Members Seminar

Topologies of nodal sets of random band limited functions

Peter Sarnak
Institute for Advanced Study; Faculty, School of Mathematics
March 3, 2014
We discuss various Gaussian ensembles for real homogeneous polynomials in several variables and the question of the distribution of the topologies of the connected components of the zero sets of a typical such random real hypersurface. For the "real -Fubini -Study ensemble" and at the other end the "monochromatic wave ensemble ", one can show that these have universal laws. Some qualitative features of these laws are also established. Joint work with I. Wigman.

Zeros of polynomials via matrix theory and continued fractions

Olga Holtz
University of California, Berkeley; Member, School of Mathematics
February 24, 2014
After a brief review of various classical connections between problems of polynomial zero localization, continued fractions, and matrix theory, I will show a few ways to generalize these classical techniques to get new results about some interesting polynomials and entire functions.

Rigidity and Flexibility of Schubert classes

Colleen Robles
Texas A & M University; Member, School of Mathematics
January 27, 2014
Consider a rational homogeneous variety \(X\). (For example, take \(X\) to be the Grassmannian \(\mathrm{Gr}(k,n)\) of \(k\)-planes in complex \(n\)-space.) The Schubert classes of \(X\) form a free additive basis of the integral homology of \(X\). Given a Schubert class \([S]\), represented by a Schubert variety \(S\) in \(X\), Borel and Haefliger asked: aside from the Schubert variety, does \([S]\) admit any other algebraic representatives? I will discuss this, and related questions, in the case that \(X\) is Hermitian symmetric.

Eigenvalues and eigenvectors of spiked covariance matrices

Antti Knowles
December 9, 2013
I describe recent results on spiked covariance matrices, which model multivariate data containing nontrivial correlations. In principal components analysis, one extracts the leading contribution to the covariance by analysing the top eigenvalues and associated eigenvectors of the covariance matrix. I give non-asymptotic, sharp, high-probability estimates relating the principal components of the true covariance matrix to those of the sample covariance matrix.

From Gromov to the Moon

Joel Fish
Massachusetts Institute of Technology; Member, School of Mathematics
December 2, 2013
I will present some recent applications of symplectic geometry to the restricted three body problem. More specifically, I will discuss how Gromov's original study of pseudoholomorphic curves in the complex projective plane has led to the construction of global surfaces of section, and more generally finite energy foliations, below and slightly above the first Lagrange point in the regularized planar circular restricted three body problem. The talk will be accessible to a general mathematical audience.

Random Cayley Graphs

Noga Alon
Tel Aviv University; Member, School of Mathematics
November 25, 2013
The study of random Cayley graphs of finite groups is related to the investigation of Expanders and to problems in Combinatorial Number Theory and in Information Theory. I will discuss this topic, describing the motivation and focusing on the question of estimating the chromatic number of a random Cayley graph of a given group with a prescribed number of generators.

Interacting Brownian motions in the Kadar-Parisi-Zhang universality class

Herbert Spohn
Technische Universitaet Muenchen; Member, School of Mathematics
November 18, 2013
A widely studied model from statistical physics consists of many (one-dimensional) Brownian motions interacting through a pair potential. The large scale behavior of this model has has been investigated by Varadhan, Yau, and others in the 90's. As a crucial property the model satisfies time-reversibility (alias detailed balance). Once this symmetry is broken, generically one crosses into the KPZ class. Two specific examples will be discussed.

cdh methods in K-theory and Hochschild homology

Charles Weibel
Rutgers University; Member, School of Mathematics
November 11, 2013

This is intended to be a survey talk, accessible to a general mathematical audience. The cdh topology was created by Voevodsky to extend motivic cohomology from smooth varieties to singular varieties, assuming resolution of singularities (for example complex varieties). The slogan is that every variety is locally smooth in this topology. Adapting this to algebraic K-theory has led to several breakthroughs in computation, and clarified connections between the "singular" part of K-theory and Hodge theory.