Dependence of the density of states on the probability distribution for discrete random Schrödinger operators, II

Speaker: 

Christoph Marx

Institution: 

Oberlin College

Time: 

Friday, April 20, 2018 - 1:00pm

Location: 

rh 340N

We prove the Hölder-continuity of the density of states measure (DOSm) and the integrated density of states (IDS) for discrete random Schrödinger operators with finite-range potentials with respect to the probability measure. In particular, our result implies that the DOSm and the IDS for smooth approximations of the Bernoulli distribution converge to the corresponding quantities for the Bernoulli-Anderson model. Other applications of the technique are given to the dependency of the DOSm and IDS on the disorder, and the continuity of the Lyapunov exponent in the weak-disorder regime for dimension one. The talk is based on joint work with Peter Hislop (Univ. of Kentucky) 

Concentration of Eigenfunctions: Sup-norms and Averages

Speaker: 

Jeffrey Galkowski

Institution: 

Stanford University

Time: 

Thursday, May 17, 2018 - 2:00pm

Host: 

Location: 

RH 340P

In this talk we relate concentration of Laplace eigenfunctions in position and momentum to sup-norms and submanifold averages. In particular, we present a unified picture for sup-norms and submanifold averages which characterizes the concentration of those eigenfunctions with maximal growth. We then exploit this characterization to derive geometric conditions under which maximal growth cannot occur. 

Exact bosonization in two spatial dimensions and a new class of lattice gauge theories

Speaker: 

Anton Kapustin

Institution: 

Caltech

Time: 

Sunday, December 10, 2017 - 5:00pm

Location: 

NS 1201

We describe a 2d analog of the Jordan-Wigner transformation which maps an arbitrary fermionic system on a 2d lattice to a lattice gauge theory while preserving the locality of the Hamiltonian. When the space is simply-connected, this bosonization map is an equivalence. We describe several examples of 2d bosonization, including free fermions on square and honeycomb lattices and the Hubbard model. We describe Euclidean actions for the corresponding lattice gauge theories and find that they contains Chern-Simons-like terms.

Spectral gaps for quasi-periodic Schrodinger operators with Liouville frequencies III

Speaker: 

Yunfeng Shi

Institution: 

Fudan University

Time: 

Thursday, December 14, 2017 - 2:00pm

Location: 

Rh 340P

We consider the spectral gaps of quasi-periodic Schrodinger operators with Liouville frequencies. By establishing quantitative reducibility of the associated Schrodinger cocycle,  we show that the size of the spectral gaps decays exponentially. This is a joint work with Wencai Liu. 

Dependence of the density of states on the probability distribution for discrete random Schrödinger operators

Speaker: 

Christoph Marx

Institution: 

Oberlin

Time: 

Thursday, February 15, 2018 - 2:00pm

Location: 

RH 340P

We prove the Hölder-continuity of the density of states measure (DOSm) and the integrated density of states (IDS) for discrete random Schrödinger operators with finite-range potentials with respect to the probability measure. In particular, our result implies that the DOSm and the IDS for smooth approximations of the Bernoulli distribution converge to the corresponding quantities for the Bernoulli-Anderson model. Other applications of the technique are given to the dependency of the DOSm and IDS on the disorder, and the continuity of the Lyapunov exponent in the weak-disorder regime for dimension one. The talk is based on joint work with Peter Hislop (Univ. of Kentucky) 

Localization and spectral statistics for Schrodinger operators with random point interactions

Speaker: 

Peter Hislop

Institution: 

U Kentucky

Time: 

Thursday, April 19, 2018 - 2:00pm

Location: 

RH 340P

We discuss localization and local eigenvalue statistics for Schr\"odinger operators with random point interactions on $R^d$, for $d=1,2,3$. The results rely on probabilistic estimates, such as the Wegner and Minami estimate, for the eigenvalues of the Schr\"odinger operator restricted to cubes. The special structure of the point interactions facilitates the proofs of these eigenvalue correlation estimates.
One of the main results is that the local eigenvalue statistics is given by a Poisson point process in the localization regime, one of the first examples of Poisson eigenvalue statistics for multi-dimensional random Schr\"odinger operators in the continuum.  This is joint work with M.\ Krishna and W.\ Kirsch.

Renormalization and rigidity of circle diffeomorphisms with breaks

Speaker: 

S. Kocic

Institution: 

U Mississippi

Time: 

Thursday, February 1, 2018 - 2:00pm

Abstract: Renormalization provides a powerful tool to approach universality and
rigidity phenomena in dynamical systems. In this talk, I will discuss
recent results on renormalization and rigidity theory of circle
diffeomorphisms (maps) with a break (a single point where the derivative
has a jump discontinuity) and their relation with generalized interval
exchange transformations introduced by Marmi, Moussa and Yoccoz. In a
joint work with K.Khanin, we proved that renormalizations of any two
sufficiently smooth circle maps with a break, with the same irrational
rotation number and the same size of the break, approach each other
exponentially fast. For almost all (but not all) irrational rotation
numbers, this statement implies rigidity of these maps: any two
sufficiently smooth such maps, with the same irrational rotation number
(in a set of full Lebesgue measure) and the same size of the break, are
$C^1$-smoothly conjugate to each other. These results can be viewed as
an extension of Herman's theory on the linearization of circle
diffeomorphisms.
 

Deviations of random matrices and applications

Speaker: 

Roman Vershynin

Institution: 

UCI

Time: 

Thursday, January 25, 2018 - 2:00pm

Host: 

Location: 

RH 340P

Uniform laws of large numbers provide theoretical foundations for statistical learning theory. This talk will focus on quantitative uniform laws of large numbers for random matrices. A range of illustrations will be given in high dimensional geometry and data science.

 

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