School of Mathematics

Most odd degree hyperelliptic curves have only one rational point

Bjorn Poonen
Massachusetts Institute of Technology
March 26, 2015
We prove that the probability that a curve of the form $y^2 = f(x)$ over $\mathbb Q$ with $\deg f = 2g + 1$ has no rational point other than the point at infinity tends to 1 as $g$ tends to infinity. This is joint work with Michael Stoll.

Framed motives of algebraic varieties (after V. Voevodsky)

Ivan Panin
Steklov Institute of Mathematics, St. Petersburg; Member, School of Mathematics
March 25, 2015
This is joint work with G .Garkusha. Using the machinery of framed sheaves developed by Voevodsky, a triangulated category of framed motives is introduced and studied. To any smooth algebraic variety $X$, the framed motive $M_{fr}(X)$ is associated in that category. Theorem. The bispectrum \[( M_{fr} X, M_{fr}(X)(1), M_{fr}(X)(2), ... ),\] each term of which is a twisted framed motive of $X$, has motivic homotopy type of the suspension bispectrum of $X$. (this result is an $A^1$-homotopy analog of a theorem due to G.Segal).

On the incidence complex of the boundary of the character variety

Carlos Simpson
University of Nice
March 24, 2015
Starting from an example in which the Hitchin correspondence can be written down explicitly, we look at what might be said relating the incidence complex of the boundary of the character variety, and the Hitchin map.

Decoupling in harmonic analysis and applications to number theory

Jean Bourgain
IBM von Neumann Professor, School of Mathematics
March 23, 2015
Decoupling inequalities in harmonic analysis permit to bound the Fourier transform of measures carried by hyper surfaces by certain square functions defined using the geometry of the hyper surface. The original motivation has to do with issues in PDE, such as smoothing for the wave equation and Strichartz inequalities for the Schrodinger equation on tori. It turns out however that these decoupling inequalities have surprizing number theoretical consequences,on which we will mainly focus.

Random walks that find perfect objects and the Lovász local lemma

Dimitris Achlioptas
University of California, Santa Cruz
March 23, 2015
At the heart of every local search algorithm is a directed graph on candidate solutions (states) such that every unsatisfactory state has at least one outgoing arc. In stochastic local search the hope is that a random walk will reach a satisfactory state (sink) quickly. We give a general algorithmic local lemma by establishing a sufficient condition for this to be true. Our work is inspired by Moser's entropic method proof of the Lovász Local Lemma (LLL) for satisfiability and completely bypasses the Probabilistic Method formulation of the LLL.

$A^1$ curves on quasi-projective varieties

Qile Chen
Columbia University
March 17, 2015
In this talk, I will present the recent joint work with Yi Zhu on $A^1$-connectedness for quasi-projective varieties. The theory of $A^1$-connectedness for quasi-projective varieties is an analogue of rationally connectedness for projective varieties. To study curves on a quasi-projective variety $U$, we compactify $U$ by a log smooth pair $(X,D)$. Using the theory of stable log maps to $(X,D)$, we were able to produce $A^1$ curves on $U$ from degeneration. This provides many interesting examples of $A^1$-connected varieties.

Chernoff bounds for expander walks

Christopher Beck
Member, School of Mathematics
March 10, 2015
Expander walk sampling is an important tool for derandomization. For any bounded function, sampling inputs from a random walk on an expander graph yields a sample average which is quite close to the true mean, and moreover the deviations obtained are qualitatively similar to those obtained from statistically independent samples. The "Chernoff Bound for Expander Walks" was first described by Ajtai, Komlos, and Szemeredi in 1987, and analyzed in a general form by Gilman in 1988.

Strong contraction and influences in tail spaces

Elchanan Mossel
University of Pennsylvania
March 9, 2015
Various motivations recently led Mendel and Naor, Hatami and Kalai to study functions in tail spaces - i.e. function all of whose low level Fourier coefficients vanish. I will discuss the questions and conjectures that they posed and our recent work which resolves some of these questions and conjectures. Based on a joint work with Steven Heilman and Krzysztof Oleszkiewicz.

On random walks in the group of Euclidean isometries

Elon Lindenstrauss
Hebrew University of Jerusalem
March 6, 2015
In contrast to the two dimensional case, in dimension $d \geq 3$ averaging operators on the $d-1$-sphere using finitely many rotations, i.e. averaging operators of the form $Af(x)= |S|^{-1} \sum_{\theta \in S} f(s x)$ where $S$ is a finite subset of $\mathrm{SO}(d)$, can have a spectral gap on $L^2$ of the $d - 1$-sphere. A result of Bourgain and Gamburd shows that this holds, for instance, for any finite set of elements in $\mathrm{SO}(3)$ with algebraic entries and spanning a dense subgroup.

Whitney numbers via measure concentration in representation varieties

Karim Adiprasito
Member, School of Mathematics
March 3, 2015
We provide a simple proof of the Rota--Heron--Welsh conjecture for matroids realizable as c-arrangements in the sense of Goresky--MacPherson: we prove that the coefficients of the characteristic polynomial of the associated matroids form log-concave sequences, proving the conjecture for a family of matroids out of reach for all previous methods. To this end, we study the Lévy--Milman measure concentration phenomenon on natural pushforwards of uniform measures on the Grassmannian to realization spaces of arrangements under a certain extension procedure on matroids.