Schedule and Abstracts

10:15 - 10:45

Welcome and Registration - Kassar Lobby

10:45 - 11:35

Mamikon Gulian - Brown

Machine Learning of Differential Equations with Gaussian Processes and
Fractional Calculus
A recent application of machine learning is the discovery of ODE and PDE models directly from data. At the same time, our modern understanding of fractional-order PDEs is that they are a natural tool for modeling anomalous diffusion, memory effects, non-locality, and systems driven by heavy-tailed probability distributions. We propose a Gaussian process framework for discovering linear fractional-order equations from data in such applications, allowing one to bypass the analytic difficulties of fractional calculus. Our framework allows for a wide-variety of space-fractional operators and covariance kernels, including the Matern family. The use of fractional derivatives has a dual benefit of allowing standard derivative terms to be learned from a single fractional-order archetype. Thus, both fractional- and integer-order equations can be learned from a simple Gaussian process regression, using continuous optimization, reducing the need to use "dictionaries'' and genetic algorithms.

11:45 - 12:35

Hong Wang - MIT

Fourier restriction problems 
If the Fourier transform of f is supported on a sphere in R^3, what can we say about f? Stein conjectured in the 1960s that the L^p-norm of f is bounded by the L^∞-norm of its Fourier transform  for any p>3. This conjecture is still open for full range of p. Guth introduced the polynomial method in the study of this problem and proved the conjecture for p>3.25. We discuss the history of this problem and how to attack it using polynomials.

12:45 - 1:30


Lunch will be provided for those who attend.

1:30 - 2:20

Monika Pichler - NU

An inverse boundary value problem for Maxwell’s equations
Inverse boundary value problems arise naturally in many physical situations where one wishes to study the properties of an object using measurements on its surface. This amounts to recovering coefficients of a partial differential equation from boundary data of its solutions. A pioneer contribution in the mathematical study of such problems was a paper by A. P. Calderón published in 1980, posing an inverse problem for the conductivity equation. This work motivated many developments in the field, first concerning the conductivity equation, and subsequently also for other partial differential equations. I will give a brief history of the study of this prototypical inverse problem, and then discuss an inverse problem for Maxwell’s equations, how it relates to that for the conductivity equation, and what tools are needed to show unique solvability of this inverse problem.

2:30 - 3:20

Linhan Li - Brown

Boundary value problems with unbounded leading coefficients
In this talk, we will discuss the boundary behavior of solutions of divergence-form operators with an elliptic symmetric part and a BMO anti-symmetric part.  We will show that many of the classical results about the boundary value problems with bounded coefficients hold for these operators.  Our results will hold in non-tangentially accessible (NTA) domains; these general domains include the class of Lipschitz domains. When specialized to Lipschitz domains, it is then possible to extend to these operators various criteria for determining mutual absolute continuity of elliptic measure with surface measure.

3:30 - 4:00


4:00 - 4:50

Patrick Lopatto - Harvard

Universality of the least singular value for sparse random matrices
Random symmetric matrices have been intensely studied since Wigner’s discovery that, as the size of a random matrix tends to infinity, its eigenvalue density becomes independent of the distribution used to generate the matrix entries. Because the same behavior arises regardless of the underlying distribution, it is said to be universal. Recent investigations have culminated in a proof of the Wigner-Dyson-Mehta conjecture, which asserts the universality of the local eigenvalue statistics in the limit. This talk will discuss the methods developed to prove the universality of local statistics and describe their many applications, which include a proof of the universality of the least singular value for sparse, non-symmetric random matrices (based on joint work with Ziliang Che)

5:00 - 5:50

Ewain Gwynne - MIT

Scaling limits of random planar maps
Random planar maps are random graphs embedded in the plane which can be thought of as discrete random surfaces. Such graphs are expected to converge to so-called Liouville quantum gravity (LQG) surfaces as the number of vertices tends to infinity. I will give an overview of known results and conjectures concerning (a) the sense in which this convergence occurs and (b) the extent to which we can make rigorous sense of Liouville quantum gravity surfaces, assuming only basic background in probability theory.