We will discuss how one can tile the hyperbolic plane with various polygons. We will focus on the tiling with the smallest possible tile, the (2,3,7)-triangle. The reflexion group associated to it turns out to have fundamental importance in the theory of Hurwitz surfaces.

Consider the tilings of an oriented surface by triangles, or squares, or hexagons, up to combinatorial equivalence. The combinatorial curvature of a vertex is 6, 4, or 3 minus the number of adjacent polygons, respectively. Tilings are naturally stratified into all such having the same set of non-zero curvatures. We outline a proof that for squares and hexagons, the generating function for the number of tilings in a fixed stratum lies in a ring of quasi-modular forms of specified level and weight. First, we rephrase the problem in terms of Hurwitz theory of an elliptic orbifold---a quotient of the plane by an orientation-preserving wallpaper group. In turn, we produce a formula for the number of tilings in terms of characters of the symmetric group. Generalizing techniques pioneered by Eskin and Okounkov, who studied the pillowcase orbifold, we express the generating function for a stratum in terms of the q-trace of an operator acting on Fock space. The key step is to compute the trace in a different basis to express it as an infinite product, and apply the Jacobi triple product formula to conclude quasi-modularity.

The Torelli subgroups of mapping class groups are fundamental objects in low-dimensional topology, through some basic questions about their structure remain open. In this talk I will describe these groups, and how to use tools from representation theory to establish patterns their homology. This project is joint with Jeremy Miller and Peter Patzt. These ``representation stability'' results are an application of advances in a general algebraic framework for studying sequences of group representations.

The moduli space of Riemann surfaces of fixed genus is one of the hubs of modern mathematics and physics. We will tell the story of how simple sounding problems about polygons, some of which arose as toy models in physics, became intertwined with problems about the geometry of moduli space, and how the study of these problems in Teichmuller dynamics lead to connections with homogeneous spaces, algebraic geometry, dynamics, and other areas. The talk will mention joint works with Alex Eskin, Simion Filip, Curtis McMullen, Maryam Mirzakhani, and Ronen Mukamel.

The AdS instability conjecture is a conjecture about the initial value problem for Einstein vacuum equations with a negative cosmological constant. Proposed by Dafermos and Holzegel in 2006, the conjecture states that generic, arbitrarily small perturbations to the initial data of the AdS spacetime, under evolution by the vacuum Einstein equations with reflecting boundary conditions on conformal infinity, lead to the formation of black holes.Following the work of Bizon and Rostworowski in 2011, a vast amount of numerical and heuristic works have been dedicated to the study of this conjecture, focusing mainly on the simpler setting of the spherically symmetric Einstein-scalar field system. In this talk, we will provide the first rigorous proof of the AdS instability conjecture in the simplest possible setting, namely for the Einstein-null dust system, allowing for both ingoing and outgoing dust. This system is a singular reduction of the spherically symmetric Einstein-massless Vlasov system, in the case when the Vlasov field is supported only on radial geodesics. In order to overcome the 'trivial' break down occurring once the null dust reaches the center $r=0$, we will study the evolution of the system in the exterior of an inner mirror with positive radius $r_0$ and prove the conjecture in this setting. After presenting our proof, we will briefly explain how the main ideas can be extended to more general matter fields, including the regular Einstein-massless Vlasov system.

The stochastic inverse problem for determining parameter values in a physics model from observational data on the output of the model forms the core of scientific inference and engineering design. We describe a recently developed formulation and solution method for stochastic inverse problems that is based on measure theory and a generalization of a contour map. In addition to a complete analytic and numerical theory, advantages of this approach include avoiding the introduction of ad hoc statistics models, unverifiable assumptions, and alterations of the model like regularization. We present a high-dimensional application to determination of parameter fields in storm surge models. We conclude with recent work on defining a notion of condition for stochastic inverse problems and the use in designing sets of optimal observable quantities.

When we choose a metric on a manifold we determine the spectrum of the Laplace operator. Thus an eigenvalue may be considered as a functional on the space of metrics. For example the first eigenvalue would be the fundamental vibrational frequency. In some cases the normalized eigenvalues are bounded independent of the metric. In such cases it makes sense to attempt to find critical points in the space of metrics. For surfaces, the critical metrics turn out to be the induced metrics on certain special classes of minimal (mean curvature zero) surfaces in spheres and Euclidean balls. The eigenvalue extremal problem is thus related to other questions arising in the theory of minimal surfaces. In this talk we will give an overview of progress that has been made for surfaces with boundary, and discuss some recent results in higher dimensions. This is joint work with R. Schoen.

The combinatorics of permutations in the symmetric group $S_n$ has deep connections to algebraic properties of the {\em coinvariant ring} (through work of Artin, Chevalley, Lusztig-Stanley, and others) and geometric properties of the {\em flag variety} whose points are complete flags in $\mathbb{C}^n$ (through work of Ehresmann, Borel, and others). We will discuss new generalizations of the coinvariant ring and flag variety indexed by two positive integers $k \leq n$. The algebraic and geometric properties of these objects are controlled by ordered set partitions of $[n]$ with $k$ blocks. There are connections between these objects and the Delta Conjecture in the theory of Macdonald polynomials. Joint with Jim Haglund, Brendan Pawlowski, and Mark Shimozono.

