Abstract: An inhomogeneous Markov chain X_n is a Markov chain whose state spaces and transition kernels change in time. A “local limit theorem” is an asymptotic formula for probabilities of th form
Prob[S_N-z_N\in (a,b)] , S_N=f_1(X_1,X_2)+....+f_N(X_N,X_{N+1})
in the limit N—>infinity. Here z_N is a “suitable” sequence of numbers.
I will describe general sufficient conditions for such results.
If time allows, I will explain why such results are needed for the study of certain problems related to irrational rotations.
This is joint work with Dmitry Dolgopyat.
Krieger’s generator theorem shows that any free invertible ergodic measure preserving action (Y,\mu, S) can be modelled by A^Z (equipped with the shift action) provided the natural entropy constraint is satisfied; we call such systems (here it is A^Z) universal. Along with Tom Meyerovitch, we establish general specification like conditions under which Z^d-dynamical systems are universal. These conditions are general enough to prove that
1) A self-homeomorphism with almost weak specification on a compact metric space (answering a question by Quas and Soo)
Title: Eigenvector correlation in the complex Ginibre ensemble
Abstract:
The complex Ginibre ensemble is a non-Hermitian random matrix on $\mathbb{C}^N$ with i.i.d. complex Gaussian entries normalized to have mean zero and variance $1/N$. Unlike the Gaussian unitary ensemble, for which the eigenvectors are orthogonal, the geometry
of the eigenbases of the Ginibre ensemble are not particularly well understood.
We will discuss a some results regarding the analytic and algebraic structure of eigenvector correlations in this matrix ensemble. In particular, we uncover an extended
Title: Fuglede's spectral set conjecture for convex polytopes.
Abstract:
A set \Omega \subset \mathbb{R}^d is called spectral if the space L^2(\Omega) admits an orthogonal basis of exponential functions. Back in 1974, B. Fuglede conjectured that spectral sets could be characterized geometrically as sets which can tile the space by translations. This conjecture inspired extensive research over the years, but nevertheless, the precise connection between the notions of spectrality and tiling, is still a mystery.
Title: A decomposition of the Laplacian on symmetric metric graphs
Abstract
The spectrum of the Laplacian on graphs which have certain symmetry properties can be studied via a decomposition of the operator as a direct sum of one-dimensional operators which are simpler to analyze. In the case of metric graphs, such a decomposition was described by M. Solomyak and K. Naimark when the graphs are radial trees. In the discrete case, there is a result by J. Breuer and M. Keller treating more general graphs.
Abstract: During the last 20 years there has been a considerable literature on a collection of related mathematical topics: higher degree versions of Poncelet’s Theorem, certain measures associated to some finite Blaschke products and the numerical range of finite dimensional completely non-unitary contractions with defect index 1. I will explain that without realizing it, the authors of these works were discussing OPUC.
The hydrogen atom system is one of the most thoroughly studied examples of a quantum mechanical system. It can be fully solved, and the main reason why is its (hidden) symmetry. In this talk I shall explain how the symmetries of the Schrödinger equation for the hydrogen atom, both visible and hidden, give rise to an example in the recently developed theory of algebraic families of Harish-Chandra modules. I will show how the algebraic structure of these symmetries completely determines the spectrum of the Schrödinger operator and sheds new light on the quantum nature of the system.
Title:
Regularity via minors and applications to conformal maps.
Abstract:
Let f:\mathbb{R}^n \to \mathbb{R}^n be a Sobolev map; Suppose that the k-minors of df are smooth. What can we say about the regularity of f?
This question arises naturally in the context of Liouville's theorem, which states that every weakly conformal map is smooth. I will explain the connection of the minors question to the conformal regularity problem, and describe a regularity result for maps with regular minors.
