This seminar runs every Monday during the semester at 2PM in Hill 705. The current organizers are Jeff Kahn, Bhargav Narayanan, Swee Hong Chan, and Natalya Ter-Saakov.
If you wish to join our listserv, please email Natalya Ter-Saakov at nt399@math.rutgers.edu.
Click here for the abstract archive for this semester, and here for older abstracts.
Next Talk:
Date: October 13, 2025 at 2:00pm
Speaker: Doron Puder (Tel Aviv University)
Title: Aldous-type spectral gaps in Unitary groups
Abstract: Around 1992, Aldous made the following bold conjecture. Let A be any set of transpositions in the symmetric group Sym(N). Then the spectral gap of the Cayley graph Cay(Sym(N),A) is identical to that of a relatively tiny N-vertex graph defined by A. So even though the spectrum of the Cayley graph contains N! eigenvalues, the largest non-trivial one always comes from a tiny pool of N of them. This conjecture was proven nearly 20 years later by Caputo, Liggett and Richthammer (JAMS, 2010).
Driven by the conviction that such a stunning phenomenon cannot possibly be isolated, Gil Alon and I found a probable parallel of this phenomenon in the unitary group U(N). We have a concrete conjecture supported by simulations, and we prove it in several non-trivial special cases. As it turns out, the corresponding spectrum in the case of U(N) contains the one in Sym(N). Moreover, the critical part of the spectrum in U(N) coincides with the spectrum of an interesting discrete process.
In the talk, I will try to convey these ideas and some of the proofs.
October 13 --- Doron Puder (Tel Aviv University)
October 20 --- Boris Bukh (Carnegie Mellon University)
October 27 --- Lior Gishboliner (University of Toronto)
November 3 --- Jonathan Tidor (Princeton University)