Monday, Dec 10 3:30 pm - 4:30 pm
Title: Mixing times and the cutoff phenomenon. --- Abstract: Markov chains are random processes that retain no memory of the past. The mixing time of a Markov chain is the time it takes for it to reach equilibrium. During the last three decades, there has been a lot of progress in developing various techniques to estimate mixing times for various chains and to understand the cutoff phenomenon which means that the Markov chain has an abrupt convergence to equilibrium. We will present recent work establishing cutoff for the random to random card shuffle which confirms a 2001 conjecture of Diaconis. We will also present a proof of uniform lower bounds for Glauber dynamics for the Ising model, extending a result of Ding and Peres. The proofs employ both probabilistic and algebraic techniques.
Special Colloquium, Professor and Head, Anantharam Raghuram, Indian Institute of Science Education and Research, BRNG B268
Tuesday, Dec 11 3:30 pm - 4:30 pm
Title: Automorphic Number Theory. -- Abstract: I will begin my talk by introducing the two major inputs to my research: (1) L-function: an L-function is a function L(s, M) of a complex variable s that is attached to some interesting arithmetic or geometric object, say, M; a basic principle is that the special values of such an L-function at critical points give structural information about M. (2) Cohomology: given a smooth manifold X and a sheaf F on X, one attempts to understand the cohomology groups H*(X, F); producing nontrivial cohomology classes when X is a locally symmetric space may be construed as constructing interesting automorphic forms on the ambient group that determines X. After explaining such inputs, I will discuss a working principle that underlies much of my recent work: given an analytic theory of L-functions, interpret the underlying integrals in terms of maps in cohomology. Finally, I will give glimpses into my recent results, ongoing work, and the work I propose to do in the near future, that concern the arithmetic properties of the special values of automorphic L-functions using analytic techniques from the Langlands program and geometric techniques from the cohomology of locally symmetric spaces.
Tuesday, Apr 16 3:30 pm - 4:30 pm