaging, $L^2$ projection or superconvergence patch recovery. Then I will show the postprocessed finite element solution superconverges to the true solution. If time permits, I will also briefly describe applications to Maxwell's equations an
d generalizations to fourth order elliptic equations.

A sunflower is a collection of distinct sets such that the intersection of any two of them is the same as the common intersection C of all of them, and |C| is smaller than each of the sets. There has been a lot of recent progress on determining the maximum size of a sunflower-free family of subsets of [n]. We consider the problems of determining the maximum sum and product of k families of subsets of [n] that contain no sunflower of size k with one set from each family. We solve the sum problem exactly and make partial progress towards the product problem.

\noindent{Joint work with Lujia Wang.}

This talk introduces a new kernel-based Maximum Mean Discrepancy (MMD) statistic for measuring the distance between two distributions given finitely-many multivariate samples. When the distributions are locally low-dimensional, the proposed test can be made more powerful to distinguish certain alternatives by incorporating local covariance matrices and constructing an anisotropic kernel. The techniques and theory touch on spectral theory of Laplacians and heat kernels, optimization, and linear algebra.
Applications to flow cytometry and diffusion MRI datasets are demonstrated, which motivate the proposed approach to compare distributions.

Entanglement is a core concept in quantum mechanics and quantum information theory. Put simply: a tensor is entangled if is not a product state. Measuring precisely how much entanglement a given tensor has is a big question with competing answers in the physics community. One natural measure is the {\bf geometric measure of entanglement}, which is a version of the Hilbert--Schmidt distance of the given tensor from the set of product states. It can also be described as the log of the spectral norm.

In 2009, Gross, Flammia, and Eisert showed that, as the mode of the tensor grows, the geometric measure of entanglement of a random tensor is, with high probability, very close to the theoretical maximum. In this talk, I will describe my joint work with Shmuel Friedland on the analogous situation for symmetric tensors. While symmetric tensors are inherently entangled, it turns out their maximum geometric measure of entanglement is exponentially smaller than for generic tensors. Using tools from representation theory and random matrix theory, we prove that, nevertheless, random symmetric tensors are, with high probability, very close to maximally entangled.

We explore the implications of molecular charge fluctuations on the kinetics of electrostatically dominated diffusion-controlled association, employing diffusion theory and Brownian computer simulations. In general, stochastically fluctuating degrees of freedom that dynamically couple to the diffusional transport can yield additional dissipative forces, i.e., friction, resonant response or feedback mechanisms. Using a minimalistc model we present how charge fluctuations due to randomly fluctuating protonation of a charged ligand influence the association kinetics to a charged membrane. Here, the exploration of the limiting and intermediate frequency domains of the fluctuations serve to clarify the use of concepts of an average charge, an average potential, and an associated capacitance for quantitative rate prediction.

Equidistribution results play an important role in dynamical
systems and their applications in number theory. Often in such
applications it is necessary for equidistribution to be effective (i.e.
the rate of convergence is known) so that number-theoretic methods such
as sieving can be applied. In this talk, we will give a brief history
of effective equidistribution results in homogeneous dynamics and their
applications to number theory. We will then present an effective
equidistribution result for certain flows on the space of lattices and
discuss a number-theoretic application regarding almost-prime times for
these flows.

In quantum mechanics, at an energy level
E, there arises an allowed region A(E), a forbidden
region F(E) and an interface C(E) between them.
Most quantities of interest, ranging from sizes of
quantum states to their nodal sets, exhibit a transition
across the interface between allowed and forbidden behavior. I will illustrate these interfaces with two
different types of problems: nodal sets of eigenfunctions
of Schrodinger equations and ``partial Bergman kernels''
for ample line bundles over Kahler manifolds. The two
settings seem quite different at first sight but they are just two types of geometry in phase space. No prior knowledge of quantum mechanics is assumed.

We consider the problem of detecting and localizing an submatrix with larger-than-usual
entries inside a large, noisy matrix. This problem arises from analysis of data in
genetics, bioinformatics, and social sciences. We consider that entries of the data matrix are
independently following distributions from a natural exponential family, which generalizes
the common Gaussian assumptions in the literature. Distribution-free methods of detection and
size-adaptive methods of both detection and localization problems are studied with their
asymptotic behaviors illustrated.

I will give a brief introduction to the Hodge conjectures and
describe some examples where the conjectures are nontrivial.

Rigidity phenomena in homogeneous dynamics have been extensively
studied over the past few decades with several striking results and
applications. In this talk we will give an overview of these results
and also the more recent activities which aim at presenting
qualitative versions of them.

The goal of the meeting is to bring together world class mathematicians with expertise in complex analysis/geometry and CR geometry, as well a provide a forum for the exchange of ideas with graduates.

The goal of the meeting is to bring together world class mathematicians with expertise in complex analysis/geometry and CR geometry, as well a provide a forum for the exchange of ideas with graduates.

Tesler matrices are certain integral matrices counted by the Kostant partition function and have appeared recently in Haglund's study of diagonal harmonics. In 2014, Drew Armstrong defined a poset on such matrices and conjectured that the characteristic polynomial of this poset is a power of $(q-1)$. We will use a method of Bruce Sagan and Joshua Hallam to prove Armstrong's conjecture and explore how this result can improve the bounds on the number of Tesler matrices.

We continue the previous studies in two papers of Huang-Yin on the flattening problem of a CR singular point of real codimension two sitting in a submanifold in $C^{n+1}$ with $n + 1 ≥ 3$, whose CR points are non-minimal. We give a very general flattening theorem for a non-degenerate CR singular point. This is joint work with Xiaojun Huang.

The theory supporting conforming finite elements on meshes of squares and cubes requires some care when extended to more general meshes of quadrilaterals and hexahedra in order to preserve desired rates of convergence. In this talk, I'll present two areas of research related to these issues. First, I'll describe a new family of methods called ``trimmed serendipity elements'' that fit within the same framework described by the Periodic Table of the Finite Elements (see https://femtable.org). The computational effort required to employ a trimmed serendipity element method is significantly less that what is required for comparable alternatives from the table, thereby presenting a host of potential benefits to the speed and accuracy of square/cube finite element methods in practice. Second, I'll present the loss of convergence issues that arise when square/cube elements are mapped non-affinely as well as some recent techniques that restore convergence order. In addition, I'll show why general quad/hex meshes are of increasing interest in application contexts, including an cardiac electrophysiology example carried out in collaboration with Andrew McCulloch's research group and NBCR.

I will discuss recent joint results with B. Lamel addressing the $C^\infty$ regularity problem for CR mappings between smooth CR submanifolds in complex spaces of possibly different dimension. We essentially show that a nowhere smooth CR map must have its image contained in the set of D’Angelo infinite type points of the target manifold, from which we derive a number of new regularity results, even in the hypersurface case. Applications to the boundary regularity of proper holomorphic mappings will also be mentioned.

Holomorphic classification of real submanifolds in complex space is one of the central goals in complex analysis in several variables. This classification is well understood and approaches are well developed for submanifolds satisfying certain bracket generating conditions of Hormander type, while very little is known in more degenerate setting. In particular, somewhat surprizingly, the classification problem is still completely open for hypersurfaces in complex 2-space. The class of hypersurfaces bringing conceptual difficulties here is the class of (Levi-nonflat) infinite type hypersurfaces.
In our joint work with Ebenfelt and Lamel, we develop the equivalence theory for infinite type hypersurfaces in $C^2$. We do so by providing a normal for for such hypersurfaces. We extensively use the newly developed approach of Associated Differential Equations. The normal form construction is performed in two steps: (i) we provide a normal for for associated ODES; (ii) we use the normal form of ODEs for solving the equivalence problem for hypersurfaces.
Somewhat similarly to the Poincare-Dulac theory in Dynamical System, our classification theory exhibits resonances, convergence and divergence phenomena, Stokes phenomenon and sectorial regularity phenomena.

An important task of olfactory sensing is the discrimination of different odors. An odor captures the chemical state of the environment in a mixture of smell molecules, called odorants. Olfactory sensing is realized by the selective binding of odorants to a set of olfactory receptors, which in turn activates the corresponding olfactory sensory neurons, constructing the brain's first representation of the odor. Despite the high-dimensional nature of olfactory sensing, recent measurements with human olfactory receptors suggest that the odorant-receptor interaction is sparse; only a small fraction of all available pairs interact. What are the optimal interaction structures for effective olfactory discrimination, and are these optimal solutions employed by the real system? We investigate these questions by combining studies of model systems and analyses of experimental data. We show that optimization depends on the statistical properties of the olfactory environment, and furthermore suggest that the human olfactory receptors are adapted to the natural odor statistics.

We will discuss a few interesting open problems related to the analysis of PDEs which model fluid motion. While some of these models were derived by Euler in the 1750s, understanding the dynamics of solutions continues to be a major challenge despite many advancements in recent years.

Consider the scaling invariant, sharp log entropy (functional) introduced by Weissler on noncompact manifolds with nonnegative Ricci curvature. It can also be regarded as a sharpened version of Perelman's W entropy in the stationary case. We prove that it has a minimizer if and only if the manifold is isometric to the Euclidean space. Using this result, it is proven that a class of noncompact manifolds with nonnegative Ricci curvature is isometric to $R^n$. Comparing with earlier well known flatness results on asymptotically flat manifolds and asymptotically locally Euclidean (ALE) manifolds, their decay or integral condition on the curvature tensor is replaced by the condition that the metric converges to the Euclidean one in $C^1$ sense at infinity. No second order condition on the metric is needed.

Solving an unconstrained quadratic program means solving a linear equation where the matrix is symmetric and positive definite. This is a fundamental subproblem in nonlinear optimization. We discuss the behavior of the method of conjugate gradients and quasi-Newton methods on a quadratic problem. We first derive the method of conjugate gradients and then give necessary and sufficient conditions for an exact line search quasi-Newton method to generate a search direction which is parallel to that of the method of conjugate gradients. We analyze update matrices and show how the secant condition fits the discussion of giving parallel search directions.

Our interest is limited-memory quasi-Newton methods tailored for interior methods. The talk describes the fundamental properties for the exact quadratic case, which is the foundation for the work.

The talk is based on joint work with David Ek and Tove Odland.

The hydrodynamic limit of the Kawasaki dynamics states that a certain stochastic evolution of a lattice system converges macroscopically to a deterministic non-linear heat equation. We will discuss how the statement of the hydrodynamic limit can be made quantitative. The key step is to introduce an additional evolution on a mesoscopic scale that emerges from projecting the microscopic observables onto splines. The hydrodynamic limit is then deduced in two steps. In the first step one shows the convergence of the microscopic to the mesoscopic evolution and in the second step one deduces the convergence of the mesoscopic to the macroscopic evolution.

The talk is about a joint work with Deniz Dizdar, Felix Otto and Tianqi Wu.

A certain infinite determinant arises as the constant in the block case of the Szego-Widom Limit Theorem for Toeplitz matrices. These are matrices whose block entries are constant on the diagonals. While the constant has a very nice form, often a more explicit form, analogous to the one found in the scalar case, is needed in applications. This talk will survey some of the ways the constant can be more explicitly computed (often using functional analysis techniques) and also some applications.

In a 1987 letter to Tate, Serre showed that the Hecke
eigensystems appearing in mod p modular forms are the same as those
appearing in mod p functions on a finite double coset constructed from
the quaternion algebra ramified at p and infinity. At the end of the
letter, he asked whether there might be a similar relation between
p-adic modular forms and p-adic functions on the quaternion algebra. We
show the answer is yes: the completed Hecke algebra of p-adic modular
forms is the same as the completed Hecke algebra of naive p-adic
automorphic functions on the quaternion algebra. The resulting p-adic
Jacquet-Langlands correspondence is richer than the classical
Jacquet-Langlands correspondence -- for example, Ramanujan's delta
function, which is invisible to the classical correspondence, appears.
The proof is a lifting of Serre's geometric argument from characteristic
p to characteristic zero; the quaternionic double coset is realized as a
fiber of the Hodge-Tate period map, and eigensystems are extended off of
the fiber using a variant of Scholze's fake Hasse invariants.

Nonconvex optimization problems are the bottleneck in many applications in science and technology. Often these problems are NP-hard and they are approached with ad hoc methods that frequently fail to yield satisfactory results. This issue becomes even more prevalent in the ``Big Data Regime''.
In my talk I will report on recent breakthroughs in solving some important nonconvex optimization problems. In particular, I will discuss the problems of phase retrieval, blind deconvolution, and blind source separation. The most notorious among these three is arguably phase retrieval, which is the century-old problem of reconstructing a function from intensity measurements, typically from the modulus of a diffracted wave. Phase retrieval problems arise in numerous areas including X-ray crystallography, differential geometry, astronomy, diffraction imaging, and quantum physics and are very difficult to solve numerically.

Combining tools from optimization, random matrix theory and harmonic analysis, we will derive rigorous mathematical methods that can solve the aforementioned problems under meaningful practical conditions. The proposed methods come with rigorous theoretical guarantees, are numerically efficient and stable in the presence of noise, and require little or no parameter tuning, thus making them useful for massive datasets. I will also discuss connections to the emerging field of self-calibration, which is based on the idea of equipping a sensor with a smart algorithm that can compensate automatically for the sensor's imperfections. The effectiveness of our methods will be illustrated in applications such as astronomy, X-ray crystallography, terahertz imaging, and the Internet-of-Things.

The gonality of a smooth projective curve is the smallest degree of a map from the curve to the projective line. If a curve is embedded in projective space, it is natural to ask whether the gonality is related to the embedding. In my talk, I will discuss recent work with James Hotchkiss. Our main result is that, under mild degree hypotheses, the gonality of a complete intersection curve in projective space is computed by projection from a codimension 2 linear space, and any minimal degree branched covering of $P^1$ arises in this way.

In this talk, we present some work from data science teams at Google to give
students an impression of how statistics and experimental design are used in
practice. Most examples will be drawn from the Search Ads Data Science team, but
many of them are representative of techniques used all around the tech industry.
One point of emphasis is that in data science techniques for creating meaningful
data sets can be as or more important than the statistical techniques that are
afterwards applied to them. Often doing a good job both in data generation and
evaluation are needed to answer interesting research questions. The talk is aimed
at graduate and undergraduate students with an interest in applied statistics.

We show that the collection of groups which satisfy the conclusion of Popa's Cocycle Superrigidity Theorem for Bernoulli actions is invariant under measure equivalence. In the case of orbit equivalence, the proof is mainly an application of a cocycle untwisting lemma of A. Furman and S. Popa, along with the fact that any orbit equivalence of free p.m.p. actions can be lifted to an orbit equivalence of the corresponding Bernoulli extensions of those actions.

