Perturbative and non-perturbative results for quasiperiodic operators with unbounded monotone potentials.

Speaker: 

ILYA KACHKOVSKIY

Institution: 

Michigan State

Time: 

Wednesday, January 2, 2019 - 2:00pm to Tuesday, January 22, 2019 - 3:00pm

Location: 

RH 440R

Abstract: The class of quasiperiodic operators with unbounded monotone potentials is a natural generalization of the Maryland model. In one dimension, we show that Anderson localization holds at all couplings for a large class of Lipschitz monotone sampling functions. The method is partially based on earlier results joint with S. Jitomirskaya on the bounded monotone case. We also establish that the spectrum is the whole real line. In higher dimensions, we tentatively establish perturbative Anderson localization by showing directly that eigenvalue and eigenfunction perturbation series are convergent. Compared to the previously known KAM localization proof by Bellissard, Lima, and Scoppola, our approach gives explicit diagram-like series for eigenvalues and eigenfunctions, and allows a larger class of potentials. The higher-dimensional results are joint with L. Parnovski and R. Shterenberg.

Diffusion in the Mean for a Periodic Schrödinger Equation Perturbed by a Fluctuating Potential. 

Speaker: 

Shiwen Zhang

Institution: 

Michigan State

Time: 

Tuesday, January 22, 2019 - 2:00pm to 3:00pm

Location: 

RH 440R

Abstract: 

We consider the solution to a tight-binding, periodic Schrödinger equation with a random potential evolving stochastically in time. If the potential evolves according to a stationary Markov process, we obtain a positive, finite diffusion constant for the evolution of the solution. More generally, we show that the square amplitude of the wave packet, after diffusive rescaling, converges to a solution of the heat equation. This a joint work with Jeffrey Schenker and  Zak Tilocco. 

Macroscopic theory of cavitation

Speaker: 

V. Shneidman

Institution: 

New Jersey Institute of Technology

Time: 

Tuesday, October 2, 2018 - 2:00pm

Location: 

rh 340p

The classical description of nucleation of cavities in a stretched fluid relies on a one-dimensional Fokker-Planck equation (FPE) in the space of their sizes, with the diffusion coefficient  constructed from macroscopic hydrodynamics and thermodynamics, as shown by Zeldovich. When additional variables (e.g., vapor pressure) are required to describe the state of a bubble, a similar approach to construct a diffusion tensor  generally works only in the direct vicinity of the thermodynamic saddle point corresponding to the critical nucleus. We show, nevertheless, that “proper” kinetic variables to describe a cavity can be selected, allowing to introduce a diffusion tensor in the entire domain of parameters. In this way, for the first time, complete FPE’s are constructed for viscous volatile and inertial fluids. 

Bright solitons in an optical lattice

Speaker: 

M. Olshanii

Institution: 

U Mass Boston

Time: 

Thursday, September 20, 2018 - 2:00pm

Location: 

rh 306

We discuss ultracold atomic gas with attractive interactions in a one-dimensional optical lattice. We find that its excitation spectrum displays a quantum soliton band, corresponding to N-particle bound states, and a continuum band of other, mostly extended, states. For a system of a finite size, the two branches are degenerate in energy for weak interactions, while a gap opens above a threshold value for the interaction strength. We find that the interplay between degenerate extended and bound states has important consequences for both static and dynamical properties of the system.

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.

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