Georgia Institute of Technology College of Sciences

In this talk, we investigate the structures of C*-algebras generated by collections of linear-fractionally-induced composition operators and either the forward shift or the ideal of compact operators. In the setting of the classical Hardy space, we present a full characterization of the structures, modulo the ideal of compact operators, of C*-algebras generated by a single linear-fractionally-induced composition operator and the forward shift. We apply the structure results to compute spectral information for algebraic combinations of composition operators. We also discuss related results for C*-algebras of operators on the weighted Bergman spaces.

In the talk we will present a mechanism of diffusion in the Planar Circular Restricted Three Body Problem. The mechanism is similar to the one that appeared in the celebrated work of V. I. Arnold [Dokl. Akad. Nauk SSSR 156 (1964), 9–12]. Arnold conjectured that this phenomenon, usually called Arnold diffusion, appears in the three body problem. The presented method is a step towards a proof of the conjecture.

We present a framework for proving lower bounds on computational problems over distributions, including optimization and unsupervised learning. The framework is based on defining a restricted class of algorithms, called statistical algorithms, that instead of directly accessing samples from the input distribution can only obtain an estimate of the expectation of any given function on the input distribution. Our definition captures many natural algorithms used in theory and practice, e.g., moment-based methods, local search, linear programming, MCMC and simulated annealing. Our techniques are inspired by the statistical query model of Kearns from learning theory, which addresses the complexity of PAC-learning. For specific well-known problems over distributions, we obtain lower bounds on the complexity of any statistical algorithm. These include an exponential lower bounds for moment maximization and a nearly optimal lower bound for detecting planted clique distributions when the planted clique has size n^{1/2-\delta} for any constant \delta > 0. Variants of the latter problem have been assumed to be hard to prove hardness for other problems and for cryptographic applications. Our lower bounds provide concrete evidence of hardness. This is joint work with V. Feldman, E. Grigorescu, L. Reyzin and Y. Xiao.

Invariant tori play a prominent role in the dynamics of symplectic maps. These tori are especially important in two dimensional systems where they form a boundary to transport. Volume preserving maps also admit families of invariant rotational tori, which will restrict transport in a d dimensional map with one action and d-1 angles. These maps most commonly arise in the study of incompressible fluid flows, however can also be used to model magnetic field-line flows, granular mixing, and the perturbed motion of comets in near-parabolic orbits. Although a wealth of theory has been developed describing tori in symplectic maps, little of this theory extends to the volume preserving case. In this talk we will explore the invariant tori of a 3 dimensional quadratic, volume preserving map with one action and two angles. A method will be presented for determining when an invariant torus with a given frequency is destroyed under perturbation, based on the stability of approximating periodic orbits.