Thursday September 14, 2017 4:00 PM  5:00 PM Math Tower P131
 Brian Lawrence, Columbia University
Diophantine Problems and $p$adic AnalysisMordell'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 P131
 Misha Skolnikov, Princeton University
A new approach to the largest eigenvalues of random matricesI 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 highdimensional 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 ChebfunWhat 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 P131
 Lionel Levine, Cornell University
Will this avalanche go on forever?In the abelian sandpile model on the ddimensional 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 P131
 Alex Gamburd, CUNY Graduate Center
Arithmetic and Dynamics on MarkoffHurwitz VarietiesMarkoff 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 MarkoffHurwitz 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 RetrospectiveThis 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 P131
 Simon Marshall, University of Wisconsin, and the Neil Chriss and Natasha Herron Chriss Founders' Circle Member, IAS
The asymptotic size of (arithmetic) eigenfunctionsConsider an L^2normalized Laplace eigenfunction f with large eigenvalue on a compact Riemannian manifold. A wellstudied question in harmonic analysis, called the supnorm 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 supnorm problem in negative curvature that one can obtain by specializing to the case of eigenfunctions with extra arithmetic properties, called HeckeMaass forms. I will also describe connections between the supnorm problem for HeckeMaass forms and the subconvexity problem for Lfunctions. Some of these results will be work of myself and Farrell Brumley.

Wednesday December 06, 2017 2:30 PM  3:30 PM Math Tower P131
 Dorian Goldfeld, Columbia University
Superpositivity of LfunctionsLfunctions are ubiquitous in number theory. Their coefficients and special values encode the most important number theoretic and geometric invariants. An Lfunction is said to be superpositive if all its derivatives at a real number $s ≥ 1/2$ are nonnegative. In this talk we discuss how superpositivity can arise, how it relates to the generalized Riemann hypothesis and the BirchSwinnertonDyer conjecture, and why it can be shown that a positive proportion of primitive selfdual Lfunctions in a suitable family have the superpositivity property.

