Dimension estimates for sets of uniformly badly approximable systems of linear forms

Speaker: 

Ryan Broderick

Institution: 

Northwestern University

Time: 

Tuesday, May 13, 2014 - 1:00pm to 2:00pm

Host: 

Location: 

RH 440R

A number is called badly approximable if there is a positive constant c such that |x-p/q| > c/q^2 holds for all rationals p/q, so that close approximation by rationals requires relatively large denominators. The set of such numbers is Lebesgue-null but has full Hausdorff dimension. This set can be viewed as the union over c of the set BA(c) of numbers which satisfy the above inequality for the fixed constant c. J. Kurzweil obtained dimension bounds on BA(c), which were later improved by D. Hensley. We will discuss recent work, joint with D. Kleinbock, in which we use homogeneous dynamics to produce dimension bounds for a higher-dimensional analog.

Products of Cantor sets and Spectral Properties of Labyrinth Model

Speaker: 

Yuki Takahashi

Institution: 

UC Irvine

Time: 

Tuesday, April 1, 2014 - 1:00pm to 2:00pm

Location: 

RH 440R

We prove that the product of two Cantor sets of large thickness is an interval in the case when one of them contains the origin. We apply this result to the Labyrinth model of a two-dimensional quasicrystal, where the spectrum is known to be the product of two Cantor sets, and show that the spectrum becomes an interval for small values of the coupling constant. We also consider the density of states measure of the Labyrinth model, and show that it is absolutely continuous with respect the Lebesgue measure for most values of coupling constants.

Information propagation in 1D quantum spin chains via linear ODEs with Hermitian field

Speaker: 

William Yessen

Institution: 

Rice University

Time: 

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

Host: 

Location: 

RH 440R

Since the early 1970's, it has been known in both, the mathematical physics and in the physics communities, that propagation of information in quantum spin chains cannot exceed the so-called Lieb-Robinson bound (effectively providing the quantum analog of the light cone from the relativity theory). Typically these bounds depend on the parameters of the model (interaction strength, external field). The recent Hamza-Sims-Stolz result demonstrates exponential localization (a la Anderson localization) of information propagation in most spin chains (in the sense of a given probability distribution with respect to which interaction and external field couplings are drawn). A natural question arises: what can be said about lower bounds on propagation of information in spin crystals (i.e. the case far from the one in which localization is expected), as well as in the intermediate case--the spin quasicrystals. This problem can be reduced to solving a linear ODE given by a Hermitian matrix, the solutions of which live on finite-dimensional complex spheres.

In this talk we shall discuss the history, give a general overview of the field, reduce the problem to an ODE problem as mentioned above, and look at some open problems. We shall also present some numerical computations with animations.

 

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