Spherical Designs are finite sets of points
on the sphere with the property that the average of
low-degree polynomials over the sphere coincides
with the average over the finite set. These objects
are very beautiful, very symmetric and have been
studied since the 1970s. We use a completely
new approach that replaces delicate combinatorial
arguments with the a simple application of the heat
equation; this approach improves the known results
and extends to other manifolds. We also discuss
some related issues in Combinatorics, Irregularities
of Distribution and Fourier Analysis.

The distribution of descents in certain conjugacy classes of $S_n$ have been previously studied, and it is shown that its moments have interesting properties. This talk provides a bijective proof of the symmetry of the descents and major indices of matchings and uses a generating function approach to prove a central limit theorem for the number of descents in matchings. We also extend this result to derangements.

Given a Cayley graph of a finitely generated group, one can consider its growth function which counts how many elements are there in a ball of radius n on the graph. We will discuss two seminal results in the subject of growth of groups proved in early 1980s: Gromov’s polynomial growth theorem and Grigorchuk’s construction of groups of intermediate growth. We will illustrate how random walks on the Cayley graphs can help to study growth.

After a short introduction to topics in nonparametric curve estimation, covered in my new 2018
Chapman & Hall’s book with the same title as the talk, three specific problems will be considered.
The first one is nonparametric regression with missing at random (MAR) responses. It will be explained that a complete case approach is optimal in this case. The second problem is nonparametric regression with missing at random (MAR) predictors. It will be explained that in general a complete case approach is inconsistent for this type of missing and a special procedure is needed for efficient estimation. The last explored problem is devoted to estimation of hazard rate functions for truncated and censored data.

I will discuss a model of nonintersecting Brownian bridges on the unit circle, which produces quite a few universal determinantal processes as scaling limits. I will focus on the tacnode process, in which two groups of particles meet at a single point in space-time before separating, and introduce a new version of the tacnode process in which a finite number of particles ``switch sides'' before the two groups separate. We call this new process the k-tacnode process, and it is defined by a kernel expressed in terms of a system of tau-functions for the Painleve II equation. Technically, our model of nonintersecting Brownian bridges on the unit circle is studied using a system of discrete orthogonal polynomials with a complex (non-Hermitian) weight, so I'll also discuss some of the analytical obstacles to that analysis.

\noindent This is joint work with Dong Wang and Robert Buckingham

This talk will focus on my works on mean-field models for chemotaxis based on kinetic theory, including pathway based mean-field models, augmented Keller-Segel model for E. coli chemotaxis, and an asymmetric model for biological aggregation. I will give mathematical derivation of the mean-field models by taking some proper moment closure of kinetic biological systems. Building biological mechanism in the models are essential to capture some interesting swarming phenomena, for example, phase-delayed traveling wave (memory effect) and soliton solution (asymmetric sensing). Connections to the chemotaxis model proposed in [G. Si, T. Wu, Q. Quyang and Y. Tu, Phys. Rev. Lett., 109 (2012), 048101] will be also discussed.

The Navier-Stokes regularity problem is well-known, but there are many related simpler equations, including equations in 1d, for which our knowledge is surprisingly incomplete. Several of them have been recently studied, and in this talk we will discuss some of the results, together with connections to the full problem.

An abstract simplicial complex $\Sigma$ is said to be $k$-connected if for each $−1 \leq d \leq k$ and each continuous map $f$ from the sphere $S$ to $\|\Sigma\|$ (the polyhedron of the geometric realization of $\Sigma$), the map $f$ can be extended to a continuous map from the ball $B_{d+1}$ to $\|\Sigma\|$. The topological connectedness of $\Sigma$ is the largest $k$ for which it is $k$-connected. In 2000 a link was discovered between the topological connectedness of the independence complex of a graph and various other important graph parameters to do with colouring and partitioning. When the graph represents some other combinatorial structure, for example when it is the line graph of a hypergraph $H$, this link can be exploited to obtain information such as lower bounds on the matching number of $H$. Since its discovery there have been various other applications of this phenomenon to other combinatorial problems, including several different types of colouring (list colouring, strong colouring, delay edge colouring, circular colouring), hypergraph packing and covering, toughness and Hamiltonicity, and job scheduling and other resource allocation problems.
In this talk we give an overview of this technique, and describe some of its applications.

It has been observed since the early work of Richard Stanley that several well-known permutation statistics are ``compatible'' with the operation of shuffling permutations. In joint work with Ira Gessel, we formalize this notion of a shuffle-compatible permutation statistic and develop a unifying framework for studying shuffle-compatibility, which has close connections to the theory of P-partitions, quasisymmetric functions, and noncommutative symmetric functions. In this talk, I will survey the main results of our work as well as several new directions of research concerning shuffle-compatibility.

By a gluing construction, we produce steady Kahler-Ricci solitons on equivariant crepant resolutions of quotient singularities $C^n/G$, with the same asymptotics as Cao's soliton on $C^n$. This is joint work with Olivier Biquard.

Let $G$ denote a $p-adic$ reductive group, and $I_1$ a $pro-p-Iwahori$ subgroup. A classical result of Borel and Bernstein shows that the category of complex $G$-representations generated by their $I_1$-invariant vectors is equivalent to the category of modules over the (pro-p-)Iwahori-Hecke algebra $H$. This makes the algebra H an extremely useful tool in the study of complex representations of $G$, and thus in the Local Langlands Program. When the field of complex numbers is replaced by a field of characteristic $p$, the equivalence above no longer holds. However, Schneider has shown that one can recover an equivalence if one passes to derived categories, and upgrades $H$ to a certain differential graded Hecke algebra. We will attempt to understand this equivalence by examining the $H$-module structure of certain higher $I_1$-cohomology spaces, with coefficients in mod-$p$ representations of $G$. If time permits, we'll discuss how these results are compatible with Serre weight conjectures of Herzig and Gee--Herzig--Savitt.

On an n-dimensional complete manifold M, consider an h-almost gradient Ricci soliton, which is a generalization of gradient Ricci solitons and $(\lambda , n + m)$-Einstein manifolds. In this talk, we show that if the manifold is Bach-flat and $dh/du > 0$, then the manifold M is either Einstein or rigid. In particular, such a manifold has harmonic Weyl curvature. When the dimension of M is four, the metric is locally conformally flat.

Optimal power flow (OPF) problems are non-convex and large-scale optimization problems which appear in operation analysis, scheduling, and energy management of power systems. Various algorithms have been developed to solve the OPF problems, while in many cases, only local optimal solutions are available. Recent studies show that semidefinite programming (SDP) can either provide an exact or near global optima for the OPF problems. In this regard, there are enormous potential for SDP in solving the OPF problems. However, SDP-based approaches are far from real-world implementations. This talk will cover our recent results that partially address the limitations of SDP-based approaches for the OPF, including scalability, incorporation of binary variables, and distributed formulation.

Motivated by local-global compatibility in the $p$-adic Langlands program, Emerton and Helm (and others) studied how the local Langlands correspondence for $GL(n)$ can be interpolated in Zariski families. In this talk I will report on joint work with C. Johansson and J. Newton on the interpolation in rigid families. We take our rigid space to be an eigenvariety $Y$ for some definite unitary group $U(n)$ which parametrizes Hecke eigensystems appearing in certain spaces of $p$-adic modular forms. The space $Y$ comes endowed with a natural coherent sheaf $\mathcal{M}$. Our main result is that the dual fibers $\mathcal{M}_y'$ essentially interpolate the local Langlands correspondence at all points $y \in Y$. This make use of certain Bernstein center elements which appear in Scholze's proof of the local Langlands correspondence (and also in work of Chenevier). In the pre-talk I will talk about the local Langlands correspondence, primarily for $GL(2)$.

The class number formula is a precise relationship between two
cornerstones of number theory -- the zeta function of a number field and
its class group. A mysterious transcendental quantity, the regulator,
serves as a bridge that links these objects. A central open question in
number theory is Beilinson's vast conjectural generalization of this
formula.

After explaining and motivating Beilinson's conjecture, we state some
new cases obtained jointly with Aaron Pollack. The key behind them is
to find a setting where one can simultaneously realize Beilinson's
regulator pairing in representation theory and algebraic geometry. We
mention some diverse ingredients involved in the proof, such as the
relative Fourier-Mukai transform and explicit solutions of systems of
vector-valued PDEs, as well as some next steps.

Calabi-Yau varieties and Fano varieties are building blocks of varieties in the sense of birational geometry. Birkar recently proved that Fano varieties with bounded singularities belong to just finitely many algebraic families. One can then ask if an analogous result holds for Calabi-Yau varieties. If one only considers rationally connected Calabi-Yau varieties with klt singularities - those Calabi-Yau varieties behaving most like Fano - Shokurov actually conjectured that also these varieties should be bounded in any fixed dimension. We show that rationally connected klt Calabi-Yau 3-folds form a birationally bounded family. In many cases, we can actually give more precise statements and we are able to relate the boundedness problem to the study of a quite mysterious birational invariant: the minimal log discrepancy. This is a joint work in progress with W. Chen, G. Di Cerbo, J. Han, and C. Jiang.

We present a technique for proving convergence of h and hp
adaptive finite element methods through comparison with certain
reference refinement schemes based on interpolation error. We then
construct a testing environment where properties of different
adaptive approaches can be evaluated and improved.

I will discuss recent results concerning the Kuramoto-Sivashinky equation in two space dimensions with periodic boundary conditions. In particular, I will present a global existence result in the Wiener algebra, when growing modes are absent, and bounds on the analyticity radius when the data is only $L^2$. This is joint work with David Ambrose (Drexel University).

We will discuss rigidity, compactness, and finite quantization of entropy for the space of self-shrinking Lagrangian surfaces. This is based on joint work with John Ma

Let $P$ be a uniform random permutation matrix of size $N$ and let $\chi_N(z)= \det(zI - P)$ denote its characteristic polynomial. We prove a law of large numbers for the maximum modulus of $\chi_N$ on the unit circle, specifically,
\[
\sup_{|z|=1}|\chi_N(z)|= N^{x_c + o(1)}
\]
with probability tending to one as $N\to \infty$, for a numerical constant $x_c\approx 0.677$. The main idea of the proof is to uncover an approximate branching structure in the distribution of (the logarithm of) $\chi_N$, viewed as a random field on the circle, and to adapt a well-known second moment argument for the maximum of the branching random walk. Unlike the well-studied \emph{CUE field} in which $P_N$ is replaced with a Haar unitary, the distribution of $\chi_N(z)$ is sensitive to Diophantine properties of the argument of $z$. To deal with this we borrow tools from the Hardy--Littlewood circle method in analytic number theory. Based on joint work with Ofer Zeitouni.

How do the choices made by individual pedestrians influence the large-scale crowd dynamics?
What are the factors that slow them down and motivate them to seek detours?
What happens when multiple crowds pursuing different targets interact with each other?
We will consider how answers to these questions shape a class of popular PDE-based models, in which a conservation law models the evolution of pedestrian density while a Hamilton-Jacobi-Bellman PDE is used to determine the directions of pedestrian flux. This presentation will emphasize the role of anisotropy in pedestrian interactions, the geometric intuition behind our choice of optimal directions, and connections to the non-zero-sum game theory. (Joint work with Elliot Cartee.)

Let G be a simple and simply connected algebraic group over a field. We can attach to a it the sheaf of conformal blocks: a vector bundle on $M_g$ whose fibres are identified with global sections of a certain line bundle on the stack of G-torsors. We generalize the construction of conformal blocks to the case in which G is replaced by a ``twisted group'' defined over curves in terms of covering data. In this case the associated conformal blocks define a sheaf on a Hurwitz stack and have properties analogous to the classical case.

The main goal of this talk is to show that geometric information captured by certain invariant CR tensors provides sufficient information to establish the closed range property for $\bar\partial$ on a domain in $\mathbb{C}^n$. A secondary goal of the talk is to provide a general construction method for establishing when a domain (or its boundary) satisfies weak Z(q).

We present a synthesis of several papers written with co-authors and related to the construction of adaptive testing procedures in non parametric statistics. We consider the examples of goodness-of-fit tests in a density model, homogeneity test for Poisson processes and the two sample testing problem. The first step in such a construction is to define a test statistic $T$ which behavior under the null hypothesis and under the alternative differs. We consider test statistics based on the estimation of the $\mathbb{L}^2$ distance of the distribution of the observations under the null hypothesis and under the alternative. The testing procedures that we propose are non asymptotic : the level is guarantied for all sample size. Several cases can occur : the most simple case is the case where the distribution of the test statistic $T$ under the null hypothesis is known, one can then determine the quantiles of this distribution and define the testing procedure. When the null hypothesis is composite, this distribution is generally unknown. One can then consider conditional tests, if it is possible to determine a random variable $Z$ such that the distribution of $T$ conditionally on $Z$ under the null is known. For the two sample testing problem in density, in regression or for Poisson processes, we use kernel procedures combined with bootstrap and permutation methods. Concerning the power of the testing procedures, we will see that the aggregation of several tests allows to adapt to the unknown structure of the alternative.

Generalized Nash equilibrium problems (GNEP) have become very important as a modeling tool during the last decades. This talk summarizes some recent advances in the research on computational methods for GNEP. For example, Newton Method, Penalty Method, VI-based method etc. The algorithms and their convergence properties are also included. Furthermore, some results on error bound are presented.

Most bacteria on our planet reside in densely packed communities, yet we have little understanding of bacterial behavior in such communities. My laboratory has uncovered ion channel-mediated long-range signaling within such bacterial communities. These action potentials give rise to unexpected emergent behaviors at the community-level that are organized in space and time. We are working to understand the underlying electrophysiology of bacterial communities that allows them to cope with stress (such as antibiotic exposure) as a collective. I will discuss our recent efforts to develop new devices, techniques and theoretical frameworks to understand and control the electrophysiology and behavior of bacterial communities.

Using Ibukiyama's conjecture on transfers from inner forms
of GSp(4) we compute paramodular forms with prime levels up to 400.
This is joint work with Jeffery Hein and Gonzalo Tornar\'\i a.

A countable group is called Schmidt if it admits an ergodic free pmp action whose full group contains a non-trivial central sequence. We discuss several permanence properties and examples of Schmidt groups.

The incompressible Euler equation of fluid mechanics has been derived in 1755. It is one of the central equations of applied analysis, yet due to its nonlinearity and non-locality many fundamental properties of its solutions remain poorly understood. In particular, global regularity vs finite time blow up question for 3D Euler equation remains open.

In two dimensions, it has been known since 1930s that solutions to Euler equation with smooth initial data are globally regular. The best available upper bound on the growth of derivatives of solutions has been double exponential in time. I will describe a construction (based on a work joint with Vladimir Sverak) that shows that such fast generation of small scales can actually happen, so that the double exponential upper bound is qualitatively sharp.

This work has been motivated by numerical experiments due to Hou and Luo who propose a new scenario for singularity formation in solutions to 3D Euler equation. The scenario is axi-symmetric, and the geometry of the scenario involves hyperbolic points of the flow located at the boundary. If time permits, I will also discuss some models that attempt to gain insight into this scenario.

