A random ensemble of cusp forms for the full modular group is introduced. For a weight-k cusp form, restricted to a compact subdomain of the modular surface, the true order of magnitude of its expected supremum is determined to be approximately \sqrt{\log(k)}, in line with the conjectured bounds. In addition, the exponential concentration of the supremum around its median is established. Contrary to the compact case, the global expected supremum, attained around the cusp, grows like k^{1/4}. This talk is based on a forthcoming work, joint with B. Huang, S. Lester and N. Yesha.

Speaker: Lassina Dembele, 14:00-14:20

Title: Quaternionic Hermitian lattices and applications.

Abstract: In this short presentation, I will report on some work on quaternionic Hermitian lattices and discuss how these lattices relate to automorphic forms.

Speaker: Igor Wigman, 14:30-14:50

Title: Around Gauss circle problem: Hardy's conjecture and the distribution of lattice points near circles

Abstract: This talk is based on a joint work with Steve Lester.

We review the Gauss circle problem, and Hardy's conjecture regarding the order of magnitude of the remainder term. It is attempted to rigorously formulate the folklore heuristics behind Hardy's conjecture. Some weaker forms of the likely statement are proved to support it.