Colloquium

from Thursday
June 01, 2017 to Sunday
December 31, 2017
Show events for:
Instructions for subscribing to Stony Brook Math Department Calendars

Thursday
September 14, 2017

4:00 PM - 5:00 PM
Math Tower P-131
Brian Lawrence, Columbia University
Diophantine Problems and $p$-adic Analysis

Mordell's Conjecture (now Faltings' Theorem) establishes a surprising link between geometry and number theory: a curve of genus at least two has only finitely many rational points. We show how $p$-adic analysis can prove finiteness results in number theory. The speaker and A. Venkatesh hope to use these ideas to give a new proof of Faltings' Theorem.


Thursday
September 21, 2017

4:00 PM - 5:00 PM
Math Tower P-131
Misha Skolnikov, Princeton University
A new approach to the largest eigenvalues of random matrices

I will discuss a new method for the study of fluctuations of the largest eigenvalues in various random symmetric matrix ensembles. In addition to striking mathematical features, these arise naturally in the principal component analysis of sample covariance matrices from high-dimensional data. The new approach is based on the moment method for tridiagonal random matrices and strong invariance principles for random walks and their local times. Based on joint works with Vadim Gorin and Pierre Yves Gaudreau Lamarre.


Thursday
September 28, 2017

4:00 PM - 5:00 PM
SCGP Rm 102
Nick Trefethen, NYU
Random functions, random ODEs, and Chebfun

What is a random function? What is noise? The standard answers are nonsmooth, defined pointwise via the Wiener process and Brownian motion. In the Chebfun project, we have found it more natural to work with smooth random functions defined by finite Fourier series with random coefficients. The length of the series is determined by a wavelength parameter $λ$. Integrals give smooth random walks, which approach Brownian paths as $λ→ 0$, and smooth random ODEs, which approach stochastic DEs of the Stratonovich variety. Numerical explorations become very easy in this framework. There are plenty of conceptual challenges in this subject, starting with the fact that white noise has infinite amplitude and infinite energy, a paradox that goes back two different ways to Einstein in 1905.


Thursday
October 05, 2017

4:00 PM - 5:00 PM
Math Tower P-131
Lionel Levine, Cornell University
Will this avalanche go on forever?

In the abelian sandpile model on the d-dimensional lattice Z^d, each site that has at least 2d grains of sand gives one grain of sand to each of its 2d nearest neighbors. An "avalanche" is what happens when you iterate this move. In https://arxiv.org/abs/1508.00161 Hannah Cairns proved that for d=3 the question in the title is algorithmically undecidable: it is as hard as the halting problem! This infinite unclimbable peak is surrounded by appealing finite peaks: What about d=2? What if the initial configuration of sand is random? I’ll tell you about the “mod 1 harmonic functions” Bob Hough and Daniel Jerison and I used to prove in https://arxiv.org/abs/1703.00827 that certain avalanches go on forever.


Thursday
October 19, 2017

4:00 PM - 5:00 PM
Math Tower P-131
Alex Gamburd, CUNY Graduate Center
Arithmetic and Dynamics on Markoff-Hurwitz Varieties

Markoff triples are integer solutions of the equation $x^2+y^2+z^2=3xyz$ which arose in Markoff's spectacular and fundamental work (1879) on diophantine approximation and has been henceforth ubiquitous in a tremendous variety of different fields in mathematics and beyond. After reviewing some of these, we will discuss joint work with Bourgain and Sarnak on the connectedness of the set of solutions of the Markoff equation modulo primes under the action of the group generated by Vieta involutions, showing, in particular, that for almost all primes the induced graph is connected. Similar results for composite moduli enable us to establish certain new arithmetical properties of Markoff numbers, for instance the fact that almost all of them are composite.
Time permitting, we will also discuss recent joint work with Magee and Ronan on the asymptotic formula for integer points on Markoff-Hurwitz surfaces $x_1^2+x_2^2 + \dots + x_n^2 = x_1 x_2 \dots x_n$, giving an interpretation for the exponent of growth in terms of certain conformal measure on the projective space.


Thursday
October 26, 2017

4:00 PM - 5:30 PM
SCGP 102
Peter Kronheimer, Harvard University
Simon Donaldson's Mathematics: A Retrospective

This talk will provide a personal perspective on some of Simon Donaldson's many important contributions to the geometry and topology of manifolds.


Thursday
November 02, 2017

4:00 PM - 5:00 PM
Math Tower P-131
Simon Marshall, University of Wisconsin, and the Neil Chriss and Natasha Herron Chriss Founders' Circle Member, IAS
The asymptotic size of (arithmetic) eigenfunctions

Consider an L^2-normalized Laplace eigenfunction f with large eigenvalue on a compact Riemannian manifold. A well-studied question in harmonic analysis, called the sup-norm problem, asks for the best bound on the pointwise norm of f that one can give in terms of its eigenvalue. This is particularly interesting when the manifold is negatively curved, as the gap between what we expect and can prove is quite large.

I will survey some results on the sup-norm problem in negative curvature that one can obtain by specializing to the case of eigenfunctions with extra arithmetic properties, called Hecke-Maass forms. I will also describe connections between the sup-norm problem for Hecke-Maass forms and the subconvexity problem for L-functions. Some of these results will be work of myself and Farrell Brumley.


Wednesday
December 06, 2017

2:30 PM - 3:30 PM
Math Tower P-131
Dorian Goldfeld, Columbia University
Super-positivity of L-functions

L-functions are ubiquitous in number theory. Their coefficients and special values encode the most important number theoretic and geometric invariants. An L-function is said to be super-positive if all its derivatives at a real number $s ≥ 1/2$ are non-negative. In this talk we discuss how super-positivity can arise, how it relates to the generalized Riemann hypothesis and the Birch-Swinnerton-Dyer conjecture, and why it can be shown that a positive proportion of primitive self-dual L-functions in a suitable family have the super-positivity property.


Show events for:
Instructions for subscribing to Stony Brook Math Department Calendars