On gradient complexity of measures on the discrete cube

Ronen Eldan
Weizmann Institute of Science
December 12, 2016
The motivating question for this talk is: What does a sparse Erdős–Rényi random graph, conditioned to have twice the number of triangles than the expected number, typically look like? Motivated by this question, In 2014, Chatterjee and Dembo introduced a framework for obtaining Large Deviation Principles (LDP) for nonlinear functions of Bernoulli random variables (this followed an earlier work of Chatterjee-Varadhan which used limit graph theory to answer this question in the dense regime).

Arithmetic and geometry of Picard modular surfaces

Dinakar Ramakrishnan
California Institute of Technology; Visitor, School of Mathematics
December 8, 2016
Of interest are (i) the conjecture of Bombieri (and Lang) that for any smooth projective surface $X$ of general type over a number field $k$, the set $X(k)$, of $k$-rational points is not Zariski dense, and (ii) the conjecture of Lang that $X(k)$, is even finite if in addition $X$ is hyperbolic, i.e., there is no non-constant holomorphic map from the complex line $C$ into $X(C)$. We can verify them for the Picard modular surfaces $X$ which are smooth toroidal compactifications of congruence quotients $Y$ of the unit ball in $\mathbb C^2$.

Approximate constraint satisfaction requires sub-exponential size linear programs

Pravesh Kothari
Member, School of Mathematics
December 6, 2016
We show that for constraint satisfaction problems (CSPs), sub-exponential size linear programming relaxations are as powerful as $n^{\Omega(1)}$-rounds of the Sherali-Adams linear programming hierarchy. As a corollary, we obtain sub-exponential size lower bounds for linear programming relaxations that beat random guessing for many CSPs such as MAX-CUT and MAX-3SAT.

Lawless Economy? Putin's Russia and the Imperfect Market

Bill Browder
Founder and Chief Executive Officer of Hermitage Capital Management
December 2, 2016
Bill Browder
World Disorder Lecture Series: Lawless Economy?
In this public lecture, Bill Browder, Founder and Chief Executive Officer of Hermitage Capital Management, will give a firsthand critical analysis of the Russian economy–—particularly the absence of the rule of law–—laden with insights derived from his personal experience.

On the number of ordinary lines determined by sets in complex space

Shubhangi Saraf
Rutgers University
December 5, 2016
Consider a set of $n$ points in $\mathbb R^d$. The classical theorem of Sylvester-Gallai says that, if the points are not all collinear then there must be a line through exactly two of the points. Let us call such a line an "ordinary line". In a recent result, Green and Tao were able to give optimal linear lower bounds (roughly $n/2$) on the number of ordinary lines determined $n$ non-collinear points in $\mathbb R^d$. In this talk we will consider the analog over the complex numbers.

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.

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