School of Mathematics

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.

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.

Intelligent learning: similarity control and knowledge transfer

Vladimir Vapnik
Columbia University
March 30, 2015
During last fifty years a strong machine learning theory has been developed. This theory includes: 1. The necessary and sufficient conditions for consistency of learning processes. 2. The bounds on the rate of convergence which in general cannot be improved. 3. The new inductive principle (SRM) which always achieves the smallest risk. 4. The effective algorithms, (such as SVM), that realize consistency property of SRM principle.

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.