I will explain a new proof of the non-linear stability of the Minkowski spacetime as a solution of the Einstein vacuum equation. The proof relies on an iteration scheme at each step of which one solves a linear wave-type equation globally. The analysis takes place on a suitable compactification of $\mathbb{R}^4$ to a manifold with corners whose boundary hypersurfaces correspond to spacelike, null, and timelike infinity; I will describe how the asymptotic behavior of the metric can be deduced from the structure of simple model operators at these boundaries. This talk is based on joint work with Andras Vasy.

We study variations of the Bergman kernel and their asymptotic behaviors at degeneration. For a holomorphic family of hyperelliptic nodal or cuspidal curves and their Jacobians, we announce our results on the Bergman kernel asymptotics near various singularities. For genus-two curves particularly, asymptotic formulas with precise coefficients involving the complex structure information are written down explicitly. Time permitting, we would like to talk about the equality part of the Suita conjecture as an application.

Entanglement is a core concept in quantum mechanics and quantum information theory. Put simply: a tensor is entangled if is not a product state. Measuring precisely how much entanglement a given tensor has is a big question with competing answers in the physics community. One natural measure is the {\bf geometric measure of entanglement}, which is a version of the Hilbert--Schmidt distance of the given tensor from the set of product states. It can also be described as the log of the spectral norm.

In 2009, Gross, Flammia, and Eisert showed that, as the mode of the tensor grows, the geometric measure of entanglement of a random tensor is, with high probability, very close to the theoretical maximum. In this talk, I will describe my joint work with Shmuel Friedland on the analogous situation for symmetric tensors. While symmetric tensors are inherently entangled, it turns out their maximum geometric measure of entanglement is exponentially smaller than for generic tensors. Using tools from representation theory and random matrix theory, we prove that, nevertheless, random symmetric tensors are, with high probability, very close to maximally entangled.

We identify the linear space spanned by the real-valued excessive functions of a Markov process with the set of those functions which are quasimartingales when we compose them with the process. Applications to semi-Dirichlet forms are given. We provide a unifying result which clarifies the relations between harmonic, co-harmonic, invariant, co-invariant, martingale and co-martingale functions, showing that in the conservative case they are all the same. The talk is based on joint works with Iulian Cimpean (Bucharest) and Michael Roeckner (Bielefeld).

We provide two asymptotic continued fraction expansions of the
prime counting function. We also develop a ``degree'' calculus that
enables us to strengthen the connections between various reformulations
and extensions of the Riemann hypothesis.

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 non-linear problems, as well as suggest further approaches to non-linear problems.

For
any open Riemann surface $X$ admitting Green functions, Suita asked about the precise relations between the Bergman kernel and the logarithmic capacity. It was conjectured that the Gaussian curvature of the Suita metric is bounded from above by $-4$, and moreover
the curvature is identically equal to $-4$ if and only if $X$ is conformally equivalent to the unit disc less a (possible) polar set. After the contributions made by B\l{}ocki
and Guan & Zhou, we provide an alternative and simplified proof of the equality part in Suita's conjecture. Our proof combines the Ohsawa-Takegoshi extension theorem and the plurisubharmonic variation properties of Bergman kernels.

I will describe recent joint work with C. Mantoulidis in which we study the properties of bounded Morse index solutions to the Allen--Cahn equation on 3-manifolds. One consequence of our work is that a generic Riemannian 3-manifold contains an embedded minimal surface with Morse index p, for each positive integer p.

Let $X$ be an elliptically fibered K3 surface with a fixed $SU(n)$ bundle $\mathcal{E}$. I will talk about degenerations of connections on $\mathcal{E}$ that are Hermitian-Yang-Mills with respect to a collapsing family of Ricci flat metrics. This can be thought of as a vector bundle analog of the degeneration of Ricci flat metrics studied by Gross-Wilson and Gross-Tosatti-Zhang. I will show that under some mild conditions on the bundle, the restriction of the connections to a generic elliptic fiber converges to a flat connection. I will also talk about some ongoing work on strengthening this result. This is based on joint work with Adam Jacob and Yuguang Zhang.

Branching Brownian motion is a prototype of a disordered system and a toy model for spin glasses and log-correlated fields. It also has an exact duality relation with the FKPP equation, a well-known reaction diffusion equation. In this talk, I will present recent results (obtained with Michel Pain) on the fluctuations of the Gibbs measure at the critical temperature. By Gibbs measure I mean here the measure whose atoms are the positions of the particles, weighted by their Gibbs weight. It is known that this Gibbs measure, after a suitable scaling, converges to a deterministic measure. We prove a non-standard central limit theorem for the integral of a function against the Gibbs measure, for a large class of functions. The possible limits are 1-stable laws with arbitrary asymmetry parameter depending on the function. In particular, the derivative martingale and the usual additive martingale satisfy such a central limit theorem with, respectively, a totally asymmetric and a Cauchy distribution.

In recent years, Popa’s deformation/rigidity theory has lead to a wealth of classification
and rigidity results for von Neumann algebras given by countable groups and their actions on
measure spaces. In this talk, I will present the first rigidity and classification results for von
Neumann algebras given by locally compact groups and their actions. I establish that the crossed
product von Neumann algebra has a unique Cartan subalgebra, when the action is
free and probability measure preserving and the group is a connected simple Lie group of real rank one, or
a group acting properly on a tree. From this, I deduce a W*-strong rigidity result for irreducible
actions of products of such groups. I also establish that the group von Neumann algebra of such
groups are strongly solid. More generally, our results hold for locally compact groups that are
non-amenable, weakly amenable and belong to Ozawa’s class S. This is joint work with Arnaud
Brothier and Stefaan Vaes.

Classical modular curves associated to GL(2) are moduli spaces of
elliptic curves with additional structure. Taking advantage of the
analogy between number fields and function fields, Drinfeld modules (of
rank 2) were introduced as a good analogue of elliptic curves. While
there are no Shimura varieties associated to the general linear group
GL(N) for N$>$2, the situation is sharply different over function fields.
The Drinfeld modular variety for GL(N) is a moduli space of Drinfeld
modules of rank N (with auxiliary level structure). It is an affine
scheme of dimension N-1. In this talk, I will explain how analogues of
well-established theories due to Hida and Coleman in the classical
p-adic context extend to Drinfeld modular varieties and their associated
modular forms.

Joint with G. Rosso (Montr\'eal).

I will talk about two problems, completely different from
each other and both quite fun, that turn out to be related to
the Riemann hypothesis. One of them concerns the universal
limiting position (after rescaling) of the zeros of polynomials
belonging to a rather general class, with as an application
a weak version of an old conjecture of Polya that in its strong
version would imply the Riemann hypothesis. The other gives
an equivalence between the generalized Riemann hypothesis and
a statement about the growth rate of the determinants of certain
matrices whose entries are elementary cotangent sums, with an
unexpected appearance of quantum modular forms as a byproduct.

In a pioneering paper, Pila and Zannier showed how one can
prove arithmetic results (the Manin-Mumford Conjecture) using
transcendental methods (the Ax-lindemann conjecture). Their approach has
since been greatly developed, and is a major ingredient in the
Andre-Oort conjecture for Shimura varieties as well as the more
general Zilber-Pink conjecture, that serves as a sort of flagship for
the field of unlikely intersections. We'll explain this story and
present a new result (joint with Pila and Mok)
proving a general transcendence theorem known as Ax-Schanuel for
arbitrary Shimura varieties.

In their 1967 seminal paper, Foias and Prodi captured precisely a notion of finitely many degrees of freedom in the context of the two-dimensional (2D) incompressible Navier-Stokes equations (NSE). In particular, they proved that if a sufficiently large spectral projection of the difference of two solutions converge to 0 asymptotically in time, then the corresponding complementary projection of their difference must also converge to 0 in the infinite-time limit. In other words, the high modes are ``eventually enslaved`` by the low modes. One could thus define the number of degrees of freedom of the flow to be the smallest number of modes needed to guarantee this convergence for a given flow, insofar as it is represented by a solution to the NSE. This property has since led to several developments in the understanding and application of the long-time behavior of solutions to the NSE, for instance, in the context of turbulence, data assimilation, and the existence of determining forms. In this talk, we will discuss this asymptotic enslavement property as it regards the issue of uniqueness of invariant measures for stochastically forced equations, particularly the damped, stochastic KdV equation.

A new path-following primal-dual augmented Lagrangian method is proposed
for solving nonlinear equality constrained optimization problems (NEP).
At each iteration, a Newton-like method is used to solve a perturbed
optimality condition that defines a penalty trajectory parameterized by
both the penalty parameter and the estimated Lagrange multipliers.
We show that this method is globally convergent and has a quadratic
convergence rate in the limit. Finally, numerical experiments on
problems from the CUTEst test collection are are used to support the
theoretical analysis.

The unit distance problem asks: given $n$ points in $\mathbb R^d$, how many pairs of points can have distance one? This problem can be re-phrased as a question in incidence geometry, and standard machinery from that field leads to certain non-optimal bounds. I will discuss some recent progress on the unit distance problem in three dimensions that goes beyond the standard tools of incidence geometry.

Amphiphilic structures such as cell membranes and lipid vesicles play essential roles in biological applications. In this talk, we will first introduce the functionalized Cahn-Hilliard (FCH) model for the free energy of amphiphilic mixtures. The FCH model admits local minimizers corresponding to amphiphilic bilayers, filaments, micelles, and other defect structures. We will describe the geometric motion of bilayers, filaments, and their competition. To capture the coexistence and localization of amphiphilic structures, we introduce a degenerate FCH functional as a modified model for the free energy of amphiphilic mixtures. We prove that the degenerate FCH functional admits geometrically localized minimizers, which correspond to localized amphiphilic structures. Specifically, we identify the leading order profile of bilayers under the assumption that the geometrically localized minimizers have bounded variations along the tangential directions.

For each $-1 < q < 1$, Bozejko and Speicher's q-Gaussian functor is a natural and important generalization of Voiculescu's free Gaussian functor. We study a class of subalgebras of the corresponding q-Gaussian von Neumann subalgebras. We construct a Riesz basis in the spirit of Radulescu in the q-Fock space. Then we use this basis and follow Popa's approach to show that when $|q| < 1/9$ , the generator subalgebras are maximal amenable inside those q-Gaussian algebras. This is joint work with Sandeepan Parekh and Koichi Shimada.

In his influential paper ``Modular curves and the Eisenstein ideal,''
Barry Mazur studied congruences modulo p between cusp forms and the
Eisenstein series of weight 2 and prime level N. In particular, he
defined the Eisenstein ideal in the relevant Hecke algebra, and showed
that it is locally principal. We'll discuss the analogous situation for
certain squarefree levels N, and show that, while the Eisenstein ideal
may not be locally principal, we can count the minimal number of
generators and explain the arithmetic significance of this number. This
is joint work with Carl Wang-Erickson.

The theta graph $\Theta_{k, t}$ consists of two vertices with $t$ internally disjoint paths of length $k$ between them. Since $\Theta_{k,2}$ is a cycle of length $2k$, the study of the Tur\'an number of this graph includes the notorious even-cycle problem. We show that for fixed $k$ and large $t$, a graph without a $\Theta_{k,t}$ contains at most $c_k t^{1-1/k}n^{1+1/k}$ edges, and we use graphs constructed via random polynomials to show that this is sharp up to the constant $c_k$ when $k$ is odd.

In the 1990's, Vafa-Witten tested S-duality of N=4 SUSY Yang-Mills theory on a complex algebraic surface X by studying modularity of a certain partition function. In 2017, Tanaka-Thomas defined Vafa-Witten invariants by constructing a symmetric perfect obstruction theory on the moduli space of Higgs pairs (E,$\phi$) on X. The instanton contribution ($\phi=0$) to these invariants is the virtual Euler number of moduli space of sheaves. I outline a method to calculate this contribution, when X is of general type, by reducing to descendent Donaldson invariants. For rank 2, this leads to verifications of a formula from Vafa-Witten. The method can be ``refined'' to virtual $\chi_y$ genus, elliptic genus, and cobordism class, which involves weak Jacobi forms and Borcherds lifts thereof. I also give a new formula for rank 3 VW invariants on general type surfaces, correcting an error in the physics literature. Joint with G$\ddot{\text{o}}$
ttsche.

The Burau representation of the braid group was introduced in the 1930s and has been used since the 1980s to understand the Jones polynomial. A critical component of this study has been trying to understand when the Burau representation is faithful. It is known to be faithful for the Braid group on fewer than 4 strands and unfaithful on more than 4 strands. If the representation is unfaithful on 4 strands, it will solve the Jones unknot conjecture. We will review the current state of knowledge and some new approaches suggested by the work of Armond, Huynh and L\^e.

We first discuss the inviscid limit of the global energy dissipation of Leray solutions of incompressible Navier-Stokes on the torus. Assuming that the solutions have Besov norms bounded uniformly in viscosity, we establish an upper bound on energy dissipation. As a consequence, Onsager-type ``quasi-singularities'' are required in the Leray solutions, even if the total energy dissipation is vanishes in the limit v $\rightarrow$ 0. Next, we discuss an extension
of Onsager's conjecture for domains with solid boundaries. We give a localized regularity condition for energy conservation of weak solutions of the Euler equations assuming (local) Besov regularity of the velocity with exponent >1/3 and, on an arbitrary thin layer around the boundary, boundedness of velocity, pressure and continuity of the wall-normal velocity. We also prove that the global viscous dissipation vanishes in the inviscid limit for Leray-Hopf solutions of the Navier-Stokes equations under the similar assumptions, but holding uniformly in a vanishingly thin viscous boundary layer. Finally, if a strong Euler solution exists, we show that equicontinuity at the boundary within a O(v) strip alone suffices to conclude the absence of anomalous dissipation.
The talk concerns joint work with G. Eyink and H.Q. Nguyen.

In this talk, space-time finite element methods will be presented as an approach to solving PDEs with time-dependence. In these methods, time is treated in the same way as the spatial variables, leading to a fully unstructured finite element discretization in space-time. The fundamental properties of this approach will be discussed, especially in contrast to the Method of Lines. In particular, the efficiency of space-time methods on massively parallel computers will be discussed, as well as the importance of this property in modern numerical software.

We will discuss two different approaches to computing the L-functions of Shimura varieties associated with GU(2,1). Both approaches employ the comparison of the Grothendieck-Lefschetz formula with the Arthur-Selberg trace formula. The first approach, carried out by the author, takes as its starting point the recent work of Laumon and Morel. The second approach is due to Flicker. In both approaches, the principal challenge is that the Shimura varieties in question are non-compact, and one must use cohomology with compact supports. Time permitting, we will discuss the prospects for extending these approaches to the non-compact Shimura varieties associated with higher-rank unitary groups.

