Seminars Sorted by Series

I will review certain stabilization phenomena in the characteristic zero representation theory of general linear and symmetric groups as the rank tends to infinity. Then I will give a survey of some results and conjectures about analogs of these in...

Let $G$ be a semi-simple algebraic group over an algebraically closed field $k$ of positive characteristic and let $B$ be a Borel subgroup. The cohomology of line bundles on the flag variety $G/B$ induced by characters of $B$ are important objects...

Abstract: in joint work with Leonardo Patimo, we investigate the compatibility of
p-cells with respect to standard parabolic subgroups. Given a p-cell in
a standard parabolic subgroup of a crystallographic Coxeter system,
one can induce the...

Reverse plane partitions - or RPPs for short - are order reversing maps of minuscule posets in types ADE. We report on joint work in progress with Elek, Kamnitzer, Libman, and Morton-Ferguson in which we give a type independent proof that RPPs form...

For a finite group $G$ one has a process of modular reduction which takes a $KG$-module, over a field $K$ of characteristic zero, and produces a $kG$-module, over a field $k$ of positive characteristic. Starting with a simple $KG$-module its modular...

Given a representation of a reductive group, Braverman-Finkelberg-Nakajima defined a Poisson variety called the Coulomb branch, using a convolution algebra construction. This variety comes with a natural deformation quantization, called a Coulomb...

(work in progress with R. Rouquier) I will present a computational (yet conjectural) method to determine some decomposition matrices for finite groups of Lie type. These matrices encode how ordinary representations decompose when they are reduced to...

I'll present joint work with Tsao-Hsien Chen on the geometry of real and symmetric matrices. For classical groups, we use hyperkahler geometry to lift the Kostant-Sekiguchi correspondence to an equivariant homeomorphism. As an application, we show...

The Hecke algebra admits an involution which preserves the standard basis and exchanges the canonical basis with its dual. This involution is categorified by "monoidal Koszul duality" for Hecke categories, studied in positive characteristic in my...

The problem of classification of perverse sheaves on the quotient $h/W$ for a semisimple Lie algebra $g$ has an explicit answer which turns out to be related to the algebraic properties of induction and restriction operations for parabolic...

I will survey some applications of Ramanujan Graphs in theoretical computer science, as well as some of the work they have inspired. 

Along the way, I'll explain how they impacted the thinking and assumptions of my generation.

I'll discuss spectral gaps in the following contexts:
- d-regular graphs
- locally symmetric spaces e.g. hyperbolic manifolds
- finite dimensional unitary representations of discrete groups e.g. free groups, surface groups
 

Of particular interest are...

I’ll speak about new joint work with Rachel Greenfeld and Marina Iliopoulou in which we address some classical questions concerning the size and structure of integer distance sets. A subset of the Euclidean plane is said to be an integer distance...

In this lecture, we will review recent works regarding  spectral  statistics of the normalized adjacency matrices of random  $d$-regular graphs on $N$ vertices.

Denote their eigenvalues by $\lambda_1=d/\sqrt{d-1}\geq \la_2\geq\la_3\cdots\geq \la_N$...

Since work of Montgomery and Katz-Sarnak, the eigenvalues of random matrices have been used to model the zeroes of the Riemann zeta function and other L-functions. Keating and Snaith extended this to also model the distribution of values of the L...

We'll discuss problems where bounds for L-functions have arisen as inputs and where techniques for estimating them through their integral representations have been useful (all of which have been shaped and influenced by Peter Sarnak’s work).

Bounds for Dirichlet polynomials play an important role in several questions connected to the distribution of primes. For example, they can be used to bound the number of zeroes of the Riemann zeta function in vertical strips, which is relevant to...

Bounds for Dirichlet polynomials play an important role in several questions connected to the distribution of primes. For example, they can be used to bound the number of zeroes of the Riemann zeta function in vertical strips, which is relevant to...

Abstract: Discrete subgroups of PSL(2,C) are called Kleinian groups and they are fundamental groups of complete oriented hyperbolic 3-manifolds/orbifolds. Except for countably many conjugacy classes, all Kleinian groups have infinite co-volume in...

Abstract: What is the densest lattice sphere packing in the d-dimensional Euclidean space? In this talk we will investigate this question as dimension d goes to infinity and we will focus on the lower bounds for the best packing density, or in other...

Abstract: The Higher order Fourier uniformity conjecture asserts that on most short intervals, the Mobius function is asymptotically uniform in the sense of Gowers; in particular, its normalized Fourier coefficients decay to zero.  This conjecture...

Abstract: Cohen, Lenstra, and Martinet have given highly influential conjectures on the distribution of class groups of number fields, the finite abelian groups that control the factorization in number fields. Malle, using tabulation of class groups...

Abstract: Starting with the "Leibniz" formula for $\pi$

$ \pi/4 = 1 - 1/3 + 1/5 - 1/7 + \ldots$

the special values of Dirichlet L-functions have long been a source of fascination and frustration. From Euler's solution in 1734 of the Basel problem to...

Abstract: What are the slopes of periodic billiard paths in a regular polygon?

We will connect this question and others to:

        - cusps of thin groups,

        - curves on Hilbert modular varieties,

        - heights from Jacobians with real...

Abstract: To study the asymptotic behavior of orbits of a dynamical system, one can look at orbit closures or invariant measures. When the underlying system has a homogeneous structure, usually coming from a Lie group, with appropriate assumptions a...

Abstract: (Joint with Samuel Grushevsky, Gabriele Mondello, Riccardo Salvati Manni) We determine the maximal dimension of a compact subvariety of the moduli space of principally polarized abelian varieties $A_g$ for any value of g. For $g<16$ the dimension is $g-1$, while for $g \geq 16$, it is determined by the larged dimensional compact Shimura subvariety, which we determine. Our methods rely on deforming the boundary using special varieties, and functional transcendence theory.

Abstract: Ratner's landmark equidstribution results for unipotent flows have had dramatic applications in many mathematical areas. Recently there has been considerable progress in the long sought for goal of getting effective equidistribution...

Abstract: The Mobius function is one of the most important arithmetic functions. There is a vague yet well known principle regarding its randomness properties called the “Mobius randomness law". It basically states that the Mobius function should be...