Shapes of Eigenvectors for 1-D Random Schrodinger Operators following Rifkind and Virag

Speaker: 

Nishant Rangamani

Institution: 

University of California, Irvine

Time: 

Thursday, October 4, 2018 - 2:30pm to 3:30pm

Location: 

RH 340P

We will discuss the recent work by Rifkind and Virag concerning the shape of eigenvectors for the one-dimensional critical random Schrodinger operator. 

https://arxiv.org/abs/1605.00118

Poncelet’s Theorem, Paraorthogonal Polynomials and the Numerical Range of Truncated GGT matrices

Speaker: 

Barry Simon

Institution: 

Caltech

Time: 

Thursday, November 1, 2018 - 2:00pm

 

 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.  This will allow us to use OPUC methods to provide illuminating proofs of some of their results and in turn to allow the insights from this literature to tell us something about OPUC.  In particular, I’ll discuss a Wendroff  theorem for POPUC.  This is joint work with Andrei Martínez-Finkelshtein  and Brian Simanek.

Three Fairy Math Stories

Speaker: 

D. Burago

Institution: 

Penn State

Time: 

Thursday, October 25, 2018 - 2:00pm

Host: 

 Three different math stories in one lecture. Only definitions, motivations, results, some ideas behind proofs, open questions. 

1. One of the greatest achievements in Dynamics in the XX century is the KAM Theory. It says that a small perturbation of a non-degenerate completely integrable system still has an overwhelming measure of invariant tori with quasi-periodic dynamics. What happens outside KAM tori remains a great mystery. The main quantitate invariants so far are entropies.  It is easy, by modern standards, to show that topological entropy can be positive. It lives, however, on a zero measure set. We are now able to show that metric entropy can become positive too, under arbitrarily small C^{infty} perturbations, answering an old-standing problem of Kolmogorov. Furthermore, a slightly modified construction resolves another long–standing problem of the existence of entropy non-expansive systems. In these modified examples  positive metric entropy is generated in arbitrarily small tubular neighborhoods of one trajectory. Joint with S. Ivanov and Dong Chen.

2. A survival guide for a feeble fish and homogenization of the G-Equation. How fish can get from A to B in turbulent waters which maybe much fasted than the locomotive speed of the fish provided that there is no large-scale drift of the water in the ocean? This is related to the G-Equation and has applications to its homogenization. The G-equation is believed to govern many combustion processes, say wood fires or combustion in combustion engines (generally, in pre-mixed media with “turbulence".  Based on a joint work with S. Ivanov and A. Novikov.

3. Just 20 years ago the topic of my talk at the ICM was a solution of a problem which goes back to Boltzmann and has been formulated mathematically by Ya. Sinai. The conjecture of Boltzmann-Sinai states that the number of collisions in a system of $n$ identical balls colliding elastically in empty space is uniformly bounded for all initial positions and velocities of the balls. The answer is affirmative and the proven upper bound is exponential in $n$. The question is how many collisions can actually occur. On the line, one sees that  there can be $n(n-1)/2$ collisions, and this is the maximum. Since the line embeds in any Euclidean space, the same example works in all dimensions. The only non-trivial (and counter-intuitive) example I am aware of is an observation by Thurston and Sandri who gave an example of 4 collisions between 3 balls in $R^2$. Recently, Sergei Ivanov and me proved that there are examples with exponentially many collisions between  $n$ identical balls in $R^3$, even though the exponents in the lower and upper bounds do not perfectly match. Many open questions left.

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. 

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