Motion by mean curvature for networks of surfaces arises in a variety of applications, such as the dynamics of foam and the evolution of microstructure in polycrystalline materials. It is steepest descent (gradient flow) for an energy: the sum of the areas of the surfaces constituting the network.

During the evolution, surfaces may collide and junctions (where three or more surfaces meet) may merge and split off in myriad ways as the network coarsens in the process of decreasing its energy. The first idea that comes to mind for simulating this evolution -- parametrizing the surfaces and explicitly specifying rules for cutting and pasting when collisions occur -- gets hopelessly complicated. Instead, one looks for algorithms that generate the correct motion, including all the necessary topological changes, indirectly but automatically via just a couple of simple operations.

A remarkably elegant such algorithm, known as threshold dynamics, was proposed by Merriman, Bence, and Osher in 1992. Extending this algorithm, while preserving its simplicity, to more general energies where each surface in the network is measured by a different, possibly anisotropic, notion of area requires new mathematical understanding of the original version, which then elucidates a systematic path to new algorithms.

I will construct a pair of Calabi-Yau threefolds which are deformation and derived equivalent, but not birationally equivalent. I will explain how this gives a counterexample to the birational Torelli problem for Calabi-Yau threefolds, as well as new examples of zero divisors in the Grothendieck ring of varieties. This is joint work with Lev Borisov and Andrei Caldararu.

We revisit the preservation of analyticity and Gevrey regularity for the Euler equation. We provide a result on preservation of Gevrey norm and analyticity in Lagrangian formulation of the Euler equation and discuss the validity of the result in the Eulerian variables. Next, we consider the Navier-Stokes equations posed on the half space, with Dirichlet boundary conditions. We give a direct energy based proof for the instantaneous space-time analyticity and Gevrey class regularity of the solutions, uniformly up to the boundary of the half space.

This is a brief introduction to mathematical programming with complementarity constraints.
This type of optimization problem is ill-posed in the usual sense. Stationarities and
constraint qualifications for this problem are introduced, together with some existing methods.

Free entropy theory is an analogue of information theory in a non-commutative setting, which has had great applications to the examination of structural properties of von Neumann algebras. I will discuss some ongoing joint work with Paul Skoufranis to extend this approach to the setting of bi-free probability which attempts to study simultaneously ``left'' and ``right'' non-commutative variables. I will speak in particular of an approach to a bi-free Fisher information and bi-free conjugate variables -- analogues of Fisher's information measure and the score function of information theory. The focus will be on constructing these tools in the non-commutative setting, and time permitting, I will also mention some results such as bi-free Cramer-Rao and Stam inequalities, and some quirks of the bi-free setting which are not present in the free setting.

Trying to understand the mechanisms of chemotaxis,during which biological cells move in the presence of chemical gradients, is a challenging and fascinating problem. Chemotaxis and cell migration are key components in a multitude of biological processes, including neuronal patterning, wound healing, embryogenesis, and cancer metastasis. Even though many of the key components, cell migration remains a poorly understood process. In this talk, I will present our modeling and experimental efforts aimed towards a better quantitative and mechanistic understanding of chemotaxis and cell motility.

Given an n-tuple of non-commutative random variables, its free Stein discrepancy relative to the semicircle law measures how ​``close '' the distribution is to the semicircle law. By considering free Stein discrepancies relative to a broader class of laws, one can define a quantity called the free Stein information. In this talk, we will discuss this and its relation to other free probabilistic quantities such as the free Fisher information and the non-microstates free entropy dimension. This is based on joint work in progress with Ian Charlesworth.

For a tracial $*$-probability space $(\mathcal{A}, \varphi)$, C\'{e}bron, Dahlqvist, and Male constructed an enveloping traffic space $(\mathcal{G}(\mathcal{A}), \tau_\varphi)$ that extends the trace $\varphi$. The CDM construction provides a universal object that allows one to appeal to the traffic probability framework in generic situations, prioritizing an understanding of its structure. We show that $(\mathcal{G}(\mathcal{A}), \tau_\varphi)$ comes equipped with a canonical free product structure, regardless of the choice of $*$-probability space $(\mathcal{A}, \varphi)$. If $(\mathcal{A}, \varphi)$ is itself a free product, then we show how this additional structure lifts into $(\mathcal{G}(\mathcal{A}), \tau_\varphi)$. Here, we find a duality between classical independence and free independence. We apply our results to study the asymptotics of large (possibly dependent) random matrices, generalizing and providing a unifying framework for results of Bryc, Dembo, and Jiang and of Mingo and Popa. This is joint work with Camille Male.

A real symmetric matrix (resp., tensor) is said to be copositive if the associated quadratic (resp., homogeneous) form is greater than or equal to zero over the nonnegative orthant. The problem of detecting their copositivity is NP-hard. This paper proposes a complete semidefinite relaxation algorithm for detecting the copositivity of a matrix or tensor. If it is copositive, the algorithm can get a certificate for the copositivity. If it is not, the algorithm can get a point that refutes the copositivity. We show that the detection can be done by solving a finite number of semidefinite relaxations, for all matrices and tensors.

In this talk I will describe a family of events arising in two related probability models, one having to do with uniformly random ``fully packed loops'' (a family of combinatorial objects which are in bijection with alternating sign matrices), and another appearing in connection with a natural random walk on noncrossing matchings. The connection between the two models is highly nonobvious and was conjectured by physicistsRazumov and Stroganov in 2001, and given a beautiful proof in 2010 by Cantini and Sportiello. Another intriguing phenomenon is that the probabilities of the events in question, known as ``connectivity events'', appear to be rational functions of a size parameter N (with the simplest such formula being $3(N^2-1)/2(4N^2+1))$, but this is only conjectured in all but a few cases. The attempts to prove such formulas by myself and others have led to interesting algebraic results on a family of multivariate polynomials known as ``wheel polynomials'', and to a family of conjectural constant term identities that is of independent interest and poses an interesting challenge to algebraic combinatorialists.

A local system on the Riemann sphere minus finitely many
points is defined to be ``rigid'' if it is determined by the conjugacy
classes of its monodromy operators along the missing points. Katz proves
a convenient cohomological criterion characterizing irreducible rigid
local systems, which is based on an analysis of the moduli of local
systems on the punctured Riemann sphere. In this talk, we will discuss
this story, and then proceed towards an analogous story in the
arithmetic setting, where, in place of local systems on the punctured
Riemann sphere, we consider overconvergent isocrystals on the punctured
projective line over a field of positive characteristic.

The moving sofa problem is a well-known open problem in geometry. Posed by Leo Moser in 1966, it asks for the planar shape of largest area that can be moved around a right-angled corner in a two-dimensional hallway of width 1. Although deceptively easy to state, it turns out to be highly nontrivial to analyze, and has a rich structure that is intriguing to amateurs and experts alike. In this talk I will survey both old and new results about the problem, including a new moving sofa shape with an interesting algebraic structure that I discovered in 2016, and new bounds on the area of a moving sofa I derived more recently in joint work with Yoav Kallus. I will conclude with a discussion of how the heavily experimental and computer-assisted nature of the recent results offers broader lessons for aspiring research mathematicians.

The essential skeleton is an invariant of a degeneration that
appears in both Berkovich geometry and minimal model theory. I will show
that for degenerations with a semistable model, the essential skeleton of a
product of degenerations is the product of their skeleta.

As an application, we are able to describe the homeomorphism type of some
degenerations of hyperkähler varieties, both for a Hilbert scheme of
degenerating K3 surfaces and for a Kummer variety associated to a
degeneration of an abelian surface. This is joint work with Enrica Mazzon.

I will discuss four families of random matrices. The first two are
classical: a Gaussian measure on the space of $N\times N$ Hermitian matrices
(\textquotedblleft Gaussian unitary ensemble\textquotedblright) and a Gaussian
measure on the space of all $N\times N$ complex matrices (\textquotedblleft
Ginibre ensemble\textquotedblright). As $N\rightarrow\infty,$ the eigenvalues
of the Gaussian unitary ensemble concentrate onto an interval with a
semicircular density, while the eigenvalues of the Ginibre ensemble become
uniformly distributed in a disk in the complex plane.

Now, the space of $N\times N$ Hermitian matrices can be identified with the
Lie algebra $u(N)$ of the unitary group $U(N),$ and the Gaussian unitary
ensemble is the distribution of Brownian motion in $u(N).$ Similarly, the
space of all $N\times N$ matrices is the Lie algebra $gl(N;\mathbb{C})$ of the
general linear group $GL(N;\mathbb{C})$ and the Ginibre ensemble is the
distribution of Brownian motion in $gl(N;\mathbb{C}).$ It is then natural to
consider also Brownian motions in the groups $U(N)$ and $GL(N;\mathbb{C})$
themselves.

The eigenvalues for Brownian motion in $U(N)$ have a known limiting
distribution in the unit circle. The eigenvalues for Brownian motion in
$GL(N;\mathbb{C})$ have received little attention up to now. Assuming that the
eigenvalues have a limiting distribution, recent results of mine with Kemp
show that the limiting distribution is supported in a certain domain
$\Sigma_{t}$ in the complex plane. The figure shows the domain for $t=3.85$,
along with a plot of the eigenvalues for $N=2,000.$ One notably feature of the
domains is that they change topology from simply connected to doubly connected
at $t=4.$ I will give background on all four families of random matrices,
describe our new results, and mention some ideas in the proof.

The theory of congruences of modular forms is a central topic in
contemporary number theory.
Congruences between modular forms play a crucial role in understanding
links between geometry and arithmetic: cornerstone example of this is
the proof of Serre's modularity conjecture by Khare and Wintenberger.
Congruences of Galois representations govern many kinds of
representations of the absolute Galois group of number fields. Even
though our understanding is improving, many aspects remain very
mysterious, some are theoretically approachable, many are not; and
amongst the latter, some allow numerical studies to reveal first insights.
In this talk I will introduce congruence graphs, which are graphs
encoding congruence relations between classical newforms. Then I will
explain first how to construct analogous graphs for congruences of
Galois representations, and then how to use these graphs to study
questions regarding Hecke algebras and Atkin-Lehner operators.

We introduce a rate balance principle for general (not necessarily Markovian) stochastic processes. Special attention is given to processes with birth and death like transitions, for which it is shown that for any state i, the rate of two consecutive transitions from i−1 to i+1, coincides with the corresponding rate from i+1 to i−1. This observation appears to be useful in deriving well-known, as well as new, results for the Mn/Gn/1 and G/Mn/1 queueing systems, such as a recursion on the conditional distributions of the residual service times (in the former model) and of the residual inter-arrival times (in the latter one), given the queue length. The talk is based on B. Oz. I Adan and M. Haviv, ``A rate balance principle and it application to queueing models,'' Queueing Systems: Theory and Applications, 87, 95-111 (2017).

A Riemannian metric gives rise to geodesics. As unparameterised curves, at each point there is one geodesic in every direction. Does this arrangement of curves determine the metric and will any such arrangement of curves determine a metric? These are classical questions considered, for example, by Roger Liouville in 1889 who made considerable progress in two dimensions. This talk will discuss these questions in two and three dimensions, mostly based on joint work with Robert Bryant and Maciej Dunajski.

There are many well-known concentration results for sums of independent random matrices, e.g. those of Rudelson, Ahlswede-Winter, Tropp, and Oliveira. We move beyond the independent setting, and prove a Chernoff-type bound for sums of matrix-valued random variables sampled via a random walk on an reversible Markov chain, confirming a conjecture due to Wigderson and Xiao. Our proof is based on a new multi-matrix extension of the Golden-Thompson inequality which follows from complex interpolation methods.
Joint work with A. Garg, Y. Lee, and Z. Song.

The motion of a flying saucer is restricted by the three-dimensional geometry of the space in which it moves. I shall discuss three possible cases, one of which may be interpreted as an explanation of Engel's 1893 construction of the exceptional Lie algebra G2. This is joint work with Pawel Nurowski.

Donaldson-Thomas (abbreviated as DT) theory is a sheaf theoretic technique of enumerating curves on a Calabi-Yau threefold. Classical DT invariants give a virtual count of Gieseker stable sheaves provided that no strictly semistable sheaves exist. This assumption was later lifted by the work of Joyce and Song who defined generalized DT invariants using Hall algebras and the Behrend function, their method being motivic in nature. In this talk, we will present a new approach towards generalized DT theory, obtaining an invariant as the degree of a virtual cycle inside a Deligne-Mumford stack. The main components are an adaptation of Kirwan’s partial desingularization procedure and recent results on the structure of moduli of sheaves on Calabi-Yau threefolds. Based on joint work with Young-Hoon Kiem and Jun Li.

Biological systems make use of feedback in an extraordinary number of ways, on scales
ranging from molecules to cells to organisms to ecosystems. In this talk I will discuss the use
of concepts from feedback control theory in the design of feedback circuits at the molecular,
cellular, and multicellular level. After a brief survey of relevant concepts from control theory
and synthetic biology, I will present some recent results that demonstrate how feedback
controllers can be implemented in cells, including ratiometric control of protein
concentrations, reference tracking of cell concentration, and fractional control of cell
populations. Using these examples and others from the literature, I will discuss some of the
common features and possible architectures for implementation of biomolecular feedback
control systems, some limits on performance predicted by the theory, and some
implementation challenges that require more effort and new approaches.

In this talk we use the \emph{unit-graphs} and the \emph{special unit-digraphs} on matrix rings to show that every $n \times n$ nonzero matrix over $\Bbb F_q$ can be written as a sum of two $\operatorname{SL}_n$-matrices when $n>1$. We compute the eigenvalues of these graphs in terms of Kloosterman sums and study their spectral properties; and prove that if $X$ is a subset of $\operatorname{Mat}_2 (\Bbb F_q)$ with size $|X| > \frac{2 q^3 \sqrt{q}}{q - 1}$, then $X$ contains at least two distinct matrices whose difference has determinant $\alpha$ for any $\alpha \in \Bbb F_q^{\ast}$. Using this result we also prove a sum-product type result: if $A,B,C,D \subseteq \Bbb F_q$ satisfy $\sqrt[4]{|A||B||C||D|}= \Omega (q^{0.75})$ as $q \rightarrow \infty$, then $(A - B)(C - D)$ equals all of $\Bbb F_q$. In particular, if $A$ is a subset of $\Bbb F_q$ with cardinality $|A| > \frac{3} {2} q^{\frac{3}{4}}$, then the subset $(A - A) (A - A)$ equals all of $\Bbb F_q$. We also recover a classical result: every element in any finite ring of odd order can be written as the sum of two units.

