## Why can't we prove tensor rank and Waring rank lower bounds?

Visu Makam

University of Michigan; Member, School of Mathematics

February 12, 2019

Visu Makam

University of Michigan; Member, School of Mathematics

February 12, 2019

Daniel Stern

Princeton University

February 12, 2019

We will describe recent progress on the existence theory and asymptotic analysis for solutions of the complex Ginzburg-Landau equations on closed manifolds, emphasizing connections to the existence of weak minimal submanifolds of codimension two. On manifolds with nontrivial first cohomology group, our results rely on new estimates for the Ginzburg-Landau energies along paths of maps connecting distinct homotopy classes of circle-valued maps, which may be of independent interest.

Francesco Maggi

The University of Texas at Austin; Member, School of Mathematics

February 12, 2019

We review various recent results aimed at understanding bubbling into spheres for boundaries with almost constant mean curvature. These are based on joint works with Giulio Ciraolo (U Palermo), Matias Delgadino (Imperial College London), Brian Krummel (Purdue), Cornelia Mihaila (U Chicago), and Robin Neumayer (Nothwestern and IAS).

Rana Mitter

Professor of the History and Politics of Modern China, St. Cross College, University of Oxford

February 6, 2019

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.

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

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.

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.

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.

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).