Georgia Institute of Technology College of Sciences

The mapping class group of the plane minus a Cantor set naturally appears in many dynamical contexts, including group actions on surfaces, the study of groups of homeomorphisms on a Cantor set, and complex dynamics. In this talk, I will present the 'ray graph', which is a Gromov-hyperbolic graph on which this big mapping class group acts by isometries (it is an equivalent of the curve graph for this surface of infinite topological type). If time allows, I will give a description of the Gromov-boundary of the ray graph in terms of long rays in the plane minus a Cantor set. This involves joint work with Alden Walker.

Abstract: Shift-invariant (SI) spaces play a prominent role in the study of wavelets, Gabor systems, and other group frames. Working in the setting of LCA groups, we use a variant of the Zak transform to classify SI spaces, and to simultaneously describe families of vectors whose shifts form frames for the SI spaces they generate.

In this talk, i shall provide some optimal PIR bounds, which confirmed a conjecture on optimal RIP bound. Furtheremore, i shall also investigate some results on signals recovery with redundant dictionaries, which are also related to statistics and sparse representation.