Activities This Week

This work studies the relation between two graph parameters, rho and Lambda. For an undirected graph G, rho(G) is the growth rate of its universal covering tree, while Lambda(G) is a weighted geometric average of the vertex degree minus one, corresponding to the rate of entropy growth for the non-backtracking random walk (NBRW).
It is well known that rho(G) >= Lambda(G) for all graphs, and that graphs with rho=Lambda exhibit some special properties. In this work we derive an easy to check, necessary and sufficient condition for the equality to hold. Furthermore, we show that the variance of the number of random bits used by a length k NBRW is O(1) if rho = Lambda and Omega(k) if $rho > \Lambda. As a consequence we exhibit infinitely many non-trivial examples of graphs with rho = Lambda.

Joint work with Idan Eisner, Tel-Hai College

n stable homotopy theory, one attempts to understand the category of spectra. This category is morally akin to the derived category of abelian groups. But it lends itself to a form of localization that is not available in classical algebra - chromatic localization. The study of chromatically localized spectra is central to stable homotopy theory. I will explain a novel approach for such study (pioneered by Barthel-Schlank-Stapleton-Weinstein): the use of the isomorphism of the Lubin-Tate and Drinfeld towers of rigid analytic geometry. I shall explain a structural result, accounting for much of the rational behavior of chromatically localized spectra, obtained in collaboration with Shay Ben-Moshe.