Date: March 26 - 28th of 2025
Venue: Dept of Mathematical Sciences, building 27 room 220, Seoul National University, Seoul, Korea. Directions
Title: On the ergodic theory of some complex continued fraction maps
Abstract: We discuss the ergodic theory of complex continued fraction maps associated with Gauss field and Eisenstein field. In particular, we construct natural extensions of continued fraction maps, which are defined on two-dimensional complex space.
Title: Large deviations and applications in homogeneous dynamics
Abstract: The Birkhoff Ergodic Theorem can be seen as a type of the law of large numbers, and it is natural to ask how quickly the measure of the exceptional set decays, which is known as large deviations. In this talk, we discuss large deviation estimates for diagonal actions on homogeneous spaces and, as an application, the effective uniqueness of the measure of maximal entropy. This is a joint work with Elon Lindenstrauss and Ron Mor.
Title: Uniform Diophantine approximation on the Hecke group H4
Abstract: Dirichlet's uniform approximation theorem is a fundamental result in Diophantine approximation that gives an optimal rate of approximation. We study uniform Diophantine approximation properties on the Hecke group $H_4$. For a given real number $\alpha$, we find the sequence of the best approximations of $\alpha$ and show that they are convergents of the Rosen continued fraction and the dual Rosen continued fraction of $\alpha$. We give analogous theorems of Dirichlet uniform approximation and the Legendre theorem with optimal constants. This is joint work with Ayreena Bakhtawar and Seul Bee Lee.
Title: Tubes in complex hyperbolic manifolds
Abstract: We will talk about a tubular neighborhood theorem for an embedded complex geodesic in a complex hyperbolic 2-manifold where the width of the tube depends only on the Euler characteristic of the embedded complex geodesic. We give an explicit estimate for this width. We supply two applications of the tubular neighborhood theorem, the first is a lower volume bound for such manifolds. The second is an upper bound on the first eigenvalue of the Laplacian in terms of the geometry of the manifold.
Title: Norm Forms, Periodic Orbits and Rigidity
Abstract: We will survey on values norm forms on lattices. Norm forms are connected with maximal toral (possibly nonsplit) orbits on the space of lattices. We will discuss rigidity problems in this setting and the connection with number theory.
Title: Classifying Hyperbolic Ergodic Stationary Measures on K3 Surfaces with Large Automorphism Groups
Abstract: Let X be a K3 surface. Consider a finitely supported probability measure $\mu$ on Aut(X) such that $\Gamma_{\mu} = \langle Supp(\mu)\rangle < Aut(X) is non-elementary. We do not assume that $\Gamma_{\mu}$ contains any parabolic elements. We study and classify hyperbolic ergodic $\mu$-stationary probability measures on X.
Title: Cantor attractor of Fibonacci-like unimodal maps
Abstract: The Fibonacci-like unimodal maps that have been studied in recent years give rise to zero-entropy minimal Cantor systems. In this talk we present an explicit construction of these Cantor systems in the real line, how to obtain Bratelli-Vershik systems from them, and some applications of these constructions. This is joint work with Semin Yoo.
Title: On the problems counting primitive lattice points under congruence conditions
Abstract: Siegel transforms are tools that translate problems related to counting lattice points in number theory into the framework of homogeneous dynamics. In this talk, we will examine a problem involving the counting of primitive lattice points under a congruence condition in a specific domain, which is an application of moment formulas for Siegel transforms. In dimensions of at least 3, this problem arises from corollaries of an S-arithmetic primitive counting theorem, which generalizes Schmidt's counting theorem (1960). For the case of dimension 2, we introduce a new Siegel transform that simultaneously considers both the primitive and congruence conditions, and we establish the first and second moment formulas for this new Siegel transform. This is joint work with Samantha Fairchild, and with Seul Bee Lee.