Bounded orbits of mixing flows on homogeneous spaces

Speaker: 

Ryan Broderick

Institution: 

UC Irvine

Time: 

Tuesday, October 21, 2014 - 1:00pm to 2:00pm

Location: 

RH 440R

Given a Lie group G and a lattice \Gamma in G, we consider a flow on G/\Gamma induced by the action of a one-parameter subgroup of G. If this flow is mixing then a generic orbit is dense, but nevertheless one can discuss the dimension of the set of exceptions. We discuss work of S. G. Dani, in which such estimates are made in certain cases and a connection to diophantine approximation is established, and also generalizations due to D. Kleinbock and G. Margulis. In particular, we outline a dynamical proof, due to Kleinbock, that the set of badly approximable systems of affine forms has full Hausdorff dimension.

Tunneling in graphene: magic angles and their origins. (On a joint work with M. Katsnelson, A. Okunev, I. Schurov, D. Zubov.)

Speaker: 

Victor Kleptsyn

Institution: 

CNRS, Institute of Mathematical Research of Rennes

Time: 

Thursday, May 15, 2014 - 11:00am to 12:00pm

Host: 

Location: 

RH 440R

My talk will be devoted to a joint work with M. Katsnelson, A. Okunev, I. Schurov and D. Zubov.

Graphene is a layer of carbon (forming a hexagonal lattice) of thickness of one or several atoms. One of its remarkable properties is that the behavior of electrons on it is described by the Dirac equation, the same equation that describes the behavior of ultrarelativistic particles. A corollary of this is the Klein tunneling: an electron (or, as it is much more appropriate to say, an wave or quasiparticle) that falls orthogonally on a flat potential barrier on a single-layer graphene, not only has a positive chance of tunneling through it (what is quite natural in quantum mechanics), but passes through it with probability one(!).

Reijnders, Tudorovskiy and Katsnelson, while modeling a transition through an n-p-n junction, have discovered the presence of other, nonzero "magic" angles, under which the falling particle (of given energy) passes through the barrier with probability one.

There are a several interesting problems that arise out of this work. On the one hand, a zero probability of reflection is a codimension two condition (the coefficient before the reflected wave is a complex coefficient that should be equal to zero). Thus, we have a system of two equations on one variable (the incidence angle) that has nonempty set of solutions, what one would not normally expect. And it is interesting to explain their origins.

On the other hand, there is a question that is interesting from the point of view of potential applications: can one invent a potential that "closes well" the transition probabilities (in particular, that has no magic angles)? This question comes from construction of transistors: that is what we should observe for a transistor in the "closed" state.

I will speak about our advances in all these problems. In particular, the tunneling problem on bilayer graphene turns out to be (vaguely) connected to the slow-fast systems on the 2-torus.

Dynamical foliations for system isotopic to Anosov diffeomophisms

Speaker: 

Todd Fisher

Institution: 

Brigham Young University

Time: 

Thursday, June 5, 2014 - 11:00am to 12:00pm

Host: 

Location: 

RH 440R

We discuss the existence of foliations that are invariant under the dynamics for systems that are isotopic to Anosov diffeomorphisms. Specifically, we examine partially hyperbolic diffeomorphisms with one dimensional center that are isotopic to a hyperbolic toral automorphism and contained in a connected component. We show in this case there is a center foliation. We will also discuss more general cases where there is a weak form of hyperbolicity called a dominated splitting. This is joint work with Jerome Buzzi, Rafael Potrie, and Martin Sambarino.

Global Rigidity of Anosov Z^r Actions on Tori and Nilmanifolds

Speaker: 

Zhiren Wang

Institution: 

Yale University

Time: 

Tuesday, April 29, 2014 - 3:00pm to 4:00pm

Host: 

Location: 

RH 440R

As part of a more general conjecture by Katok and Spatzier, it was asked if all smooth Anosov Z^r-actions on tori, nilmanifolds and infranilmanifolds without rank-1 factor actions are, up to smooth conjugacy, actions by automorphisms. In this talk, we will discuss a recent joint work with Federico Rodriguez Hertz that affirmatively answers this question.

On the Statistical Properties of Direct Product Systems

Speaker: 

Marks Ruziboev

Institution: 

ICTP/SISSA

Time: 

Tuesday, May 27, 2014 - 1:00pm

Host: 

Location: 

RH 440R

We consider direct product of finitely many Young towers with the tails decaying at certain rate and show that the product map admits a Young tower whose tail can be estimated in terms of the rates of component towers. It has been shown that many systems admit such a towers and our results therefore imply statistical properties such as decay of correlations, central limit theorem, large deviations, local limit theorem for large class of product systems.

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