School of Mathematics

Eulerianity of Fourier coefficients of automorphic forms

Henrik Gustafsson
Member, School of Mathematics
April 30, 2020
The factorization of Fourier coefficients of automorphic forms plays an important role in a wide range of topics, from the study of L-functions to the interpretation of scattering amplitudes in string theory. In this talk I will present a transfer theorem which derives the Eulerianity of certain Fourier coefficients from that of another coefficient. I will also discuss some applications of this theorem to Fourier coefficients of automorphic forms in minimal and next-to-minimal representations.

Ellipses of small eccentricity are determined by their Dirichlet (or, Neumann) spectra

Steven Morris Zelditch
Northwestern University
April 28, 2020
In 1965, M. Kac proved that discs were uniquely determined by their Dirichlet (or, Neumann) spectra. Until recently, disks were the only smooth plane domains known to be determined by their eigenvalues. Recently, H. Hezari and I proved that ellipses of small eccentricity are also determined uniquely by their Dirichlet (or, Neumann) spectra. The proof uses recent results of Avila, de Simoi, and Kaloshin, proving that nearly circular plane domains with rationally integrable billiards must be ellipses.

A Framework for Quadratic Form Maximization over Convex Sets

Vijay Bhattiprolu
Member, School of Mathematics
April 28, 2020
We investigate the approximability of the following optimization problem, whose input is an
n-by-n matrix A and an origin symmetric convex set C that is given by a membership oracle:
"Maximize the quadratic form as x ranges over C."

This is a rich and expressive family of optimization problems; for different choices of forms A
and convex bodies C it includes a diverse range of interesting combinatorial and continuous
optimization problems. To name some examples, max-cut, Grothendieck's inequality, the

Graph and Hypergraph Sparsification

Luca Trevisan
Bocconi University
April 27, 2020
A weighted graph H is a sparsifier of a graph G if H has much fewer edges than G and, in an appropriate technical sense, H "approximates" G. Sparsifiers are useful as compressed representations of graphs and to speed up certain graph algorithms. In a "cut sparsifier," the notion of approximation is that every cut is crossed by approximately the same number of edges in G as in H. In a "spectral sparsifier" a stronger, linear-algebraic, notion of approximation holds. Similar definitions can be given for hypergraphs.

The Geography of Immersed Lagrangian Fillings of Legendrian Submanifolds

Lisa Traynor
April 24, 2020
Given a smooth knot K in the 3-sphere, a classic question in knot theory is: What surfaces in the 4-ball have boundary equal to K? One can also consider immersed surfaces and ask a “geography” question: What combinations of genus and double points can be realized by surfaces with boundary equal to K? I will discuss symplectic analogues of these questions: Given a Legendrian knot, what Lagrangian surfaces can it bound? What immersed Lagrangian surfaces can it bound?

Deep Generative models and Inverse Problems

Alexandros Dimakis
University of Texas at Austin
April 23, 2020
Modern deep generative models like GANs, VAEs and invertible flows are showing amazing results on modeling high-dimensional distributions, especially for images. We will show how they can be used to solve inverse problems by generalizing compressed sensing beyond sparsity. We will present the general framework, new results and open problems in this space.

Geodesically Convex Optimization (or, can we prove P!=NP using gradient descent)

Avi Wigderson
Herbert H. Maass Professor, School of Mathematics
April 21, 2020
This talk aims to summarize a project I was involved in during the past 5 years, with the hope of explaining our most complete understanding so far, as well as challenges and open problems. The main messages of this project are summarized below; I plan to describe, through examples, many of the concepts they refer to, and the evolution of ideas leading to them. No special background is assumed.

A variational approach to the regularity theory for the Monge-Ampère equation

Felix Otto
Max Planck Institute Leipzig
April 20, 2020
We present a purely variational approach to the regularity theory for the Monge-Ampère equation, or rather optimal transportation, introduced with M. Goldman. Following De Giorgi’s philosophy for the regularity theory of minimal surfaces, it is based on the approximation of the displacement by a harmonic gradient, which leads to a One-Step Improvement Lemma, and feeds into a Campanato iteration on the C1,α-level for the displacement, capitalizing on affine invariance.