The geometry of division algebras

Speaker: 

Daniel Krashen

Institution: 

University of Georgia

Time: 

Tuesday, November 22, 2016 - 4:00pm to 5:00pm

Host: 

Location: 

RH 306

The study of division algebras has, from its inception, been closely tied to geometry of various kinds. This relationship has become richer with developments in both algebra and algebraic geometry. In this talk I will discuss some of the history of the theory of division algebras and some of its interactions with geometry as well as introduce some modern perspectives.

Vector bundles of conformal blocks on the moduli space of curves

Speaker: 

Angela Gibney

Institution: 

University of Georgia

Time: 

Tuesday, November 8, 2016 - 3:00pm to 4:00pm

Host: 

Location: 

RH 306

In this talk I will introduce the moduli space of curves, and a class of vector bundles “of conformal blocks” on the moduli space of curves.   I’ll give a nonspecialist definition of these bundles, which have connections to algebraic geometry, representation theory and mathematical physics.  I’ll talk about how by studying the bundles we can learn about the moduli space of curves, and vice versa, focusing on just a few recent results, and open problems.

Growth and singularity in 2D fluids

Speaker: 

Andrej Zlatos

Institution: 

University of Wisconsin-Madison

Time: 

Thursday, February 18, 2016 - 4:00pm to 5:00pm

Host: 

Location: 

RH 306

The question of global regularity remains open for many fundamental models of fluid dynamics.  In two dimensions, solutions to the incompressible Euler equations have been known to be globally regular since the 1930s, although their derivatives can grow double-exponentially with time.  On the other hand, this question has not yet been resolved for the more singular surface quasi-geostrophic (SQG) equation, which is used in atmospheric models.  The latter state of affairs is also true for the modified SQG equations, a family of PDE which interpolate between these two models.

I will present two results about the patch dynamics version of these equations on the half-plane.  The first is global-in-time regularity for the Euler patch model, even if the patches initially touch the boundary of the half-plane.  The second is local-in-time regularity for those modified SQG patch equations that are only slightly more singular than Euler, but also existence of their solutions which blow up in finite time. The latter appears to be the first rigorous proof of finite time blow-up in this type of fluid dynamics models.

Locally symmetric spaces and torsion classes

Speaker: 

Ana Caraiani

Institution: 

Princeton University

Time: 

Tuesday, January 19, 2016 - 3:00pm to 4:00pm

Host: 

Location: 

RH 306

The Langlands program is an intricate network of conjectures, which are meant to connect different areas of mathematics, such as number theory, harmonic analysis and representation theory. One striking consequence of the Langlands program is the Ramanujan conjecture, which is a statement purely within harmonic analysis, about the growth rate of Fourier coefficients of modular forms. It turns out to be intimately connected to the Weil conjectures, a statement about the cohomology of projective, smooth varieties defined over finite fields. 

I will explain this connection and then move towards a mod p analogue of these ideas. More precisely, I will explain a strategy for understanding torsion occurring in the cohomology of locally symmetric spaces and how to detect which degrees torsion will contribute to. The main theorem is joint work with Peter Scholze and relies on a p-adic version of Hodge theory and on recent developments in p-adic geometry.

The dynamics of Type II solutions to energy critical wave equations

Speaker: 

Hao Jia

Institution: 

University of Chicago

Time: 

Thursday, January 14, 2016 - 4:00pm to 5:00pm

Host: 

Location: 

RH 306

The study of dynamics of energy critical wave equations has seen remarkable progresses in recent years, resulting in deeper understanding of the singularity formation, soliton dynamics, and global large data theory. I will firstly review some of the landmark results, with emphasis on the channel of energy inequalities discovered by Duyckaerts, Kenig and Merle. Applications in the study of global dynamics of defocusing energy critical wave equation with a trapping potential in the radial case will be presented in some detail. We remark that the channel of energy argument provides crucial control on the global dynamics of the solution, and seems to be the only tool currently available to measure dispersion in this context, when we do not assume any smallness condition. The channel of energy argument is however sensitive to dimensions, and in higher dimensions, it is less powerful. We will mention a new approach to eliminate the dispersive energy when the channel of energy argument fails. Lastly, a new Morawetz estimate in the context of focusing energy critical wave equations will be discussed. This estimate allows us to study the singularity formation in more details in the non-radial case, without size restriction. As a result, we can characterize the solution along a sequence of times approaching the singular time, up to every nontrivial scale, as modulated solitons. 

Part of the talk is based on joint works with C. Kenig, and with B.P. Liu, W. Schlag, G.X. Xu.
 