Studying invariant theory of commutative polynomial rings has motivated many developments in commutative algebra and algebraic geometry. For a finite group acting on a polynomial ring, the remarkable Chevalley-Shephard-Todd Theorem proves that the fixed subring is isomorphic to a polynomial ring if and only if the group is generated by pseudo-reflections. Related questions try to find properties of the fixed ring under some special group actions. In recent years, progress was made in work of Jing, Jorgensen, Kirkman, Kuzmanovich, Walton, Zhang, and others to extend the theory to regular algebras which are a noncommutative generalization of polynomial rings. Naturally, the question arises if the theory generalizes further to non-connected noncommutative algebras. This talk answers the question what conditions need to be satisfied by the fixed ring in order to make a rich theory possible. Moreover, we construct a homological determinant for preprojective algebras and discuss how it being trivial for all elements of a finite group affects the related fixed ring.

In neuroscience, functional connectivity describes the connectivity between
brain regions that share functional properties. It is often characterized by a time
series of covariance matrices between functional measurements of distributed
neuron areas. An effective statistical model for functional connectivity and
its changes over time is critical for better understanding brain functions and
neurological diseases. To this end, we propose a matrix-log mean model with an
additive heterogeneous noise for modeling random symmetric positive definite
matrices that lie in a Riemannian manifold. We introduce the heterogeneous
error terms to capture the curved nature of the nonlinear manifold. A scan
statistic is then developed for the purpose of multiple change point detection.
Theoretically, we establish the sure coverage property. Simulation studies and
an application to the Human Connectome Project lend further support to the
proposed methodology.

In this talk we investigate the Erd\H{o}s-Renyi model for random graphs, $G(n,p)$. Our focus will be on determining the probability that $G(n,p)$ is connected, which was the first problem that Erd\H{o}s and Renyi considered in their original paper. As time permits we will also discuss the ``phase transition'' of $G(n,p)$.

In many binary classification applications, such as disease diagnosis and spam detection, practitioners commonly face the need to limit type I error (that is, the conditional probability of misclassifying a class 0 observation as class 1) so that it remains below a desired threshold. To address this need, the Neyman-Pearson (NP) classification paradigm is a natural choice; it minimizes type II error (that is, the conditional probability of misclassifying a class 1 observation as class 0) while enforcing an upper bound, alpha, on the type I error. Although the NP paradigm has a century-long history in hypothesis testing, it has not been well recognized and implemented in classification schemes. Common practices that directly limit the empirical type I error to no more than alpha do not satisfy the type I error control objective because the resulting classifiers are still likely to have type I errors much larger than alpha. This talk introduces the speaker's work on NP classification algorithms and their applications and raises current challenges under the NP paradigm.

For fixed degree $d$, one could ask for a meaningful compactification of the moduli space of smooth degree $d$ surfaces in $\mathbb{P}^3$. In other words, one could ask for a parameter space whose interior points correspond to [isomorphism classes of] smooth surfaces and whose boundary points correspond to degenerations of these surfaces. Motivated by Hacking's work for plane curves, I will discuss a KSBA compactification of this space by considering a surface $S$ in $\mathbb{P}^3$ as a pair $(\mathbb{P}^3, S)$ satisfying certain properties. We will study an enlarged class of these pairs, including singular degenerations of both $S$ and the ambient space. The moduli space of the enlarged class of pairs will be the desired compactification and, as long as the degree $d$ is odd, we can give a rough classification of the objects on the boundary of the moduli space.

A long-standing open question, formalized by Blackadar and Kirchberg in the mid 90's, asks for an abstract characterization of C$^*$-subalgebras of AF-algebras. I will discuss some recent progress on this question: every separable, exact C$^*$-algebra which satisfies the UCT and admits a faithful, amenable trace embeds into an AF-algebra. Moreover, the AF-algebra may be chosen to be simple and unital with unique trace and the embedding may be taken to be trace-preserving. Modulo the UCT, this characterizes C$^*$-subalgebras of simple, unital AF-algebras. As an application, for any countable, discrete, amenable group $G$, the reduced C$^*$-algebra of $G$ embeds into a UHF-algebra.

The motivating problems in geometry are classification problems. One tries to understand the isomorphism classes of certain types of geometric objects. A peculiar feature of algebraic geometry is that there is often a variety or scheme, called the moduli space, whose points correspond to isomorphism classes of the objects we want to study. In this talk, I will talk about the history of moduli problems and spaces, and explain some of the mathematics behind the contributions Grothendieck made to the subject.

In the celebrated work of Bershadsky--Cecotti--Ooguri--Vafa the genus one string partition function in the B-model is identified with certain analytic torsion of the Hodge Laplacian on a K$\ddot{\text{a}}$hler manifold. In a joint work with Shu Shen (IMJ-PRG) and Jianqing Yu (USTC) we study the analogous torsion in Landau--Ginzburg models. I will explain the corresponding index theorem based on the asymptotic expansion of the heat kernel of the Schr$\ddot{\text{o}}$dinger operator. I will also explain the rigorous definition of the BCOV torsion for homogeneous polynomials on ${\mathbb C}^N$. Lastly I will explain the conjecture stating that in the Calabi--Yau case the BCOV torsion solves the holomorphic anomaly equation for marginal deformations.

We consider $\sigma_k$-curvature equation with $H_k$-curvature condition on a compact manifold with boundary $(X^{n+1}, M^n, g)$. When restricting to the closure of the positive $k$-cone, this is a fully nonlinear elliptic equation with a fully nonlinear Robin-type boundary condition. We prove a general bifurcation theorem in order to study nonuniqueness of solutions when 2k is less than n. We explicitly give examples of product manifolds with multiple solutions. It is analogous to Schoen example for Yamabe problem on $S^1\times S^{n-1}$. This is joint work with Jeffrey Case and Ana Claudia Moreira.

Full Approximation Scheme (FAS) is a widely used multigrid method for nonlinear problems. In this talk, we shall provide a new framework to analyze FAS for convex optimization problems and improve the original method. We view FAS as an inexact version of nonlinear multigrid methods based on space decomposition and subspace correction. The local problem in each subspace can be simplified to be linear and one gradient decent iteration is enough to ensure a linear convergence.

This is a joint work with Steve Wise (University of Tennessee) and Xiaozhe Hu (Tuffs University).

Book proofs for preferential attachment models are found through continuous
time processes with exponential waiting times. In turning these arguments around
we find large deviation results and a description of the process conditional on remaining
stable.
Joint work with Subhabrata Sen and Svante Janson

Today the main emphasis in local number theory (i.e the Local
Langlands) is on the finite places. In charactacteristic 0 the infinite
place is the ``elephant in the room''. This is especially true in the
Whittaker Theory in which serious difficulties separate the infinite from
the finite places. Whittaker models were developed to help the study of
Fourier coefficients at cusps of non-holomophic cusp forms (i.e Maass cusp
forms) through representation theory. The first of these lectures will start
with an explanation of the role of Whittaker models in the theory of
automorphic forms. It will continue with a description of the main results.
The second lecture will explain the proof of the Whittaker Plancherel
Theorem.

The Schottky problem is a classical and fundamental question in arithmetic algebraic geometry,
about the characterization of Jacobian varieties among abelian varieties.
This question is equivalent to studying the Torelli locus (i.e., the image of moduli of curves under the Torelli map) inside Siegel modular varieties.
In positive characteristics, a first approximation to this problem is understanding the discrete invariants (e.g., p-rank, Newton polygon, Ekedahl--Oort type) occurring for Jacobians of smooth curves. The Coleman--Oort conjecture predicts that if the genus is large, then up to isomorphism, there are only finitely many smooth projective curves over the field of complex numbers, of genus g and Jacobian an abelian variety with complex multiplication. An effective version of the Colemann--Oort conjecture proposes 8 as an explicit lower bound.

After introducing the framework for these problems, I will discuss recent progress towards the Schottky problem in positive characteristics which is inspired by the Coleman--Oort conjecture, and which relies on our understanding of special subvarieties (a.k.a, Shimura subvarieties) of Siegel varieties.

Birkar and Zhang recently introduced the notion of generalized pair. These pairs are closely related to the canonical bundle formula and have been a fruitful tool for recent developments in birational geometry. In this talk, I will introduce a version of the canonical bundle formula for generalized pairs. This machinery allows us to develop a theory of adjunction and inversion thereof for generalized pairs. I will conclude by discussing some applications to a conjecture of Prokhorov and Shokurov.

Maximizing computer system performance relies on careful resource management: how to best allocate resources among jobs. Effective resource allocation is most difficult in regimes with uncertainty. This talk examines three common types of uncertainty. We consider uncertainty in job sizes and ask how to optimally schedule jobs to minimize response time in such regimes. We next turn to uncertainty in the arrival rate and ask how we should adapt capacity provisioning and power management in data centers to handle unexpected load fluctuations. Finally, we consider uncertainty in the system state and look at how job replication can help curtail unpredictability. A common thread in this talk is stochastic performance modeling and the insights it illuminates.

Motivated by questions in pointwise ergodic theory, modern discrete harmonic analysis, as developed by Bourgain, has focused on understanding the oscillation of averaging operators -- or related singular integral operators -- along polynomial curves. In this talk we present the first example of a discrete analogue of polynomially modulated oscillatory singular integrals; this begins to unify the work of Bourgain, Stein, and Stein-Wainger. The argument combines a wide range of techniques from Euclidean harmonic analysis and analytic number theory.

We give a systematic and self-contained account of the construction of
geometrically decomposed bases and degrees of freedom in finite element
exterior calculus. In particular, we elaborate upon a previously overlooked
basis for one of the families of finite element spaces, which is of interest
for implementations. Moreover, we give details for the construction of
isomorphisms and duality pairings between finite element spaces. These
structural results show, for example, how to transfer linear dependencies
between canonical spanning sets, or give a new derivation of the degrees of
freedom.

Loosely defined, a sieve is a mathematical technique for finding the rate of growth of a set of objects with a quantifiable (and hopefully small) error term. Sieve techniques have wide applications in number theory and combinatorics. We will first present the idea behind sieving and present a toy example of calculating the probability that an integer is squarefree. Then we will discuss Poonen’s recently developed algebro-geometric sieve, which allows one to compute the probability that a hypersurface intersects smoothly with a given projective variety X over a finite field.

An $(\alpha,\beta)$-spanner of a graph G is a subgraph H that distorts distances in G up to a multiplicative factor of $\alpha$ and an additive factor of $\beta$, where the goal is to construct an H with as few edges as possible. When $\beta=0$ we call H a multiplicative spanner, and when $\alpha=1$ we call H an additive spanner. It is known how to construct multiplicative spanners of essentially optimal size, but much less is known about additive spanners. In this talk we discuss a recent result which shows how to construct a (0,6)-additive spanner for any graph G.

High-dimensional high-order data arise in many modern scientific applications including genomics, brain imaging, and social science. In this talk, we consider the methods, theories, and computations for tensor singular value decomposition (tensor SVD), which aims to extract the hidden low-rank structure from high-dimensional high-order data. First, comprehensive results are developed on both the statistical and computational limits for tensor SVD under the general scenario. This problem exhibits three different phases according to signal-noise-ratio (SNR), and the minimax-optimal statistical and/or computational results are developed in each of the regimes. In addition, we further consider the sparse tensor singular value decomposition which allows more robust estimation under sparsity structural assumptions. A novel sparse tensor alternating thresholding algorithm is proposed. Both the optimal theoretical results and numerical analyses are provided to guarantee the performance of the proposed procedure.

At the beginning of section 11 in Perelman's celebrated paper ``The entropy formula for the Ricci flow and its geometric applications'', he made the assertion that for an ancient solution to the Ricci flow with bounded nonnegative curvature operator, bounded entropy is equivalent to noncollapsing on all scales. We give a proof for this assertion.

Today the main emphasis in local number theory (i.e the Local
Langlands) is on the finite places. In charactacteristic 0 the infinite
place is the ''elephant in the room''. This is especially true in the
Whittaker Theory in which serious difficulties separate the infinite from
the finite places. Whittaker models were developed to help the study of
Fourier coefficients at cusps of non-holomophic cusp forms (i.e Maass cusp
forms) through representation theory. The first of these lectures will start
with an explanation of the role of Whittaker models in the theory of
automorphic forms. It will continue with a description of the main results.
The second lecture will explain the proof of the Whittaker Plancherel
Theorem.

In this talk, we will prove a classical theorem which states that the simple random walk on the integer lattice $\mathcal{Z}^d$ is
recurrent in the case where $d=1,2$ and transient in the case where $d \geq 3$. In particular, we plan to focus on the combinatorial nature of the proof.

A numerical semigroup is a subset of the natural numbers which is closed under addition. Consider a numerical semigroup S selected via the following random process: fix a probability p and a positive integer M , and select a generating set for S from the integers 1, 2, . . . , M where each generator has probability p of being selected. What properties can we expect the numerical semigroup S to have? For instance, how many minimal generators do we expect S to have? In this talk, we answer several such questions, and describe some surprisingly deep geometric and combinatorial structures that arise naturally in this process.
No familiarity with numerical semigroups or probability will be assumed for this talk.

We study regularity properties of the free boundary for solutions of the porous medium equation with the presence of drift. We show that if the initial data has super-quadratic growth at the free boundary, then the support strictly expands relative to the streamline, and that the movement is Holder continuous in time. Under additional information of directional monotonicity in space, we derive nondegeneracy of solutions and $C^{1,\alpha}$ regularity of free boundaries. Finally several examples of singularities are given that illustrate differences from the zero drift case.

``Who doesn't like those?''

Combinatorial neural codes are 0/1 vectors that are used to model the co-firing patterns of a set of place cells in the brain. One wide-open problem in this area is to determine when a given code can be algorithmically drawn in the plane as a Venn diagram-like figure. A sufficient condition to do so is for the code to have a property called k-inductively pierced. Gross, Obatake, and Youngs recently used toric algebra to show that a code on three neurons is 1-inductively pierced if and only if the toric ideal is trivial or generated by quadratics. No result is known for additional neurons in the same generality.

In this talk, we study two infinite classes of combinatorial neural codes in detail. For each code, we explicitly compute its universal Gröbner basis. This is done for the first class by recognizing that the codewords form a Lawrence-type matrix. With the second class, this is done by showing that the matrix is totally unimodular. These computations allow one to compute the state polytopes of the corresponding toric ideals, from which all distinct initial ideals may be computed efficiently. Moreover, we show that the state polytopes are combinatorially equivalent to well-known polytopes: the permutohedron and the stellohedron.

\textbf{Small KAM perturbations of integrable systems which are entropy expansive.} One of the greatest achievements in Dynamics in the XX century is the KAM Theory. It says that after a small perturbation of a non-degenerate completely integrable system it still has an overwhelming measure of invariant tori with quasi-periodic dynamics. What happens outside KAM tori remains a great mystery. It is easy, by modern standards, to show that topological entropy can be positive. It lives, however, on a zero measure set. We are now able to show that metric entropy can become infinite too, under arbitrarily small $C^{\infty}$ perturbations, answering an old-standing problem of Kolmogorov. Furthermore, a slightly modified construction resolves another long standing problem of the existence of entropy non-expansive systems. In these modified examples positive metric entropy is generated in arbitrarily small tubular neighborhoods of one trajectory. Joint with S. Ivanov and Dong Chen.

