I will present the construction of the Poisson Boundary of a group, originally defined by Furstenberg, and explain its various properties and applications. The Poisson Boundary can be thought of as the exit boundary of a random walk on the group and can be identified with the space of harmonic functions on the group. The first talk will focus on the construction of the Poisson Boundary and various results due primarily to Furstenberg and Zimmer about boundaries. The second talk will focus on the dynamical behavior of the boundary and its applications to ergodic theory.

Frames provide a mathematical framework for stably representing signals as linear combinations of basic building blocks that constitute an overcomplete collection. Finite frames are frames for finite dimensional spaces, and are especially suited for many applications in signal processing. The inherent redundancy of frames can be exploited to build compression and transmission algorithms that are resilient not only to lost of information but also to noise. For instance, tight frames constitute a particular class of frames that play important roles in many applications. After giving an overview of finite frame theory, I will consider the question of modifying a general frame to generate a tight frame by rescaling its frame vectors. A frame that positively answers this question will be called scalable. I will give various characterizations of the set of scalable frames, and present some topological descriptions of this set. (This talk is based on joint work with G. Kutyniok, F. Philipp and E. Tuley). Tea at 3:30 pm in SC 1425.