Abstract:  A classical result of Khinchin says that for almost all real numbers α, the geometric mean of the first n digits ai(α) in the continued fraction expansion of α converges to a number K ≈ 2.6854520 . . . (Khinchin’s constant) as n → ∞. On the other hand, for almost all α, the arithmetic mean of the first n continued fraction digits ai(α) approaches infinity as n → ∞. There is a sequence of refinements of the AM-GM inequality, known as Maclaurin’s inequalities, relating the 1/kthpowers of the kth elementary symmetric means of n numbers for 1 ≤ k ≤ n. On the left end (when k = n) we have the geometric mean, and on the right end (k = 1) we have the arithmetic mean. We analyze what happens to the means of continued fraction digits of a typical real number in the limit as one moves f (n) steps away from either extreme. We also study the limiting behavior of such means for quadratic irrational α.

(Joint work with Francesco Cellarosi, Doug Hensley and Steven J. Miller)

Abstract: The Hall effect is the production of a voltage difference across a conductor, transverse to an electric current, in a presence of a magnetic field in the normal direction. At very low temperatures,  the (quantum) Hall conductance as a function of the strength of the magnetic field exhibited a staircase sequence of wide plateaus. The successive values of the Hall conductance turn out to be integer multiples of e^2/h, with remarkable precision (here e is the elementary charge and h is Planck's constant). This quantization can be understood in terms of topological invariant given by the Kubo-Streda formula. I will discuss the properties of the Kubo-Streda formula and its derivation in the adiabatic setting.  
 

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.