\noindent \textbf{Metric approximations of length spaces by graphs with uniformly bounded local structure.} How well can we approximate an (unbounded) space by a metric graph whose parameters (degrees of vertices, lengths of edges, density of vertices etc) are uniformly bounded? We want to control the ADDITIVE error. Some answers are given (the most difficult case is $\mathbb{R}^2$) using dynamics and Fourier series. Joint with S. Ivanov.

\noindent \textbf{On Busemann's problem on minimality of flats in normed spaces for the Buseman-Hausdorff surface area.} Busemann asked if regions in affine subspaces of normed spaces are area minimizers with respect to the Busemann-Hausdorff measure. This has been known for long for hyperplanes (codim=1), this is a classic result in Convex Geometry. Sergei Ivanov and me were able to prove this for 2-dimensional subspaces.

Grothendieck's inequality guarantees that a certain discrete optimization problem-optimizing a bilinear form over +1, -1 inputs—is equivalent up to a constant, called Grothendieck's constant, to a continuous optimization problem -- optimizing the same bilinear form over unit vectors in a Hilbert space. This is important for convex relaxation, because the former contains NP-hard problems such as max-cut, whereas the latter is a semidefinite program and can be solved in polynomial time. A world apart from convex
relaxation is algebraic computational complexity, where it is well-known that the exponent of matrix multiplication is exactly the sharp lower bound for the (log of) tensor rank of the Strassen matrix multiplication tensor. We show that Grothendieck's constant is the sharp upper bound on a tensor norm of Strassen matrix multiplication tensor. Hence these two important quantities from disparate areas of theoretical computer science—Strassen's exponent of matrix multiplication and Grothendieck's constant—are just different measures of the same underlying tensor. This allows us to rewrite Grothendieck's inequality as a norm inequality for a 3-tensor, which in turn gives a family of natural generalized inequalities. We show that these are locally sharp and prove that Grothendieck's inequality is unique.

\textbf{Counting collisions.}20 years ago the topic of my talk at the ICM was a solution of a problem which goes back to Boltzmann and Ya. Sinai. The conjecture of Boltzmann-Sinai states that the number of collisions in a system of $n$ identical balls colliding elastically in empty space is uniformly bounded for all initial positions and velocities of the balls. The answer is affirmative and the proven upper bound is exponential in $n$. The question is how many collisions can actually occur. On the line, there can be $n(n-1)/2$ collisions, and this is he maximum. Since the line embeds in any Euclidean space, the same example works in all dimensions. The only non-trivial (and counter-intuitive) example I am aware of is an observation by Thurston and Sandri who gave an example of 4 collisions between 3 balls in $R^2$. Recently, Sergei Ivanov and me proved that there are examples with exponentially many collisions between $n$ identical balls in $R^3$, even though the exponents in the lower and upper bounds do not match.

\noindent \textbf{A survival guide for a feeble fish and homogenization of the G-Equation.} How fish can get from A to B in turbulent waters which maybe much fasted than the locomotive speed of the fish provided that there is no large-scale drift of the water? This is related to G-Equation and has applications to its homogenization. G-equation which is believed to govern many combustion processes. Based on a joint work with S. Ivanov and A. Novikov.

Regarding time as a scaling parameter, we prove the one-point, lower tail Large Deviation Principle (LDP) of the KPZ equation, with an explicit rate function. This result confirms existing physics predictions. We utilize a formula from [Borodin Gorin 16] to convert LDP of the KPZ equation to calculating an exponential moment of the Airy point process, and analyze the latter via stochastic Airy operator and Riccati transform

Smoothing spline mixed-effects density models are proposed for the nonparametric
estimation of density and conditional density functions with clustered data.
The random effects in a density model introduce within-cluster correlation and
allow us to borrow strength across clusters by shrinking cluster specific density
function to the population average, where the amount of shrinkage is decided
automatically by data. Estimation is carried out using the penalized likelihood
and computed using a Markov chain Monte Carlo stochastic approximation algorithm.
We apply our methods to investigate evolution of hemoglobin density functions
over time in response to guideline changes on anemia management for dialysis
patients.

In 1970, Manin observed that the Brauer group Br(X) of a variety
X over a number field K can obstruct the Hasse principle on X. In other
words, the lack of a K-point on X despite the existence of points over
every completion of K is sometimes explained by non-trivial elements in
Br(X). This so-called Brauer-Manin obstruction may not always suffice to
explain the failure of the Hasse principle but it is known to be sufficient
for some classes of varieties (e.g. torsors under connected algebraic
groups) and conjectured to be sufficient for rationally connected varieties
and K3 surfaces.
A zero-cycle on X is a formal sum of closed points of X. A rational point
of X over K is a zero-cycle of degree 1. It is interesting to study the
zero-cycles of degree 1 on X, as a generalisation of the rational points.
Yongqi Liang has shown that for rationally connected varieties, sufficiency
of the Brauer-Manin obstruction to the Hasse principle for rational points
over all finite extensions of K implies sufficiency of the Brauer-Manin
obstruction to the Hasse principle for zero-cycles of degree 1 over K. In
this talk, I will discuss joint work with Francesca Balestrieri where we
extend Liang's result to Kummer varieties.

The ligand-receptor binding/unbinding is a complex biophysical process in which water plays a critical role. To understand the fundamental mechanisms of such a process, we have developed a new and efficient approach that combines our level-set variational implicit-solvent model with the string method for transition paths, and have studied the pathways of dry-wet transition in a model ligand-receptor system. We carry out Brownian dynamics simulations as well as Fokker-Planck equation modeling with our efficiently calculated potentials of mean force to capture the effect of solvent fluctuations to the binding and unbinding processes. Without the description of individual water molecules, we have been able to predict the binding and unbinding kinetics quantitatively in comparison with the explicit-water molecular dynamics simulations. Our work indicates that the binding/unbinding can be controlled by a few key parameters, and provides a tool of efficiently predicting molecular recognition with application to drug design.

I will discuss the Einstein-Klein-Gordon coupled system of General Relativity, and the problem of global stability of the Minkowski space-time. This is joint work with Benoit Pausader.

The minimal log discrepancy of an algebraic variety is an invariant which measures the singularites of the variety. For mild singularities the minimal log discrepancy is a non-negative real value; the closer to zero this value is, the more singular the variety. It is conjectured that in a fixed dimension, this invariant satisfies the ascending chain condition. In this talk we will show how boundedness of Fano varieties imply some local statements about the minimal log discrepancies of klt singularities. In particular, we will prove that the minimal log discrepancies of klt singularities which admit an $\epsilon$-plt blow-up can take only finitely many possible values in a fixed dimension. This result gives a natural geometric stratification of the possible mld's on a fixed dimension by finite sets. As an application, we will prove the ascending chain condition for minimal log discrepancies of exceptional singularities in arbitrary dimension.

Consider the vanishing viscosity limit for the 2D steady Navier-Stokes equations in the region $0\leq x \leq L$ and $0 \leq y<\infty$ with no slip boundary conditions at $y=0.$ For $L<<1,$ we justify the validity of the steady Prandtl layer expansion for scaled Prandtl layers, including the celebrated Blasius boundary layer. This is joint work with Yan Guo.

We are currently witnessing an explosion of intelligent systems. The year 2012 is often cited as the beginning of the deep learning revolution which transformed the artificial intelligence industry. That year, many notable achievements were accomplished using deep neural networks, large training datasets, and powerful GPUs. In 2015, machine learning finally beat humans in classifying images after attempting for many years in the annual ImageNet competition. In this talk, we explain how deep neural networks work and show the progression of intelligent systems from the year 2000 until the present. We will focus on convolutional neural networks and image recognition tasks.

In this talk, we will explore the problem of the largest singular value of a random sign matrix. We will use the method of $\epsilon$-nets to show that there exists a $C>0$ so that the largest singular value of a random sign matrix of size $n$ is at least $C \sqrt{n}$ with exponentially high probability. While this is a highly combinatorial problem, the method of $\epsilon$-nets could of interest to those in other fields. This talk is based off Tao's book on Random Matrix theory and a recent talk he gave at the 27th Annual PCMI Summer Session on Random Matrices.

To each graded poset one can associate two sequences of numbers: the Whitney numbers of the first kind and the Whitney numbers of the second kind. One sequence keeps track of the M$\ddot{\text{o}}$bius function at each rank level and the other keeps track of the number of elements at each rank level. We say if $P$ and $Q$ are Whitney duals if the Whitney numbers of the first kind of $P$ are the Whitney numbers of the second kind of $Q$ and vice-versa. In this talk, we will discuss a method to construct Whitney duals. This method uses a new type of edge labeling as well as quotient posets. For posets which have this type of labeling, one can construct a simplicity complex whose $f$-vector encodes the Whitney numbers of the second kind of this poset. Time permitting, we will discuss this complex. This is joint work with Rafael S. Gonz\'alez D'Le\'on.

Gaussian mixture model(GMM) is a fundamental tool in applied statistics and machine learning given data from a weighted sum of several Gaussian distributions. The current practice for learning mixture of Gaussians inevitably has high computational and sample complexity which is exponential in the number of Gaussian components. It has been shown in recent work that such estimation can be reduced to the problem of decomposing a symmetric tensor derived from the moments. The decomposition of these specially structured tensors can be solved efficiently by several methods.

Parabolic flows are smoothing for short time however, over long time, singularities are typically unavoidable and can be very nasty. The key to understand such flows is to understand their singularities and the set where those singularities occur. We begin with discussing mean curvature flow and will explain which singularities are generic and what one can say about the short and long time dynamics near singularities. After that we turn to the question of optimal regularity of geometric flows in general. We will see that these seemingly different questions turn out to be related. The ideas draws inspiration from an number of different fields, including Geometry, Analysis, Dynamical Systems and Real Algebraic Geometry.

The polynomial method is a recent trend in combinatorics which draws from methods of algebraic geometry over finite fields. Instances of the theory have been known for some time, and include Stepanov's method for counting points on curves over finite fields or Alon's combinatorial nullstellensatz. In this talk we will follow an expository article of Tao [1] to present basic ideas behind the polynomial method, as well as several applications. Following Tao, ``we will assume as little prior knowledge of algebraic geometry as possible.''

In an exciting paper, J. Bedrossian and N. Masmoudi established the stability of the 2D Couette flow in Gevrey spaces of index greater than 1/2. I will talk about recent joint work with N. Masmoudi, which proves, in the opposite direction, the instability of the Couette flow in Gevrey spaces of index smaller than 1/2. This confirms, to a large extent, that the transient growth predicted heuristically in earlier works does exist and has the expected strength. The proof is based on the framework of the stability result, with a few crucial new observations. I will also discuss related works regarding Landau damping, and possible extensions to infinite time.

This talk focuses on developing a generalized bootstrap algebraic
multigrid algorithm for solving sparse matrix equations. As a
motivation of the proposed generalization, we consider an optimal form
of classical algebraic multigrid interpolation that has as columns
eigenvectors with small eigenvalues of the generalized eigen-problem
involving the system matrix and its symmetrized smoother. We use this
optimal form to design an algorithm for choosing and analyzing the
suitability of the coarse grid. In addition, it provides insights into
the design of the bootstrap algebraic multigrid setup algorithm that we
propose, which uses as a main tool a multilevel eigensolver to compute
approximations to these eigenvectors. A notable feature of the approach
is that it allows for general block smoothers and, as such, is well
suited for systems of partial differential equations. In addition, we
combine the GAMG setup algorithm with a least-angle regression
coarsening scheme that uses local regression to improve the choice of
the coarse variables. These new algorithms and their performance are
illustrated numerically for scalar diffusion problems with highly
varying (discontinuous) diffusion coefficient and for the linear
elasticity system of partial differential equations.

It is frequently the case that certain objects in differential topology/geometry can be described in purely algebraic terms, where the algebraic structures involved are constructed using the smooth structure(s) of the underlying manifold(s). For example, a common equivalent characterization of a smooth vector field on a smooth manifold $M$ is a derivation of the $\mathbb{R}$-algebra of smooth real-valued functions on $M$. I shall discuss this example and describe how it led me to a considerably more involved one: linear differential operators on $M$. This talk should be of interest to anyone who likes differential topology/geometry, algebraic geometry, or algebra.

We present an alternative approach to the theory of free Gibbs
laws with convex potentials developed by Dabrowski, Guionnet, and Shlyakhtenko.
Instead of solving SDE's, we combine PDE techniques with a notion of
asymptotic approximability by trace polynomials for a sequence of
functions on $M_N(\mathbb{C})_{sa}^m$ to prove the following. Suppose
$\mu_N$ is a probability measure on on $M_N(\mathbb{C})_{sa}^m$ given by
uniformly convex and semi-concave potentials $V_N$, and suppose that the
sequence $DV_N$ is asymptotically approximable by trace polynomials in a
certain sense. Then
the moments of $\mu_N$ converge to a non-commutative law $\lambda$.
Moreover, the free entropies $\chi(\lambda)$, $\underline{\chi}(\lambda)$,
and $\chi^*(\lambda)$ agree and equal the limit of the normalized
classical entropies of $\mu_N$. An upcoming paper will use the same
techniques to obtain transport maps from $\lambda$ to a free semicircular
family as the limit of transport maps for the matrix models $\mu_N$.

We analyze the crystal bases for the quantum queer superalgebra recently introduced by Grantcharov et. al.. Like crystals of type $A$, this crystal can be described by explicit operators on words in the alphabet $\{1, 2, \dots, n\}$. Like crystals of type $A$, each connected component of a queer supercrystal has a unique highest (and lowest) weight. In the type $A$ case, if one is given a certain highest weight, one can reconstruct the connected component containing it using simple axioms introduced by Stembridge. However, given a highest weight, it is much more difficult to reconstruct the queer connected component containing it in this way. Nevertheless, a set of axioms has been conjectured by Assaf and Oguz to do just this. Unfortunately, these axioms are not sufficient, in fact they already fail to uniquely characterize the queer connected component containing highest weight $(4,2,0)$. In this talk, we provide the additional information which is needed to reconstruct the connected queer crystal which contains a given highest weight.

Here I shall introduce the two curvatures and discuss their relations with existing ones and their implications, including comparison theorems, vanishing theorems, projective embeddings and hyperbolicity. If time permits, I shall also discuss some open problems related.

In this talk, we first summarize recent advances in the research on non-convex generalized Nash equilibrium problems(GNEP). That is, the cost functions of some players are non-convex, or the strategy sets are non-convex. Non-convex GNEP has wide applications. Some computational methods and theoretical results are given. We will present a half-space projection method with inertial step for convex GNEP.

