Time: Wednesday 4:00 pm (unless otherwise noted)
Schedule for the semester (Fall 2025)
Sep 17: Dong Han Kim (Dongguk University)
Oct 1: Seung uk Jang (IRMAR, University of Rennes)
Oct 29: Wooyeon Kim (KIAS)
Nov 12: Daniel Glasscock (University of Massachuestts Lowell) - 11 am
Nov 26: Seul Bee Lee (Seoul National University)
9.17: Dong Han Kim
Title: Markoff spectrum of Hecke groups
Abstract: After the seminal work of Markoff on the classical theory of the Markoff spectrum, many studies have been devoted to its generalizations. In this talk, we discuss the Markoff spectrum for Hecke groups, which has been investigated by A. Schmidt, Vulakh, Lehner, Haas, Series, and others. We introduce an expansion of real numbers associated with the Hecke group and describe how spectral points in the discrete part of the Markoff spectrum correspond to these numbers. This is joint work with Byungchul Cha.
10. 1: Seung uk Jang
Title: Dynamics on Markov Surfaces over Non-archimedean Fields
Abstract: Markov surfaces, which appears in the context of character varieties of free groups in SL2, is an example of log K3 surface which carries a large automorphism group. The dynamics of this automorphism group over the field of complex numbers is well-known: either it diverges to infinity or rests on a compact subset.
The same dichotomy applies for non-archimedean fields, especially for the field of p-adic numbers: it diverges to infinity, rests on a finite orbit, or fills up the points over the valuation ring (for the p-adic field, we mean by the ring of p-adic integers). We discuss about how this trichotomy is verified.
In the meantime, as the time permits, we mention briefly how this non-archimedean interest links to some geometric questions, like finding a quasi-isometric and equivariant copy of the hyperbolic plane in "natural geometric objects" over non-archimedean fields.
10. 29: Wooyeon Kim
Title: Local statistics of the Laplace spectrum on 3D rectangular flat tori
Abstract: In this talk, we show that the pair correlation of the Laplacian eigenvalues on 3-dimensional rectangular flat tori follows Poissonian statistics. Similar to the earlier work of Eskin, Margulis, and Mozes on 2D flat tori, these eigenvalues are represented by values of positive definite quadratic forms at integer points. In the 3D case, the problem reduces to a special case of the quantitative Oppenheim conjecture for rapidly shrinking intervals. Our approach reformulates the problem in terms of homogeneous dynamics via theta functions on (SL_2(R)/SL_2(Z))^3, and relies on a sharp quantitative estimate for escape of mass on this space. This is joint work in progress with Jens Marklof and Matthew Welsh.
11. 12: Daniel Glasscock - 11 am
Title: On sets of topological pointwise recurrence
Abstract: Recurrence of points and sets is a central topic in dynamics that has, for the last half century, found numerous applications in combinatorics and additive number theory. “Sets of topological recurrence” – times at which any open set is guaranteed to return to itself under a minimal action of the positive integers on a compact metric space by continuous maps – are well studied. In this talk, we will discuss the class of “sets of pointwise topological recurrence,” integer times at which any point is guaranteed to return to a neighborhood of itself. This class is markedly different from its set analogue: all sets of pointwise topological recurrence are combinatorially large and “good” for multiple set recurrence, for example. We will explain these facts and some of the combinatorial and number-theoretic connections. This talk is based on joint work with Anh Le (University of Denver).
11. 26: Seul Bee Lee
Title:
Abstract: