Modulo $p$ representations of reductive $p$-adic groups: functorial properties

Marie-France Vignéras
Institut de Mathématiques de Jussieu
November 30, 2016
Let $F$ be a local field with finite residue characteristic $p$, let $C$ be an algebraically closed field of characteristic $p$, and let $\mathbf G$ be a connected reductive $F$-group. With Abe, Henniart, Herzig, we classified irreducible admissible $C$-representations of $G=\mathbf G(F)$ in terms of supercuspidal representations of Levi subgroups of $G$. For a parabolic subgroup $P$ of $G$ with Levi subgroup $M$ and an irreducible admissible $C$-representation $\tau$ of $M$, we determine the lattice of subrepresentations of $\mathrm{Ind}_P^G \tau$.

Noncommutative geometry, smoothness, and Fukaya categories

Sheel Ganatra
Member, School of Mathematics
November 30, 2016
Noncommutative geometry, as advocated by Konstevich, proposes to replace the study of (commutative) varieties by the study of their (noncommutative) dg/A-infinity categories of perfect complexes. Conveniently, these techniques can then also be applied to Fukaya categories. In this mini-course, we will review some basic properties and structures in noncommutative geometry, with an emphasis on the notion of "smoothness" of a category and its appearance in topology and both sides of homological mirror symmetry.

$C^0$ Hamiltonian dynamics and a counterexample to the Arnold conjecture

Sobhan Seyfaddini
Member, School of Mathematics
November 29, 2016
After introducing Hamiltonian homeomorphisms and recalling some of their properties, I will focus on fixed point theory for this class of homeomorphisms. The main goal of this talk is to present the outlines of a $C^0$ counterexample to the Arnold conjecture in dimensions four and higher. This is joint work with Lev Buhovsky and Vincent Humiliere.

Asymptotic representation theory over $\mathbb Z$

Thomas Church
Stanford University; Member, School of Mathematics
November 28, 2016
Representation theory over $\mathbb Z$ is famously intractable, but "representation stability" provides a way to get around these difficulties, at least asymptotically, by enlarging our groups until they behave more like commutative rings. Moreover, it turns out that important questions in topology/number theory/representation theory/... correspond to asking whether familiar algebraic properties hold for these "rings".

Stochastic block models and probabilistic reductions

Emmanuel Abbe
Princeton University
November 28, 2016
The stochastic block model (SBM) is a random graph model with planted clusters. It has been popular to model unsupervised learning problems, inhomogeneous random graphs and to study statistical versus computational tradeoffs. This talk overviews the recent developments that establish the thresholds for SBMs, the algorithms that achieve the thresholds, and the techniques (genie reduction, graph splitting, nonbacktracking propagation) that are likely to apply beyond SBMs.