I will discuss the rational cohomology of $SL_n(R), Sp_{2n}(R)$, and their principal congruence subgroups for $R$ a number ring. Borel--Serre showed that these groups satisfy a (co)homological duality that lets us study their cohomology groups via certain representations called the `Steinberg modules’, which have a beautiful combinatorial description in terms of Tits buildings. I will describe a conjecture of Lee--Szczarba on the top cohomology of principal congruence subgroups of $SL_n(Z)$, and its resolution due to Miller--Patzt--Putman. I will then discuss forthcoming work on generalizations of this to other Euclidean rings, and also to symplectic groups.

Hyperrigidity is an interesting and important approximation-theoretic property of generating sets of C*-algebras. It plays a key role in, for example, the theory of strong convergence. In this talk, I will discuss a new characterization of hyperrigid generating sets in terms of the solvability of a certain noncommutative Dirichlet problem. I will also demonstrate how this result can be applied in practice.

Classical Choquet theory plays a key role in the study of classical Dirichlet problems, so it is perhaps not surprising that our results utilize noncommutative Choquet theory. I will provide a brief overview of some of these ideas. 

This is joint work with Eli Shamovich.

Regular multitypes are CR invariants used to exhibit (non-)degeneracies in the Levi form of a pseudoconvex hypersurface in $\mathbb{C}^n$. We study the properties of the regular multitypes as defined by Bloom, and understand how the notion of finite type implies global regularity in the $\bar{\partial}$-Neumann problem.

Lam, Lee, and Shimozono introduced the double Stanley symmetric functions in their study of the equivariant geometry of the affine Grassmannian. They proved that the assocaited double Edelman– Greene coefficients, the double Schur expansion of these functions, are positive, a result later refined by Anderson. They further asked for a combinatorial proof of this positivity. We provide the first such proof, together with a combinatorial formula that manifests the finer positivity established by Anderson. Our formula is built from two combinatorial models: bumpless pipedreams and increasing chains in the Bruhat order. The proof relies on three key ingredients: a correspondence between these two models, a natural subdivision of bumpless pipedreams, and a symmetry property of increasing chains. This talk is based on joint work with Jack Chou.
 

Convex subsets in higher-rank symmetric spaces are pretty rigid compared to rank 1 symmetric spaces, as proved by Kleiner and Leeb. In this talk, I will talk about convex subsets in products of negatively curved Hadamard manifolds. In particular, we show that the limit cone is 1-dimensional if the diagonal action is convex cocompact, which induces some rigidity-type results of the diagonal representation. This is joint work with Subhadip Dey.

What makes a game mathematical? In this talk I will not attempt to answer that question, but teach you to play a few games that are simple to learn, surprisingly thought-provoking, and feel somewhat mathematical. Come prepared to learn and play!

In physical models of fluids, a singularity occurs when a quantity of interest (velocity, pressure, vorticity…) becomes infinite at some finite time, starting from a "nice" initial configuration. In this talk, we will first introduce some classical models of compressible and incompressible fluids. We will then describe several mechanisms by which fluids can form singularities in finite time, allowing us to discuss the singularity formation problem for the incompressible three-dimensional Euler equations and the Navier--Stokes equations. We will finally touch upon computer-assisted methods and their applications to singularity formation in fluids.
 

The theory of majorization was introduced by Hardy, Littlewood and Pólya in order to formalize the intuitive idea of one set of numbers being more "spread out" than another. They established a surprising characterization of this property in terms convex functions, which allowed them to provide a unified approach to a number of seemingly disparate inequalities from in the literature from that era. The theory of majorization has subsequently found important applications throughout mathematics, mathematical economics and, more recently, quantum information theory. In this talk, I will discuss these developments and introduce a generalized theory of majorization, where numbers are replaced by (not necessarily commuting) matrices. This is joint work with Paul Skoufranis.

Dynamical sampling of probability distributions based on models or data (i.e., generative modeling) is a central task in scientific computing and machine learning. I will present some recent work on understanding and improving algorithms in high-dimensional settings. This includes a novel "delocalization of bias" phenomenon in Langevin dynamics, where biased methods are shown to achieve dimension-free scaling for low-dimensional marginals while unbiased methods cannot—a finding motivated by molecular dynamics simulations. I will also briefly discuss a new unbiased affine-invariant Hamiltonian sampler that outperforms popular samplers in the emcee package in high dimensions, and introduce a design of optimal Lipschitz energy for measure transport in generative modeling that leads to dimension-robust numerical performance with respect to resolution, offering an alternative to the optimal kinetic energy used in optimal transport. These examples demonstrate how dimensional scaling may be flattened, enabling efficient stochastic algorithms for high-dimensional sampling and generative modeling in relevant scientific applications.