Riemann--Hilbert problems, computation and universality

Speaker: 

Thomas Trogdon

Institution: 

Courant Institute of Mathematical Sciences

Time: 

Monday, January 11, 2016 - 4:00pm to 5:00pm

Host: 

Location: 

RH 306

This talk will concern two topics.  The first topic is the applications of Riemann--Hilbert (RH) problems.   RH problems provide a powerful and rigorous tool to study many problems in pure and applied mathematics.  Important problems in integrable systems and random matrix theory have been solved with the aid of RH problems. RH problems can also be approached numerically with applications to the numerical solution of PDEs and the sampling of random matrix ensembles.  The resulting methods are seen to have accuracy and complexity advantages over previously existing methods.  The second topic is recent progress on the statistical analysis of numerical algorithms.  In particular, with appropriate randomness, the fluctuations of the iteration count of numerous numerical algorithms have been demonstrated to be universal, see Pfrang, Deift and Menon (2014).  I will discuss simple algorithms where universality is provable and the wide persistence of this phenomenon.

Torsion in families of abelian varieties and hyperbolicity of moduli spaces

Speaker: 

Benjamin Bakker

Institution: 

Humboldt-Universität zu Berlin

Time: 

Tuesday, January 5, 2016 - 4:00pm to 5:00pm

Host: 

Location: 

RH 306

The group of rational points is an important but subtle invariant of an abelian variety defined over a number field.  In the case of an elliptic curve over Q, a celebrated theorem of Mazur asserts that there are only finitely many possibilities for the torsion part; the same is conjectured to be true for all abelian varieties over number fields though very little has been proven in higher dimensions.  The natural geometric analog, known as the geometric torsion conjecture, asks for a bound on the torsion sections of a family of abelian varieties over a complex curve, and can be interpreted as the nonexistence of low genus curves in congruence towers of Siegel modular varieties.  We will discuss a general method for bounding the genus of curves in locally symmetric varieties using hyperbolic geometry and apply it to some special cases of the torsion conjecture as well as some related problems.  Along the way we will also deduce some results about the global geometry of these moduli spaces.  This is joint work with J. Tsimerman.

Random walk parameters and the geometry of groups

Speaker: 

Tianyi Zheng

Institution: 

Stanford University

Time: 

Thursday, December 3, 2015 - 3:00pm to 4:00pm

Host: 

Location: 

RH306

The first characterization of groups by an asymptotic description of random walks on their Cayley graphs dates back to Kesten’s criterion of amenability. I will first review some connections between the random walk parameters and the geometry of the underlying groups. I will then discuss a flexible construction that gives solution to the inverse problem (given a function, find a corresponding group) for large classes of speed, entropy and return probability of simple random walks on groups of exponential volume growth. Based on joint work with Jeremie Brieussel.
 

KdV equation with almost periodic initial data

Speaker: 

Milivoje Lukic

Institution: 

University of Toronto

Time: 

Wednesday, December 2, 2015 - 3:00pm to 4:00pm

Host: 

Location: 

RH306

In the 1960s, the KdV equation was discovered to have infinitely many conserved quantities, explained by a "Lax pair" formalism. Due to this, the KdV equation is often described as completely integrable. Similar features were soon found in other nonlinear equations, spurring the field of integrable PDEs in which the KdV equation continues to be one of the flagship models.

These ideas were originally implemented for sufficiently fast decaying initial data and, in the 1970s, for periodic initial data. In this talk, we will describe recent progress for almost periodic initial data, centered around a conjecture of Percy Deift that the solution is almost periodic in time. The case of almost periodic initial data is strongly motivated by the periodic case but carries significant challenges so, beyond a class of algebro-geometric solutions, rigorous results have remained scarce. We will discuss the proof of existence, uniqueness, and almost periodicity in time, in the regime of absolutely continuous and sufficiently "thick" spectrum (in a sense made precise in the talk), and in particular, the proof of Deift's conjecture for small analytic quasiperiodic initial data.

Nonstandard methods in Lie theory and additive combinatorics

Speaker: 

Isaac Goldbring

Institution: 

University of Illinois at Chicago

Time: 

Monday, November 30, 2015 - 3:00pm to 4:00pm

Host: 

Location: 

RH 306

During the early development of calculus, eminent mathematicians such as Leibniz and Cauchy freely used infinitesimals in their calculations. Once the mathematical community became dubious of their status, the use of infinitesimals was replaced by the now familiar epsilon-delta rigor. In the 1960s, Abraham Robinson used techniques from model theory to rescue infinitesimals from their squalid state and instead put them on a firm foundation in what he called nonstandard analysis. Since its inception, nonstandard techniques have proven useful in many diverse areas of mathematics, from geometry to functional analysis to mathematical finance. Besides allowing one to give precise meaning to intuitive, heuristic arguments involving “ideal” elements, nonstandard analysis offers new techniques such as hyperfinite approximation and Loeb measure. In this talk, I will survey some uses of nonstandard analysis in Lie theory and additive combinatorics. Some highlights of the talk will be the nonstandard solution to Hilbert’s fifth problem (and its extension to the local case), the Breuillard-Green-Tao structure theorem for approximate groups, and some progress on a sumset conjecture of Erdos.

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