## 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

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.

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.

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

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).