Chern classes of Schubert cells and varieties

June Huh
Princeton University; Veblen Fellow, School of Mathematics
March 30, 2015
Chern-Schwartz-MacPherson class is a functorial Chern class defined for any algebraic variety. I will give a geometric proof of a positivity conjecture of Aluffi and Mihalcea that Chern classes of Schubert cells and varieties in Grassmannians are positive. While the positivity conjecture is a purely combinatorial statement, a combinatorial 'counting' proof is known only in very special cases. In addition, the current geometric argument do not work for Schubert varieties in more general flag varieties.

On the geometry and topology of zero sets of Schrödinger eigenfunctions

Yaiza Canzani
Member, School of Mathematics
March 30, 2015
In this talk I will present some new results on the structure of the zero sets of Schrödinger eigenfunctions on compact Riemannian manifolds. I will first explain how wiggly the zero sets can be by studying the number of intersections with a fixed curve as the eigenvalue grows to infinity. Then, I will discuss some results on the topology of the zero sets when the eigenfunctions are randomized. This talk is based on joint works with John Toth and Peter Sarnak.

Kolmogorov width of discrete linear spaces: an approach to matrix rigidity

Sergey Yekhanin
Microsoft Research
March 31, 2015
A square matrix $V$ is called rigid if every matrix obtained by altering a small number of entries of $V$ has sufficiently high rank. While random matrices are rigid with high probability, no explicit constructions of rigid matrices are known to date. Obtaining such explicit matrices would have major implications in computational complexity theory.

Proof of the Grothendieck-Serre conjecture on principal bundles over regular local rings containing a field

Ivan Panin
Steklov Institute of Mathematics, St. Petersburg; Member, School of Mathematics
March 31, 2015
Let $R$ be a regular semi-local domain, containing a field. Let $G$ be a reductive group scheme over $R$. We prove that a principal $G$-bundle over $R$ is trivial, if it is trivial over the fraction field of $R$. If the regular semi-local domain $R$ contains an infinite field this result is proved in a joint work with R. Fedorov. The result has the following corollary: let $X$ be a smooth affine irreducible algebraic variety over a field $K$ and let $G$ be a reductive group over $K$.

Natural algorithms for flow problems

Nisheeth Vishnoi
École polytechnique fédérale de Lausanne
April 6, 2015
In the last few years, there has been a significant interest in the computational abilities of Physarum Polycephalum (a slime mold). This drew from a remarkable experiment which showed that this organism can compute shortest paths in a maze. Subsequently, the workings of Physarum were mathematically modeled as a dynamical system and algorithms inspired by this model were proposed to solve several graph problems.

Fredholm theory for higher order elliptic boundary value problems in non-smooth domains

Irina Mitrea
Temple University; von Neumann Fellow, School of Mathematics
April 6, 2015
One of the most effective methods for solving boundary value problems for basic elliptic equations of mathematical physics in a given domain is the method of layer potentials. Its essence is to reduce the entire problem to an integral equation on the boundary of the domain which is then solved using Fredholm theory. Until now, this approach has been primarily used in connection with second order operators for which a sophisticated, far-reaching theory exists.

Counting and dynamics in $\mathrm{SL}_2$

Michael Magee
Member, School of Mathematics
April 6, 2015
In this talk I'll discuss a lattice point count for a thin semigroup inside $\mathrm{SL}_2(\mathbb Z)$. It is important for applications I'll describe that one can perform this count uniformly throughout congruence classes. The approach to counting is dynamical---with input from both the real place and finite primes. At the real place one brings ideas of Dolgopyat concerning oscillatory functions into play​. At finite places the necessary expansion property follows from work of Bourgain and Gamburd (at one prime) or Bourgain, Gamburd and Sarnak (at squarefree moduli).

Interleaved products in special linear groups: mixing and communication complexity

Emanuele Viola
Northeastern University
April 7, 2015
Let $S$ and $T$ be two dense subsets of $G^n$, where $G$ is the special linear group $\mathrm{SL}(2,q)$ for a prime power $q$. If you sample uniformly a tuple $(s_1,\ldots,s_n)$ from $S$ and a tuple $(t_1,\ldots,t_n)$ from $T$ then their interleaved product $s_1.t_1.s_2.t_2 \ldots s_n.t_n$ is equal to any fixed $g$ in $G$ with probability $(1/|G|)(1 + |G|^{-\Omega(n)})$.

Enigmatic Enceladus

Peter Goldreich
California Institute of Technology & Institute for Advanced Study
April 7, 2015

Saturn’s satellite Enceladus displays a bewildering array of thermal activity. I will describe our attempts to understand these phenomena in terms of tidal heating associated with the 2:1 mean motion resonance between Enceladus and Dione. Then I will argue that Enceladus and the other midsize satellites of Saturn are young.