Math

School of Mathematics

Integrable Stochastic Particle Systems and Macdonald Processes

Alexei Borodin
Massachusetts Institute of Technology
April 25, 2013 - 2:00pm

A large class of one dimensional stochastic particle systems are predicted to share the same universal long-time/large-scale behavior. By studying certain integrable models within this (Kardar-Parisi-Zhang) universality class we access what should be universal statistics and phenomena. In this talk we focus on two different integrable exclusion processes: q-TASEP and ASEP.


Solvability in Polynomials of Pell Equations in a Pencil and a Conjecture of Pink

Umberto Zannier
Scuola Normale Superiore de Pisa, Italy
April 10, 2013 - 2:00pm

The classical Pell equation $X^2-DY^2=1$, to be solved in integers $X,Y\neq 0$, has a variant for function fields (studied already by Abel), where now $D=D(t)$ is a complex polynomial of even degree and we seek solutions in nonzero complex polynomials $X(t),Y(t)$. In this context solvability is no longer ensured by simple conditions on $D$ and may be considered `exceptional'.


Statistics of the Zeros of the Zeta Function: Mesoscopic and Macroscopic Phenomena

Brad Rodgers
University of California, Los Angeles
March 27, 2013 - 4:30pm

We review the well known microscopic correspondence between random zeros of the Riemann zeta-function and the eigenvalues of random matrices, and discuss evidence that this correspondence
extends to larger mesoscopic collections of zeros or eigenvalues. In addition, we discuss interesting phenomena that appears in the statistics of even larger macroscopic collections of zeros. The terms microscopic, mesoscopic, and macroscopic are from random matrix theory and will be defined in the talk.


Mean Values of L-Functions for the Hyperelliptic Ensemble

Julio Andrade
Brown University
March 27, 2013 - 3:45pm

Thanks to the work of Katz and Sarnak on L-functions over function fields, we know that the Frobenius classes associated to L-functions of hyperelliptic curves over a finite field with $q$ elements, $F_{q}$, becomes equidistributed in the unitary symplectic group in the limit as the genus of the curve is fixed and $q$ is large.


A Zero-Density Approach to Smooth Numbers

Adam Harper
University of Montreal
March 27, 2013 - 2:30pm

A number is said to be $y$-smooth if all of its prime factors are less than $y$. Such numbers appear in many places throughout analytic and combinatorial number theory, and much work has been done to investigate their distribution.


Special Lectures in Analysis/Number Theory

Institute for Advanced Study
March 27, 2013 (All day)

Norm Convergence of Nonconventional Ergodic Averages

Miguel Walsh
Buenos Aires
March 27, 2013 - 1:30pm

Consider a group of measure preserving transformations acting on a probability space. The limiting behavior of the nonconventional ergodic averages associated with this action has been the subject of much attention since the work of Furstenberg on Szemerédi’s theorem. We will discuss this problem, and how to establish the convergence of these averages whenever the group is nilpotent.


The Hypoelliptic Laplacian: An Introduction

Jean-Michel Bismut
Universite de Paris-Sud
March 26, 2013 - 11:00am

The Codimension Barrier in Incidence Geometry

Larry Guth
Massachusetts Institute of Technology
March 14, 2013 - 2:00pm

Incidence geometry is a part of combinatorics that studies the intersection patterns of geometric objects. For example, suppose that we have a set of L lines in the plane. A point is called r-rich if it lies in r different lines from the set. For a given L and a given r, how many r-rich points can there be? This question is answered by a theorem of Szemer\'edi and Trotter from the early 80's. Different generalizations of this theorem are a central topic in incidence geometry.


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