School of Mathematics

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). 

Lorentzian polynomials

June Huh
Visiting Professor, School of Mathematics
February 19, 2019

Lorentzian polynomials link continuous convex analysis and discrete convex analysis via tropical geometry. The class of Lorentzian polynomials contains homogeneous stable polynomials as well as volume polynomials of convex bodies and projective varieties. I will give several combinatorial applications. No specific background will be needed to enjoy the talk. Joint work with Petter Brändén (https://arxiv.org/abs/1902.03719).

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).

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.

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

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.