Since the invention of generalized polynomial chaos in 2002, uncertainty quantification has impacted many engineering fields. However, almost all existing generalized polynomial chaos methods have a strong assumption: the uncertain parameters are mutually independent or Gaussian correlated. This assumption rarely holds in many realistic applications, and it has been a long-standing challenge for both theorists and practitioners. In this talk, I will presented two algorithms for handling this task. The first one is stochastic collocation algorithm, and the second one is sparse optimization approach. We provide some rigorous proofs for the complexity and error bound of our proposed method. Numerical experiments on synthetic, electronic and photonic integrated circuit examples show the nearly exponential convergence rate and excellent efficiency of our proposed approaches.
This is a joint work with Prof. Zheng Zhang from UC Santa Barbara.

A topological dynamical system (i.e. a group acting by homeomorphisms on a compact topological space) is said to be proximal if for any two points p and q we can simultaneously push them together i.e. there is a sequence $g_n$ such that $lim g_n(p)=lim g_n (q)$. In his paper introducing the concept of proximality Glasner noted that whenever $Z$ acts proximally that action will have a fixed point. He termed groups with this fixed point property ``strongly amenable'' and showed that non-amenable groups are not strongly amenable and virtually nilpotent groups are strongly amenable. In this talk I will discuss recent work precisely characterizing which (countable) groups are strongly amenable.

A countable discrete group is said to be Choquet-Deny if it has a trivial Poisson boundary for every non-degenerate probability measure on the group. In other words, a countable discrete group is Choquet-Deny if non-degenerate random walks on the group have trivial behavior at infinity. For example, all abelian groups are Choquet-Deny. It has been long known that all Choquet-Deny groups are amenable. I will present a recent result classifying countable discrete Choquet-Deny groups: a countable discrete group is Choquet-Deny if and only if none of its quotients have the infinite conjugacy class property. As a corollary, a finitely generated group is Choquet-Deny if and only if it is virtually nilpotent. This is a joint work with Joshua Frisch, Yair Hartman, and Omer Tamuz.

A conjecture of Serre (now a theorem of Gross, Edixhoven, and
Coleman-Voloch) classifies pairs of weights where one finds modular forms
congruent modulo a prime p in terms of local behavior at p. We discuss a
generalization of this conjecture in higher rank. A key step in our work
is the study of a certain subscheme of Gaitsgory's $A^1$ affine Grassmannian
which shares properties with some affine Springer fibers. This is joint work
with B. Le Hung, B. Levin, and S. Morra.

In the 1970's Stark made precise conjectures about the leading term of the Taylor series at s=0 for Artin L-functions. In the rank one setting when the order vanishing is exactly one, these conjectures relate the derivative of the L-function at s=0 to the logarithm of a unit in an abelian extension of the base field. In this talk, we will define a p-adic L-function and state a p-adic Stark conjecture in the rank one setting when the base field is a quadratic field. We prove our conjecture in the case when the base field is imaginary quadratic and the prime p is split, and discuss numerical evidence in the other cases.

CR geometry studies boundaries of domains in $\mathbb{C}^n$ and their generalizations. In characterizing CR structures, a central role is played by the Levi form $L$ of a CR manifold $M$, which measures the failure of the CR bundle to be integrable, so that when $L$ has a nontrivial kernel of constant rank, $M$ is foliated by complex manifolds. If the local transverse structure to this foliation still determines a CR manifold $N$, $M$ is called straightenable, and the Tanaka-Chern-Moser classification of CR hypersurfaces with nondegenerate Levi form can be applied to $N$. It remains to classify those $M$ for which $L$ is degenerate and no such straightening exists. This was accomplished in dimension 5 by Ebenfelt, Isaev-Zaitzev, Medori-Spiro, and Pocchiola. I will discuss their results, my progress on the problem in dimension 7, and my work (joint with Igor Zelenko) modifying Tanaka's prolongation procedure to treat the equivalence problem in arbitrary dimension.

Biological data sets, such as gene expressions or protein levels, are often high-dimensional, and thus difficult to interpret. Finding important structural features and identifying clusters in an unbiased fashion is a core issue for understanding biological phenomena. In this talk, we describe the dynamical behavior of the important tumor suppressor gene p53 in a data-driven manner. By using simulations from nonlinear models that describe the experimentally observed oscillatory behavior of p53, we first identify parameters which qualitatively change the behavior of the system. Focusing on these parameters, we then show that the effective dimension of the parameter and state space can be recovered from time-series data, providing a minimal realization of the underlying nonlinear system. To this end, we apply the methods of bifurcation analysis and diffusion maps. This is joint work with M. Kooshkbaghi, D. Sroczynski, Z. Belkhatir, M. Pouryahya, A. Tannenbaum, I. Kevrekid is.

A unital $C^\ast$-algebra $A$ is tracially stable if maps on $A$ that are approximately (in trace) unital $\ast$-homomorphisms can be
approximated (in trace) by honest unital $\ast$-homomorphisms on $A$. Tracial stability is closed under free products and tensor products
with abelian $C^\ast$-algebras. In this talk we expand these results to show that for a graph from a certain class, the corresponding
graph product (a simultaneous generalization of free and tensor products) of abelian $C^\ast$-algebras is tracially stable. We will
then discuss two applications of this result: a selective version of Lin’s Theorem and a characterization of the amenable traces
on certain right-angled Artin groups.

Ergodic theory deals with a particular type of measure-preserving transformation on a probability space, and are mostly used in the study of dynamical systems. On the other hand, continued fractions are just a way to represent any real number as an infinte/finite sequence of integers such that this sequence when written as a 'fraction' gives this real number. In this talk, I shall set up the preliminaries of ergodic theory and state the ergodic theorem that we shall need, and then move on to define some notions in continued fractions, and then bridge these two seemingly unrelated subjects. The main application would be to find out the rate of convergence of these 'fractions' to the real number, and thereby coming across some interesting constants like the Khinchin's constant and the Lévy's constant. The talk will not require any prerequisites except for being familiar with the notion of a measure space, which again, is not too necessary.
If time permits, I shall just mention some interesting patterns observed in the continued fraction expansion of e and its $1/n$ and $2/n$ th power, or give an idea how working with the continued fraction of $\sqrt{n}$ is helpful in finding all the solutions of the Pell's equations, i.e. $x^2 - n y^2 = 1$ (or -1).

In this talk, I will present a general upper bound for the number of incidences with k-dimensional varieties in R$^d$ such that their incidence graph does not contain K$_{s,t}$ for fixed positive integers s,t,k,d (where s,t$>$1 and k$<$d). The leading term of this new bound generalizes previous bounds for the special cases of k=1, k=d-1, and k=d/2. Moreover, we find lower bounds showing that this leading term is tight (up to sub-polynomial factors) in various cases. To prove our incidence bounds, we define k/d as the dimension ratio of an incidence problem. This ratio provides an intuitive approach for deriving incidence bounds and isolating the main difficulties in each proof. If time permits, I will mention other incidence bounds with traversal varieties and hyperplanes in complex spaces. This is joint work with Adam Sheffer.

Cyclic sieving, identified in work with Dennis Stanton and Dennis White, is a happy situation, where counting how many among some objects enjoy cyclic symmetry is as easy as $q$-counting all of the objects. We will illustrate this with two kinds of examples: old ones that still plague us with only uninsightful proofs, and new ones that have joined our list of favorites.

A bounded strictly pseudoconvex domain in C$^n$, n$>1$, supports a unique complete Kahler-Einstein metric determined by the Cheng-Yau solution of Fefferman's Monge-Ampere equation. The smoothness of the solution of Fefferman's equation up to the boundary is obstructed by a local CR invariant of the boundary called the obstruction density. In the case n=2 the obstruction density is especially important, in particular in describing the logarithmic singularity of the Bergman kernel. For domains in C$^2$ diffeomorphic to the ball, we motivate and consider the problem of determining whether the global vanishing of this obstruction implies biholomorphic equivalence to the unit ball. (This is a strong form of the Ramadanov Conjecture.)

For two fixed graphs $T$ and $H$, a positive integer $n$ and a real number $p$ in $[0, 1]$ let $ex(G(n, p), T, H)$
be the random variable counting the maximum number of copies of $T$ in an $H$-free subgraph of the random graph $G(n, p)$. In this talk we discuss this variable, its phase transition as a function of $p$
and its connection to the deterministic function counting the maximum number of copies of $T$ in an $H$-free graph on $n$ vertices.

Based on joint works with N. Alon, A. Kostochka and W. Samotij.

We prove a pointwise ergodic theorem for locally countable ergodic quasi-pmp (nonsingular) graphs, which gives an increasing sequence
of Borel subgraphs with finite connected components, averages over which converge a.e. to the expectations of $L^1$-functions.
This can be viewed as a random analogue of pointwise ergodic theorems for group actions: instead of taking a (deterministic) sequence
of subsets of the group and using it at every point to compute the averages, we allow every point to coherently choose such a sequence
at random with a strong condition that the sets in the sequence determine aconnected subgraph of the Schreier graph of the action.

Every tensor is a sum of rank-1 tensors and the decomposition in a minimal number is called canonical polyadic decompositions. In this talk, we will introduce some decomposition methods for third-order tensors based on standard linear algebra. They all reduce tensor decomposition problems to matrix decomposition problems. Generalized Schur decomposition, simultaneous matrix diagonalization, and generalized eigenvalue decomposition will be used respectively.

In 1999, David Aldous conjectured that a certain natural 'random walk' on the space of binary combinatorial trees should have a continuum analogue, which would be a diffusion on the Gromov-Hausdorff space of continuum trees. This talk discusses ongoing work by F-Pal-Rizzolo-Winkel that has recently verified this conjecture with a path-wise construction of the diffusion. This construction combines our work on dynamics of certain projections of the combinatorial tree-valued random walk with our previous construction of interval-partition-valued diffusions.

For each function field multiple zeta value (defined by Thakur), we construct a t-module with an attached logarithmic vector such that a specific coordinate of the logarithmic vector is a rational multiple of that multiple zeta value. We then show that the other coordinates of this logarithmic vector contain hyperderivatives of a deformation of these multiple zeta values, which we call t-motivic multiple zeta values. This allows us to give a logarithmic expression for monomials of multiple zeta values. Joint work with Chieh-Yu Chang and Yoshinori Mishiba.

A C$^*$-algebra consists of an algebra of bounded linear operators acting on a Hilbert space which is closed the adjoint operation (roughly, the transpose) and is complete in a certain metric. Typical examples include the ring of $n \times n$ complex matrices and the ring $C(X)$ of representation of continuous functions from a compact space $X$ to the complex numbers. Many more interesting examples arise from various dynamical objects (e.g. group and group actions) and from various geometric/topological constructions.

The structure of finite dimensional C*-algebras is well understood: they are finite direct sums of complex matrix algebras. The class of approximately finite-dimensional (AF) C*-algebras, ones which may be written as (the closure of) an increasing union of f1inite-dimensional subalgebras, are also well understood: they are determined up to isomorphism by their module structure. However, the class of subalgebras of AF-algebras is still rather mysterious; it includes, for instance, all commutative C*-algebras and all C*-algebras generated by amenable groups. It is a long-standing problem to find an abstract characterization of subalgebras of AF-algebras.

I will discuss the AF-embedding problem for C$^*$-algebras and a recent partial solution to this problem which gives a nearly complete characterization of C$^*$-subalgebras of simple AF-algebras.

One method of building extremal objects is a random construction subject to constraints. For instance, one can build a tree by randomly adding edges as long as they do not form a cycle. However, analyzing these constructions is rarely so simple and even finding good asymptotic bounds can be difficult. The Differential Equations Method provides a powerful tool for analyzing random constructions subject to constraints by building a tractable system of differential equations out of a combinatorial construction, solving the system, and then proving that the random process is 'close' to the system solution with high probability. We present the differential equations method and give an application in finding lower bounds for graph Ramsey number asymptotics. Following the treatment in Bohman and Keevash (2013), we sketch the proof that R(3,t) $>$ ((1/4) - o(1))t$^2$/log(t).

A log Del Pezzo surface is a normal log terminal surface with anti-ample canonical bundle. Over the complex numbers, Keel and McKernan have classified all but a bounded family of the simply connected log Del Pezzo surfaces of rank one. In this talk we extend their classification in positive characteristic, and in particular we prove that for $p>5$ every log Del Pezzo surface of rank one lifts to characteristic zero with smooth base. As a consequence, we see that Kawamata-Viehweg vanishing holds in this setting. Finally, we exhibit some counter-examples in characteristic two, three and five.

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

Integer partitions have been an interesting combinatorial object for centuries but we still have a lot left to understand. In this talk we will see an uncommon way to view partitions using 'Abaci' and use this to illustrate partition division (defining t-cores and t-quotients of a given integer partition). I am planning to go in more detail with a special kind of partition called t-core partitions and discuss different ways to enumerate them. I will end my talk by stating an interesting results by Prof. Ken Ono, which shows how the related objects enumerates some important algebraic invariants like class groups of certain imaginary quadratic number fields.

The Hull-Strominger system is a system of nonlinear PDEs describing the geometry of compactification of heterotic strings with torsion to 4d Minkowski spacetime, which can be regarded as a generalization of Ricci-flat K$\ddot{\text{a}}$hler metrics coupled with Hermitian Yang-Mills equation on non-K$\ddot{\text{a}}$hler Calabi-Yau 3-folds. The Anomaly flow is a parabolic approach to understand the Hull-Strominger system initiated by Phong-Picard-Zhang. We show that in the setting of generalized Calabi-Gray manifolds, the Hull-Strominger system and the Anomaly flow reduce to interesting elliptic and parabolic equations on Riemann surfaces. By solving these equations, we obtain solutions to the Hull-Strominger system on a class of compact non-K$\ddot{\text{a}}$hler Calabi-Yau 3-folds with infinitely many topological types and sets of Hodge numbers. This talk is based on joint work with Zhijie Huang and Sebastien Picard.

Suppose a function $\{1,\dots,N\} \to \mathbb R$ has the property that when we take discrete derivatives $k$ times, the result is identically zero. It is fairly well-known that this is equivalent to being a polynomial of degree $k-1$. It's not too unnatural to ask: what does the function look like if, instead, the iterated derivative is required to be zero just a positive proportion of the time? Such \emph{approximate polynomials} have a richer structure, related to nilpotent Lie groups.

On an unrelated note: given an $n \times n$ chessboard, how many ways are there to arrange $n$ queens on it, so that no two attack each other?

I'll outline how both these questions are connected to what's known as \emph{higher order Fourier analysis}, and explain more generally what higher order Fourier analysis is and what it can be used for (other than potentially placing queens on chessboards).

