School of Mathematics

Constructible sheaves in mirror symmetry

David Treumann
Boston College; von Neumann Fellow, School of Mathematics
January 20, 2017
I will survey the coherent-constructible correspondence of Bondal, which embeds the derived category of coherent sheaves on a toric variety into the derived category of constructible sheaves on a compact torus. The tools of the first lecture turn this into a homological mirror theorem -- a pretty strong one, after recent results of Ike and Kuwagaki.

Noncommutative geometry, smoothness, and Fukaya categories

Sheel Ganatra
Member, School of Mathematics
December 2, 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.

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.

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 $\mathrm{Ind}_P^G \tau$.

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.

Theory of accelerated methods

Zeyuan Allen-Zhu
Member, School of Mathematics
November 22, 2016

In this talk I will show how to derive the fastest coordinate descent method [1] and the fastest stochastic gradient descent method [2], both from the linear-coupling framework [3]. I will relate them to linear system solving, conjugate gradient method, the Chebyshev approximation theory, and raise several open questions at the end. No prior knowledge is required on first-order methods.

On the effect of randomness on planted 3-coloring models

Uri Feige
Weizmann Institute of Science
November 21, 2016
The random planted 3-coloring model generates a 3-colorable graph $G$ by first generating a random host graph $H$ of average degree $d$, and then planting in it a random 3-coloring (by giving each vertex a random color and dropping the monochromatic edges). For a sufficiently large constant $c$, Alon and Kahale [SICOMP 1997] presented a spectral algorithm that finds (with high probability) the planted 3-coloring of such graphs whenever $d > c\log n$.