School of Mathematics

On the topology and index of minimal surfaces

Davi Maximo
University of Pennsylvania; Member, School of Mathematics
February 5, 2019
For an immersed minimal surface in $R^3$, we show that there exists a lower bound on its Morse index that depends on the genus and number of ends, counting multiplicity. This improves, in several ways, an estimate we previously obtained bounding the genus and number of ends by the index. Our new estimate resolves several conjectures made by J. Choe and D.

Spacetime positive mass theorem

Lan-Hsuan Huang
University of Connecticut; von Neumann Fellow, School of Mathematics
February 5, 2019
It is fundamental to understand a manifold with positive scalar curvature and its topology. The minimal surface approach pioneered by R. Schoen and S.T. Yau have advanced our understanding of positively curved manifolds. A very important result is their resolution to the Riemannian positive mass theorem. In general relativity, the concepts of positive scalar curvature and minimal surfaces naturally extend. The extensions connect to a more general statement, so-called the spacetime positive mass conjecture.

Non-commutative rank

Visu Makam
University of Michigan; Member, School of Mathematics
February 5, 2019

A linear matrix is a matrix whose entries are linear forms in some indeterminates $t_1,\dots, t_m$ with coefficients in some field $F$. The commutative rank of a linear matrix is obtained by interpreting it as a matrix with entries in the function field $F(t_1,\dots,t_m)$, and is directly related to the central PIT (polynomial identity testing) problem. The

Drinfeld's lemma for schemes

Kiran Kedlaya
University of California, San Diego; Visiting Professor, School of Mathematics
February 4, 2019
In the course of constructing the Langlands correspondence for GL(2) over a function field, Drinfeld discovered a surprising fact about the interaction between étale fundamental groups and products of schemes in characteristic p. We state this result, describe a new approach to it involving a generalization to perfectoid spaces, and mention an application in p-adic Hodge theory (from joint work with Carter and Zabradi).

The Sample Complexity of Multi-Reference Alignment

Philippe Rigollet
Massachusetts Institute of Technology; Visiting Professor, School of Mathematics
February 4, 2019
How should one estimate a signal, given only access to noisy versions of the signal corrupted by unknown cyclic shifts? This simple problem has surprisingly broad applications, in fields from aircraft radar imaging to structural biology with the ultimate goal of understanding the sample complexity of Cryo-EM. We describe how this model can be viewed as a multivariate Gaussian mixture model whose centers belong to an orbit of a group of orthogonal transformations.

Near-Optimal Strong Dispersers

Dean Doron
The University of Texas at Austin
February 4, 2019

Randomness dispersers are an important tool in the theory of pseudorandomness, with numerous applications. In this talk, we will consider one-bit strong dispersers and show their connection to erasure list-decodable codes and Ramsey graphs. 

Upper bounds for constant slope p-adic families of modular forms

John Bergdall
Bryn Mawr College
January 31, 2019
This talk is concerned with the radius of convergence of p-adic families of modular forms --- q-series over a p-adic disc whose specialization to certain integer points is the q-expansion of a classical Hecke eigenform of level p. Numerical experiments by Gouvêa and Mazur in the nineties predicted the general existence of such families but also suggested, in spirit, the radius of convergence in terms of an initial member. Buzzard and Calegari showed, ten years later, that the Gouvêa--Mazur prediction was false. It has since remained open question how to salvage it.

Analyticity results for the Navier-Stokes Equations

Guher Camliyurt
Member, School of Mathematics
January 31, 2019
We consider the Navier–Stokes equations posed on the half space, with Dirichlet boundary conditions. We give a direct energy based proof for the instantaneous space-time analyticity and Gevrey class regularity of the solutions, uniformly up to the boundary of the half space. We then discuss the adaptation of the same method for bounded domains.