Seminar

The analogy between inequalities in convex geometry and information theory has been recognized by now for several decades. Nonetheless, recent years have seen significant advances in our understanding of how far this analogy extends. One way of better understanding these phenomena is through the study of how the family of Rényi entropies behaves for sums of independent random variables. This question makes sense not just on R^n but also on groups— and there are useful things that can be done both for R^n and for some discrete abelian groups. As a by-product, one also obtains inequalities for the maximum of a convolution that are of interest in small ball probability; in discrete settings, these yield somewhat surprising generalizations of Erdös’s results about extremizers for Littlewood-Offord-type phenomena. We will attempt to use the two hours to give an overview of this area of research, describing not only the speaker’s work with various collaborators but also relevant background at a level accessible to graduate students and the relevant results of other researchers.