Title: The (in)compatibility of 3 and 5 dimensional Heisenberg geometry with Lebesgue spaces
Abstract: The 3-dimensional (discrete) Heisenberg geometry is the shortest-path metric on the infinite graph whose vertex set is the integer grid $\Z^3$ and the neighbors of each integer vector $(a,b,c)$ are the four integer vectors $$(a+ 1,b,c), (a- 1,b,c), (a,b+ 1,c+ a), (a,b- 1,c- a).$$
Abstract: A Markov chain over a finite state space is said to exhibit the total variation cutoff phenomenon if, starting from some Dirac measure, the total variation distance to the stationary distribution drops abruptly from near maximal to near zero. It is conjectured that simple random walks on the family of $k$-regular, transitive graphs with a two sided $\epsilon$ spectral gap exhibit total variation cutoff (for any fixed $k$ and $\epsilon). This is known to be true only in a small number of cases.
Manchester Building (Hall 2), Hebrew University Jerusalem
Abstract: We will prove a sharp average-case variant of a classical embedding theorem of John through the theory of nonlinear spectral gaps. We will use this theorem to provide a new answer to questions of Johnson and Lindenstrauss (1983) and Bourgain (1985) on metric dimension reduction, and explain how it leads to algorithms for approximate nearest neighbor search.
Classical group representation theory deals with group actions on linear spaces; we consider group actions on compact convex spaces, preserving topological and convex structure. We focus on irreducible actions, and show that for a large class of groups - including connected Lie groups - these can be determined. There is a close connection between this and the theory of bounded harmonic functions on symmetric spaces and their boundary values.
