On minimizers and critical points for anisotropic isoperimetric problems

Robin Neumayer
Member, School of Mathematics
February 19, 2019

Anisotropic surface energies are a natural generalization of the perimeter functional that arise in models in crystallography and in scaling limits for certain probabilistic models on lattices. This talk focuses on two results concerning isoperimetric problems with anisotropic surface energies. In the first part of the talk, we will discuss a weak characterization of critical points in the anisotropic isoperimetric problem (joint work with Delgadino, Maggi, and Mihaila). 

An Application of a Conjecture of Mazur-Tate to Supersingular Elliptic Curves

Emmanuel Lecouturier
Tsinghua University
February 14, 2019

In 1987, Barry Mazur and John Tate formulated refined conjectures of the "Birch and Swinnerton-Dyer type", and one of these conjectures was essentially proved in the prime conductor case by
Ehud de Shalit in 1995. One of the main objects in de Shalit's work is the so-called refined $\mathscr{L}$
invariant, which happens to be a Hecke operator. We apply some results of the theory of Mazur's
Eisenstein ideal to study in which power of the Eisenstein ideal $\mathscr{L}$ belongs. One corollary of our

Min-max solutions of the Ginzburg-Landau equations on closed manifolds

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.

Isoperimetry and boundaries with almost constant mean curvature

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).

Interactive Coding Over the Noisy Broadcast Channel

Gillat Kol
Princeton University
February 11, 2019

A set of n players, each holding a private input bit, communicate over a noisy broadcast channel. Their mutual goal is for all players to learn all inputs. At each round one of the players broadcasts a bit to all the other players, and the bit received by each player is flipped with a fixed constant probability (independently for each recipient). How many rounds are needed? 

This problem was first suggested by El Gamal in 1984. In 1988, Gallager gave an elegant noise-resistant protocol requiring only

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