Products of two Cantor sets I

Speaker: 

Yuki Takahashi

Institution: 

UC Irvine

Time: 

Tuesday, November 4, 2014 - 1:00pm to 2:00pm

Location: 

RH 440

We consider product of two Cantor sets, and obtain the optimal estimates in terms of their thickness that guarantee that their product is an interval. This problem is motivated by the fact that the spectrum of the Labyrinth model, which is a two dimensional quasicrystal model, is given by the product of two Cantor sets. We also discuss the connection between our problem and the ”intersection of two Cantor sets” problem, which is a problem considered in several papers before.

Diophantine approximation and bounded orbits of mixing flows on homogeneous spaces

Speaker: 

Ryan Broderick

Institution: 

UC Irvine

Time: 

Tuesday, October 28, 2014 - 1:00pm to 2:00pm

We sketch a proof of a theorem due to Kleinbock, and generalizing previous work of Dani and of Margulis and Kleinbock, regarding the size of the set of bounded orbits of a mixing flow on a homogeneous space. We then discuss connections to number theory, specifically the fact, proved in the same paper of Kleinbock, that the set of badly approximable systems of affine forms has full Hausdorff dimension.

Mixing Flows on Homogeneous Spaces

Speaker: 

Ryan Broderick

Institution: 

UC Irvine

Time: 

Tuesday, October 14, 2014 - 1:00pm to 2:00pm

Location: 

RH 440R

We will lay the groundwork needed to discuss some results that use homogeneous dynamics to bound the Hausdorff dimension of sets arising in number theory. Specifically, we will define mixing flows, Lie groups and algebras, homogeneous spaces, and expanding horospherical subgroups, and illustrate these concepts with a few basic examples.

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.

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