Titles and abstracts

김민성

Title: The existence of invariant distributions for C^5 circle diffeomorphisms

Abstract: Irrational circle diffeomorphisms are uniquely ergodic, but whether they carry invariant distributions beyond this measure depends both on the regularity of the map and on the order of the distribution (the regularity of the observables it acts on). Avila and Kocsard proved the non-existence of such distributions for $C^k$ maps ($k \ge 5$) at the level of $k$-invariant distributions, while Navas and Triestino established a similar result for $C^{1+\text{BV}}$ maps at the level of 1-invariant distributions. This led Navas and Triestino to ask whether a $C^2$ map can carry a 2-invariant distribution (dual to $C^2$ observables) distinct from its unique invariant measure.

In this talk, we construct a non-trivial 2-invariant distribution for $C^5$ circle diffeomorphisms with Fibonacci-type rotation numbers. This gives a partial answer to the question raised by Navas and Triestino.

This is ongoing joint work with Marco Martens and Liviana Palmisano.

김우연

Title: Distribution of determinants at lattice points of matrices 

Abstract: In this talk, we study the distribution of determinant values taken by lattice points in the space of d by d real matrices. Unless the lattice has an additional multiplicative structure like the lattice of integer matrices, it turns out that the determinant values are dense in the real line, as a consequence of Ratner's theorem. A natural question is how these determinant values are distributed in the real line. We give a complete answer to this question for d=3. For d=2 our approach gives an alternative proof for the quantitative version of the Oppenheim conjecture for quadratic forms of signature (2,2), obtained by Eskin-Margulis-Mozes (2005). This is joint work with Hee Oh.

이민주

Title: Discrete subgroups with finite Bowen-Margulis-Sullivan measure in higher rank 

Abstract: Let G be a connected semisimple real algebraic group and D be its Zariski dense discrete subgroup. We prove that if D\G admits any finite Bowen-Margulis-Sullivan measure, then D is virtually a product of higher rank lattices and discrete subgroups of rank one factors of G. This may be viewed as a measure-theoretic analogue of classification of convex cocompact actions by Kleiner-Leeb and Quint, which was conjectured by Corlette in 1994. This is joint work with Mikolaj Fraczyk. We will then discuss its application on the bottom of the L^2 spectrum, in joint work with Samuel Edwards, Mikolaj Fraczyk and Hee Oh.

이정훈

Title: Hyperbolicity, invertibility, and stability
Abstract: In 1983, Mañé, Sad, and Sullivan established a celebrated result regarding the structural stability of the chaotic locus (the Julia set) in complex dynamics. A direct consequence of their theorem is that the set of structurally stable rational functions is open and dense. A quintessential class of maps satisfying this structural stability are hyperbolic, which are characterized by geometrically expanding dynamics on their chaotic locus.

Exploring the non-Archimedean analogue of the Mañé–Sad–Sullivan theorem, Lee (2019) proved that dynamics on the chaotic locus are structurally stable if they are hyperbolic. This result was recently extended to much weaker conditions by Benedetto and Lee (2022). In this talk, we will introduce a different approach to understanding the relationship between hyperbolicity and stability, based on joint work with Prof. Tomoki Kawahira.

한지영

Title: Behavior of primitive transform in large dimensions 

Abstract: Consider the number of lattice points contained in a ball of large volume in a vector space of a given dimension. If we fix the dimension and let the volume diverge to infinity, the proportion of lattices for which the number of lattice points in the ball is close to the volume of the ball increases. What happens if we instead fix the volume and let the dimension diverge? Rogers showed in 1956 that, for a fixed volume, the distribution of the number of lattice points in the ball converges to a Poisson distribution. Later, Strömbergsson and Södergren (2011) asked a similar question, but with the volume growing subexponentially, and proved that the sequence of distributions of the number of lattice points converges in normal distribution. 

On the other hand, counting primitive lattice points is one of the fundamental problems in the geometry of numbers. As the dimension grows, the proportion of primitive lattice points in a given lattice converges to 1. Thus one expects the distribution of the number of primitive lattice points behave similarly to that of lattice points. In this talk, we examine that using method of moments, together with moment formulas for primitive Siegel transform, one can show that our expectation indeed holds.

홍순기

Title: Quantitative Isolation of Closed Orbits in the Frame Bundle of a P-Adic Schottky Manifold

Abstract: Let \(p>2\), let \(K/\mathbb{Q}_p\) be an unramified quadratic extension, and set

\[

G=\operatorname{PGL}_2(K),

\qquad

H=\operatorname{PGL}_2(\mathbb{Q}_p).

\]

Following the setting of Jeoung--Lim, we consider a Schottky subgroup \(\Gamma<G\) and the homogeneous space \(\Gamma\backslash G\). Closed \(H\)-orbits in this space correspond to properly immersed \(H\)-subtrees arising from translates of the Bruhat--Tits tree of \(H\) inside the Bruhat--Tits tree of \(G\).

The goal of this talk is to formulate and investigate a \(p\)-adic analogue of the quantitative isolation theorem of Mohammadi--Oh for closed \(\operatorname{PSL}_2(\mathbb{R})\)-orbits in the frame bundle of a geometrically finite hyperbolic \(3\)-manifold. More precisely, we seek quantitative bounds on how closely two distinct closed \(H\)-orbits can approach each other. We discuss possible non-Archimedean counterparts of the tight area, shadow constant, and Margulis function introduced by Mohammadi--Oh, together with averaging operators arising from the horospherical action on Bruhat--Tits trees. We also describe the main difficulties caused by the ultrametric geometry, discrete distance structure, and recurrence properties of the infinite-volume Schottky quotient. This provides a framework for extending the quantitative isolation method of Mohammadi--Oh to the \(p\)-adic setting.