Abstracts

Week of March 5, 2023

March 6, 2023
Group Actions, Geometry and Dynamics Arithmetic Quantum Unique Ergodicity for 3-dimensional hyperbolic manifolds 4:00pm -

The Quantum Unique Ergodicity conjecture of Rudnick and Sarnak says that eigenfunctions of the Laplacian on a compact manifold of negative curvature become equidistributed as the eigenvalue tends to infinity.
In the talk I will discuss a recent work on this problem for arithmetic quotients of the three dimensional hyperbolic space. I will discuss our key result that Hecke eigenfunctions cannot concentrate on certain proper submanifolds. Joint work with Lior Silberman. 
Geometry, Symmetry and Physics Moduli of one-dimensional sheaves on P^2: cohomology, perversity, and BPS invariants 4:30pm -
LOM 214

The moduli spaces of one-dimensional sheaves on P^2 are first studied by Simpson and Le Potier, and they admit a Hilbert-Chow morphism to a projective base that behaves like a completely integrable system. Following a proposal of Maulik-Toda, one expects to obtain certain BPS invariants from the perverse filtration on cohomology induced by this morphism, which motivates us to study the cohomology ring structure of these moduli spaces. In this talk, we present some recent progress on this cohomology ring, including a minimal set of tautological generators, and a “Perverse = Chern” conjecture which specializes to an asymptotic product formula for refined BPS invariants of local P^2. This can be viewed as an analogue of the recently proved P=W conjecture for Hitchin systems. Based on joint work with Junliang Shen, and with Yakov Kononov and Junliang Shen.
March 8, 2023
Applied Mathematics A simple and accurate numerical method for singular and near-singular integration 1:00pm -
AKW 200

Boundary Value Problems (BVPs) are ubiquitous in engineering and scientific applications. One of the most robust and accurate methods for solving BVPs is the Boundary Integral Equation Method, which has the great advantage of dimensionality reduction: all of the unknowns reside on the boundary surface instead of in the volume. A key challenge when solving integral equations is that special quadrature methods are required to discretize the underlying singular and near-singular integral operators. Accurate discretization of these operators is especially important in, for example, problems that involve close structure-structure or fluid-structure interactions. In this talk, we present some recent advancements on singular and near-singular numerical integration based on one of the simplest quadrature methods -- the Trapezoidal rule.
Colloquium Ergodic approaches to arithmetic Ramsey theory 4:15pm -
LOM 214
Ramsey theory is a branch of combinatorics which seeks to find patterns in disorganized situations. One of its main achievements, Szemeredi’s theorem on arithmetic progressions, got an ergodic theoretic proof in 1977 when Furstenberg created a Correspondence Principle to encode combinatorial information about sets of integers into a dynamical system. Since then ergodic methods have been very successful in obtaining new Ramsey theoretic results, some of which still have no purely combinatorial proof.
I will survey some of the history of how ergodic theory and Ramsey theory are interconnected, leading to a recent result involving infinite sumsets.
March 9, 2023
Analysis On the analyticity of the Muskat equation 4:00pm -
LOM 205

TBA
March 10, 2023
Friday Morning Seminar Friday Morning Seminar 9:30am -
LOM 215

A relaxed-pace seminar on impromptu subjects related to the interests of the audience.

Everyone is welcome.

The subjects are geometry, probability, combinatorics, dynamics, and more!
Geometric Analysis and Application A Heintze-Karcher inequality with free boundaries and applications to capillarity theory 2:00pm -
LOM 215

 In volume-constrained capillarity problems, minimizers may have free boundaries adhering to the container. Recent work in the study of capillarity problems has utilized stability theory for the volume-constrained isoperimetric problem to classify the shape of global minimizers and (in the case without free boundary) critical points. In this talk, I will discuss joint work with Matias Delgadino extending this program to classifying the shape of local minimizers in the presence of free boundaries.