Many important maps in algebraic combinatorics (the RSK bijection, the Schutzenberger involution, etc.) can be described by piecewise-linear formulas. These formulas can then be ``de-tropicalized,'' or ``lifted,'' to subtraction-free rational functions on an algebraic variety, and certain properties of the combinatorial maps become more transparent in the algebro-geometric setting. I will illustrate how this works in the case of the promotion map on semistandard tableaux of rectangular shape. I will also indicate how promotion can be viewed as the combinatorial manifestation of a symmetry coming from representation theory, and how its geometric lift fits into Berenstein and Kazhdan's theory of geometric crystals.

Free boundary minimal surfaces in the ball are proper branched minimal immersions of a surface into the ball that meet the boundary of the ball orthogonally. Such surfaces have been extensively studied, and they arise as extremals of the area functional for relative cycles in the ball. They also arise as extremals of an eigenvalue problem on surfaces with boundary. In this talk we will discuss existence and uniqueness theorems for such surfaces, focusing on a uniqueness result for free boundary minimal annuli. This is joint work with M. Li and R. Schoen.

We construct a new quasi-local mass in space-time and show that this mass is non-decreasing along any null flow of doubly convex 2-spheres. As a result, we prove the Penrose conjecture for conical null slices, or null cones, under fairly generic conditions.

Grothendieck’s standard conjectures, Voevodsky’s nilpotence conjecture, and Tate’s conjecture, play a key central role in algebraic geometry. Notwithstanding the effort of several generations of mathematicians, the proofs of these celebrated
conjectures remain ellusive. The aim of this talk, prepared for a broad audience, is to give an overview of a recent topological/noncommutative approach which has led to the proof of the aforementioned important conjectures in several new cases.

The alternating direction method of multipliers (ADMM) has been successfully used in solving problems arising from different fields such as machine learning, image processing, statistics and so on. However, most existing works on analyzing the convergence and complexity of ADMM are for convex problems. In this talk, we discuss several recent results on convergence behavior of ADMM for solving nonconvex problems. We consider two nonconvex models. The first model allows the objective function to be nonconvex and nonsmooth, but the constraints are convex. The second model allows the constraints to be Riemannian manifolds. For both models, we propose ADMM variants for solving them and analyze their iteration complexities for obtaining an $\epsilon$-stationary solution. Numerical results on tensor robust PCA, maximum bisection problem and community detection problem are reported to demonstrate the efficiency of the proposed methods.

I will begin by introducing the Gaussian, Laguerre and Jacobi ensembles and their corresponding eigenvalue densities. The moments of these eigenvalue densities are generated by the corresponding resolvent, R(x). When investigating large matrices of size N, it is natural to expand R(x) as a series in 1/N, as N tends to infinity. The loop equation formalism enables one to compute R(x) to any desired order in 1/N via a triangular recursive system. This formalism is equivalent to the topological recursion, the Schwinger-Dyson equations and the Virasoro constraints, among other things. The loop equations provide a relatively accessible entry-point to these topics and my derivation will rely on nothing more than integration by parts, as Aomoto applied to the Selberg integral. Time permitting, I may also explore links to the topological recursion and/or some combinatorics.

All original results will be from joint work with Peter Forrester and Nicholas Witte.

In this talk, I will present several works related to tensor data analysis. Firstly, hypergraph matching (HGM) is a popular tool in establishing corresponding relationship between two sets of points, which becomes a central problem in computer vision. We reformulate HGM as a sparse constrained model, and show its relaxation problem can also recover the global optimizer. A quadratic penalty method is presented to solve the relaxation model. Secondly, the analytic connectivity (AC) is an important quantity in spectral hypergraph theory. The definition of AC involves a series of polynomial optimization problem (POP). The number of POPs can be reduced by the structure of hypergraphs. Further, we proposed a simplex constrained model, a equality constrained model and a sparse constrained model for computing AC under different situations. Thirdly, identifying new indications for known drugs, i.e., drug repositioning (DR), attracts a lot of attentions in bioinformatics. We develop a novel method for DR based on projection onto convex sets.

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.

Many real-world examples of fluid-structure interaction, including the transit of red blood cells through the narrow slits in the spleen, involve the near-contact of elastic structures separated by thin layers of fluid. Motivated by such problems, we introduce an immersed boundary method that uses elements of lubrication theory to resolve thin fluid layers between immersed boundaries. We apply this method to two-dimensional flows of increasing complexity, including eccentric rotating cylinders and elastic vesicles near walls in shear flow, to show its increased accuracy compared to the classical immersed boundary method.

Associated to a finite volume hyperbolic 3-manifold is a
number field and a quaternion algebra over that number field. Closed
hyperbolic 3-manifolds arising from Dehn surgeries on a hyperbolic
knot complement provide a family of number fields and quaternion
algebras that can be viewed as ``varying'' over a
certain curve component (the so-called canonical component) of the
$SL(2,C)$-character variety of the knot group. This talk will give
examples of different behavior and survey recent work on how the
varying behavior can be explained using the language of Azumaya
algebras over the canonical curve.

We will discuss recent work with Z. Zhang on Torelli theorems for bidouble covers of a smooth quintic curve and 2 lines in the plane, and cubic 4-folds arising from a cubic 3-fold and a hyperplane intersecting transversely in $P^4$.
The talk for graduate students will be, ``Abelian Varieties and the Torelli Theorem''. I will explain what an Abelian variety is, and discuss the Torelli theorem for curves.