The minimal superpermutation problem asks how many symbols a string must have before it contains all $n!$ permutations of $\{1,\ldots,n\}$ as substrings. The Haruhi problem asks for the most efficient way to watch the anime ``The Melancholy of Haruhi Suzumiya'' in every way possible. Remarkably, these two problems turn out to be equivalent. Even more remarkably, the anime community has made more progress on this problem than the mathematicians! In this talk we will discuss an improved lower bound to the minimal superpermutation problem that was recently discovered on 4chan, as well as the anime theoretic motivation for considering this problem. As time permits we will also briefly discuss De Brujin sequences. This talk will be entirely self contained and assume no prior knowledge of anime.

Given a unitary Modular Tensor Category M, the reconstruction program asks for the construction of a rational conformal field theory such that its representation category is isomorphic to M. This is a low dimensional version of Dolpicher-Roberts/Deligne theory which is much richer due to nontrivial representations of braid group, and is strongly motivated by recent development in subfactor theory. In this talk we will describe questions around this program and present recent progress.

A drawing of a graph on a surface is {\em independently even} if every pair of nonadjacent edges in the drawing crosses an even number of times. The strong Hanani-Tutte theorem states that a graph admitting an independently even drawing in the plane must be planar.

The {\em genus} $g(G)$ of a graph $G$ is the minimum $g$ such that $G$ has an embedding on the orientable surface $M_g$ of genus $g$. The {\em $\mathbb{Z}_2$-genus} of a graph $G$, denoted $g_0(G)$, is the minimum $g$ such that $G$ has an independently even drawing on the orientable surface of genus $g$. Clearly, every graph $G$ satisfies $g_0(G) \leq g(G)$, and the strong Hanani-Tutte theorem states that $g_0(G) = 0$ if and only if $g(G) = 0$. Although there exist graphs $G$ for which the values of $g(G)$ and $g_0(G)$ differ, several recent results suggest that these graph parameters are closely related. We provide further evidence of their similarity.

For complete bipartite graphs $K_{n,m}$ with $n \geq 3$, we prove the following:
$$
g_0(K_{n,m}) \geq \lceil \frac{1}{2} \left( \lceil \frac{(n-2)(m-2)}{2} \rceil - (n-3) \right) \rceil
$$
The value of $g(K_{n,m})$ was determined by Ringel in 1965, and equals
$\lceil \frac{(n-m)(m-2)}{4} \rceil$.

Joint work with J. Kyncl.

The classical Schur(-Weyl) duality relates the representation theory of general linear Lie algebras and symmetric groups. Drinfeld and Jimbo independently introduced quantum groups in their study of exactly solvable models, which leads to a quantization of the Schur duality relating quantum groups of general linear Lie algebras and Hecke algebras of symmetric groups.

In this talk, I will explain the generalization of the (quantized) Schur duality to other classical types, algebraically, geometrically, and categorically. This new duality leads to a theory of canonical bases arising from quantum symmetric pairs generalizing Lusztig’s canonical bases on quantum groups.

The classical Borsuk-Ulam theorem states that a
continuous map from a n-dimensional sphere to m-dimensional sphere
which preserves the antipodal Z/2-actions only exists when m is
greater than or equal to n. One can ask a similar question, by
replacing the antipodal Z/2-action with an action of the Lie group
Pin(2).

On a seemingly unrelated side, the Geography Problem of 4-manifolds
asks which simply connected topological 4-manifolds admits a smooth
structure. By the celebrated works of Kirby-Siebenmann, Freedman,
Donaldson, Seiberg-Witten and Furuta, there is a surprising connection
between these two questions. In this talk, I will:

1. Explain this beautiful connection between the two problems.

2. Present a solution to the Pin(2)-equivariant Borsuk–Ulam problem.

3. State its application to the Geography Problem. In particular, a
partial result on the famous 11/8-conjecture.

4. Describe the ideas of our proof, which uses Pin(2)-equivariant
stable homotopy theory.

This talk is based on a joint work with Mike Hopkins, XiaoLin Danny
Shi and Zhouli Xu. No familiarity of homotopy theory or 4-dimensional
topology will be assumed.

In this talk I will introduce two different notions of solutions to the Plateau Problem, called Area and Size minimizers, due respectively to Federer-Fleming and Almgren. The fundamental difference between them is wether multiplicity/orientation plays a role or not, and they were originated respectively to describe integral homology class and soap films. I will then explain how different types of singularities arise in both formulation and some recent progress made on the structure of the singular set and of minimizers near singularities. If time permits I will also explain some possible future developments.

Branching-style models were proposed in the 1960's as a logician's
tool to study combinations of tenses and modalities, as in ``it is
still possible that it will rain in SD tomorrow'' but ``it is already
settled that last Summer was hot in SD''. A current theory of
Branching Space-Times (BST), put forward by N.~Belnap in 1992, is an
axiomatic framework that aims to describe how indeterminism plays out
in a spatio-temporal world. To this end it postulates a set of
relativistic space-times, any two of which are pasted together in some
particular region. Although a BST structure is continuous, it is
possible to discretise it, by focusing on particular objects, known as
transitions, and interpreted as places at which chancy actions
happen. A discretised structure, defined as a partially ordered set of
transitions, recovers then much, but not all information about the
initial structure. As these ideas are reminiscent of the Causal Set
Program, I will end up the talk by discussing some connections between
the two frameworks.

We discuss a new method to bound 5-torsion in class groups using
elliptic curves. The most natural ``trivial'' bound
on the n-torsion is to bound it by the size of the entire class group, for
which one has a global class number formula. We explain how to
make sense of the n-torsion of a class group intrinsically as a ``dimension
0 selmer group'', and by embedding it into an appropriate Elliptic curve we
can bound its size
by the Tate-Shafarevich group which we can bound using the BSD conjecture.

This fits into a general paradigm where one bounds ``dimension 0 selmer
groups'' by embedding into global objects, and using class number formulas.

In this talk I will present a new method for studying the regularity of minimizers of some variational problems, including in particular some classical free-boundary problems. Using as a model case the so-called Obstacle problem, I will explain what regularity of the free-boundary means and how we obtain it by using a new tool, called (Log) -epiperimetric inequality. This technique is very general, and much like Caffarelli's 'improvement of flatness' for regular points, it allows for a uniform treatment of singularities in many different free-boundary problems. Moreover it is able to deal with logarithmic regularity, which in the case of the Obstacle problem is optimal due to an example of Figalli-Serra. If time permits I will explain how such an inequality is linked to the behavior of a gradient flow at infinity.

Minimal surfaces are critical points of the area functional on the space of surfaces. Thus, it is natural to try to construct them via Morse theory. However, there is a serious issue when carrying this out, namely the occurrence of ``multiplicity.'' I will explain this issue and recent joint work with C. Mantoulidis ruling this out for generic metrics.

Because of their connections with the representation theory of the symmetric group, Hopf algebras have been used to study the enumerative properties of many combinatorial objects. In this talk we will give an introduction to this topic with the intention of defining Hopf monoids, a generalization that ends up being a better set up for the study of objects like graphs, matroids, posets, and generalized permutahedra.

I'll discuss joint work with M. Eichmair and V. Moraru in which we prove a natural minimal surface analogue of the splitting theorem for 3-manifolds with non-negative scalar curvature.

Logistic regression is arguably the most widely used and studied non-linear model in statistics. It has found widespread applicability in varied domains, such as genetics, health care, e-commerce, etc. Classical maximum-likelihood theory for this model hinges on the fundamental results---(1) the maximum-likelihood-estimate (MLE) is asymptotically unbiased (2) its variability can be quantified via the inverse Fisher Information (3) the likelihood-ratio-test (LRT) is asymptotically a Chi-Square. These results are universally used for statistical inference. Our findings reveal, however, when the number of features p and the sample size n both diverge, with the ratio p/n converging to a positive constant, classical results are far from accurate. For a certain class of logistic models, we observe (1) the MLE is biased, (2) its variability is much higher than classically estimated, and (3) the LRT is not distributed as a Chi-Square. We develop a new theory that quantifies the asymptotic bias and variance of the MLE, and characterizes asymptotic distribution of the LRT under certain assumptions on the covariate distribution. Empirical findings demonstrate that our results provide extremely accurate inference in finite samples. These novel results depend on the underlying regression coefficients through a single scalar, the overall signal strength, and we discuss a procedure to estimate this parameter accurately. This is based on joint work with Emmanuel Candes and Yuxin Chen.

Regularity results for linear elliptic and parabolic systems
with measurable coefficients play an important role in the calculus of
variations. Morrey showed that in two dimensions, solutions to linear
elliptic systems are continuous. We will discuss some surprising recent
examples of discontinuity formation in the plane for the parabolic
problem.

In the theory of von Neumann algebras, fundamental unsolved problems going back to the 1930s have seen remarkable progress in the last two decades due to Sorin Popa's breakthrough deformation/rigidity theory. Popa's discovery hinges on the fact that, just as stirring a soup allows one to locate its most rigid (and desirable) hidden components, 'deformability' of an algebra $M$ allows one to precisely locate 'rigid' subalgebras known to exist only via a supposed isomorphism $M \cong N$.

This talk will focus on joint work with Rolando de Santiago, Ben Hayes, and Thomas Sinclair, which shows that any diffuse subalgebra which is rigid with respect to a mixing $s$-malleable deformation is in fact contained in subalgebra which is uniquely maximal with respect to that rigidity. In particular, an algebra generated by a family of rigid subalgebras which intersect diffusely must itself be rigid with respect to that deformation. The case where this family has two members answers a question of Jesse Peterson asked at the American Institute of Mathematics (AIM), but the result is most striking when the family is infinite.

Nowadays, data is exploding at a faster rate than computer architectures can handle. For that reason, mathematical techniques to analyze large-scale data need be developed. Stochastic iterative algorithms have gained interest due to their low memory footprint and adaptability for large-scale data. In this talk, we will present the Randomized Kaczmarz algorithm for solving extremely large linear systems of the form Ax=y. In the spirit of large-scale data, this talk will act under the assumption that the entire data matrix A cannot be loaded into memory in a single instance. We consider different settings including when a only factorization of A is available, when A is missing information, and a time-varying model. We will also present applications of these Kaczmarz variants to problems in data science.

It is not obvious why natural selection would've selected for animals that can compute integrals, yet we exist. In this talk we'll look at some explanations from cognitive science, examples from the wild, and perhaps a look at the future of how maths could be done.

Bernstein operators are vertex operators that create and annhilate Schur polynomials. These operators play a significant role in the mathematical formulation of the Boson-Fermion correspondence due to Kac and Frenkel. The role of this correspondence in mathematical physics has been widely studied as it bridges the actions of the infinite Heisenberg and Clifford algebras on Fock space. Cautis and Sussan conjectured a categorification of this correspondence within the framework of Khovanov's Heisenberg category. I will discuss how to categorify the Bernstein operators and settle the Cautis-Sussan conjecture, thus proving a categorical Boson-Fermion correspondence.

Li-Chung Chen and Mark Haiman studied a family of symmetric functions called Catalan (symmetric) functions which are indexed by pairs consisting of a partition contained in the staircase $(n-1, \dots, 1, 0)$ (of which there are Catalan many) and a composition weight of length $n$. They include the Schur functions, the Hall-Littlewood polynomials and their parabolic generalizations. They can be defined by a Demazure-operator formula, and are equal to the $GL$-equivariant Euler characteristics of vector bundles on the flag variety by the Borel-Weil-Bott Theorem. We have discovered various properties of Catalan functions, providing new insight on the existing theorems and conjectures inspired by the Macdonald Positivity Conjecture.

A key discovery in our work is an elegant set of ideals of roots whose associate Catalan functions are $k$-Schur functions, proving that graded $k$-Schur functions are $GL$-equivariant Euler characteristics of vector bundles on the flag variety, settling a conjecture of Chen-Haiman. We exposed a new shift invariance property of the graded $k$-Schur functions and resolved the Schur positivity and $k$-branching conjectures by providing direct combinatorial formulas using strong marked tableaux. We conjectured that Catalan functions with a partition weight are $k$-Schur positive which strengthens the Schur positivity of the Catalan function conjecture by Chen-Haiman and resolved the conjecture with positive combinatorial formulas in cases which capture and refine a variety of problems.

This is joint work with Jonah Blasiak, Jennifer Morse, and Daniel Summers.

Interacting particle systems are models that involve many randomly evolving agents (i.e., particles). These systems are widely used in describing real-world phenomena. In this talk we will walk through three paradigmatic facets of interacting particle systems, namely the law of large numbers, random fluctuations, and large deviations. Within each facet, I will explain how Partial Differential Equations (PDEs) play a role in understanding the systems.

Modern statistical analysis often requires an integration of statistical thinking and algorithmic thinking. In many problems, statistically sound estimation procedures (e.g., the MLE) may be difficult to compute, at least in the naive form. This challenge calls for a new look into simple statistical methods such as the spectral methods (including PCA), as well as an examination of optimization algorithms from the statistical lens.

In this talk, I will sample two typical modern statistical problems: one addresses network type data (community detection), and the other involves pairwise comparison data (phase synchronization). I will show that in high dimensions, spectral methods exhibit a very interesting new phenomenon in entrywise behavior, which leads to new theoretical insights and has practical relevance. Also, for a complex nonconvex problem, I will show how algorithmic analysis can benefit from classical statistical ideas.

This talk features joint work with (alphabetically) Emmanuel Abbe, Nicolas Boumal, Jianqing Fan, and Kaizheng Wang.

In this talk, we present some recent work on discontinuous Galerkin (DG) methods for waves and fluid flow.
Three topics will be covered, including (1) new energy-conserving DG methods for linear hyperbolic waves, and nonlinear dispersive waves, (2) globally divergence-free DG methods for incompressible flow, and (3) hybridizable DG methods for Darcy flow on polygonal meshes. The key idea of each DG scheme will be addressed.

Let k be a perfect field of characteristic p, and let Gal(k)
denote the absolute Galois group of k. By a classical result of Katz, the
category of
finite-dimensional $F_p-vector spaces$ with an action of Gal(k) is equivalent
to the category of finite-dimensional vector spaces over k with a
Frobenius-semilinear automorphism. In this talk, I'll discuss some joint
work with Bhargav Bhatt which generalizes Katz's result, replacing the
field k by an arbitrary $F_p-scheme X$. In this case, there is a
correspondence relating p-torsion etale sheaves on X to quasi-coherent
sheaves on X equipped with a Frobenius-semilinear automorphism, which can
be viewed as a ``mod p'' version of the Riemann-Hilbert correspondence for
complex algebraic varieties.

Professor Curto will talk about her work on mathematics for neuroscience and her transition from work as a pure mathematician to work on mathematics applied to and arising form theoretical and computational neuroscience.

I will discuss applications of limits of combinatorial structures to geometry and their connection to measure theory. Some motivating problems that I will touch upon are: the rectilinear crossing number of the complete graph, the probability that a random point set is in convex position and the expected number of vertices of the convex hull of a random sample.