Location:
Rice University, Herman Brown Hall 227
Date:
October 31st, 2026
Speakers: Jayadev Athreya, Jon Chaika, Jens Marklof, and Alireza Salehi Golsefidy.
Schedule:
9am: Coffee and light refreshments in 4th Floor Lounge (HBH 438)
9:30am: Jayadev Athreya (Fordham)
Title: Heisenberg Translation Flows
Abstract: In joint work with Francisco Arana-Herrera and Giovanni Forni, we study ergodic theoretical properties of flows on circle bundles over translation surfaces that arise via prequantization, generalizing the theory of Heisenberg nilflows to base surfaces more general than tori; these flows are among the most fundamental examples of parabolic dynamical systems with non-trivial central directions. In particular, we show that such flows are relatively mixing, i.e., they exhibit decay of correlations in the orthogonal complement of functions constant along fibers. We discuss applications of this result to the dynamics of such flows, to the ergodic theory on the corresponding space of wave functions, and, via surface of section constructions, to the study of affine skew products over interval exchange transformations, in the spirit of Furstenberg's classification program for measurable dynamical systems.
10:30am: Break.
11am: Jon Chaika (Rice)
Title:
Abstract:
12pm: Break for lunch.
1:30pm: Jens Marklof (Bristol)
Title: Values of ternary quadratic forms at integers and the Berry-Tabor conjecture for 3-tori
Abstract: Berry and Tabor conjectured in 1977 that spectra of generic integrable quantum systems have the same local statistics as a Poisson point process. We verify their conjecture in the case of the two-point spectral density for a quantum particle in a three-dimensional box, subject to a Diophantine condition on the domain's proportions. This extends previous work of Eskin, Margulis and Mozes (Annals of Math., 2005) in dimension two, where the problem reduces to the quantitative Oppenheim conjecture for quadratic forms of signature (2,2). The difficulty in three and higher dimensions is that we need to consider the distribution of indefinite forms in shrinking rather than fixed intervals, which we are able to resolve for special diagonal forms of signature (3,3) in various scalings, including a rate of convergence. A key step of our approach is to represent the relevant counting problem as an average of a theta function on SL(2,ℤ)^3∖SL(2,ℝ)^3 over an expanding family of one-parameter unipotent orbits. The asymptotic behaviour of these unipotent averages follows from Ratner's measure classification theorem and subtle escape of mass estimates. The talk is based on joint work with Wooyeon Kim and Matthew Welsh.
2:30pm: Break.
3pm: Alireza Salehi Golsefidy (UCSD)
Title: Near-minimal expanding generating sets
Abstract: For a given prime \(p\) and integer \(n_0>2\), it is well known that the minimum number of generators of \(G_N:={\rm SL}_{n_0}(\mathbb{F}_p)^N\) is at least \(C_{n_0,p} \log N\) where \(C_{n_0,p}\) is a positive constant depending only on \(n_0\) and \(p\). In this talk, I will present a recent joint result with Haoyu Li, in which we prove the existence of a generating set \(S_N\) for \(G_N\) satisfying \(|S_N|< C_{n_0,p}'\log N\) such that the family of Cayley graphs \({\rm Cay}(G_N;S_N)\) forms an expander family. Roughly speaking, an expander graph is a highly connected sparse graph. Equivalently, this result implies that a random element of \(G_N\) can be generated by multiplying approximately \(N\) randomly chosen elements of \(S_N\). This answers a question posed by Alex Lubotzky.
Parking:
If you are driving, please park in the North Annex Lot. You will have to use a credit card to enter the lot, but can pick up a voucher during the conference to exit for free. Vouchers can be picked up during the coffee break or lunch from Lisa Geda. The North Annex Lot is located at: https://maps.app.goo.gl/Q4riR3PXnSmUQvdr5
Organizing committee: