Workshop Dates: June 25-26th, 2026, Venue: SNU Building 27, Room 220
This workshop is a workshop associated to the Focus session on Homogeneous Dynamics and Number Theory in the 80-th anniversary international conference of the KMS, which brings together experts to explore the deep interplay between dynamics and arithmetic.
We will look into recent breakthroughs in the equidistribution of flows on locally symmetric spaces and their applications to Diophantine problems. Workshop speakers who are also the speakers of the focus session of KMS will present their second talks of their 25-minute talks at the KMS meeting.
Note also that there will be a plenary talk by Hee Oh on Circles, fractals, and dynamics at the KMS meeting at 11:20-12:20, on the 23rd of June, 2026.
Time table:
25th of June:
(Coffee and snacks will be served from 10:30.)
11:00-12:00 Sanghoon Kwon, Equidistribution in homogeneous dynamics and exact rates over function fields
12:00-13:40 Lunch (Rakgujeong) & Coffee (catered, available at 27-220)
13:40-14:40 Minsung Kim, Rapid mixing for random walks on nilmanifolds II
15:00-16:00 Taehyeong Kim, Large deviations and applications in homogeneous dynamics II
16:20-17:20 Michael Bersudsky, Dynamics on the moduli space of k-lattices of n-space II
17:50-19:50 Workshop banquet
26th of June:
(Coffee and snacks will be served from 09:20.)
09:45-10:45 Minju Lee, On measures invariant under horospherical subgroups in rank one Lie groups II
(11:00-12:00 Muju-Ssen Colloquium by Martin Hairer)
12:00-13:30 Lunch (Duremidam) & Coffee (catered, available at 27-220)
13:30-14:30 Dongryul Kim, Classification of horospherical invariant infinite measures II
14:45-15:45 René Pfitscher, Integrability of Siegel transforms and an application
16:20-17:20 Wooyeon Kim, Representations of binary forms by quaternary quadratic forms II
23rd of June (Time table for the Focus Session of the KMS meeting, venue: ST center Auditorium 4)
9:00-9:25 Wooyeon Kim, Representations of binary forms by quaternary quadratic forms
9:30-9:55 Dongryul Kim, Classification of horospherical invariant infinite measures
10:00-10:25 Michael Bersudsky, Dynamics on the moduli space of k-lattices of n-space
10:30-10:55 Minsung Kim, Rapid mixing for random walks on nilmanifolds
11:20-12:20 Hee Oh (plenary lecture, main auditorium 1)
Lunch (With a KMS lunch coupon, you can dine at the ST Center B1 or B2 restaurants.)
(13:30-14:10 Invited lecture by Gye-Seon Lee (Medium Conference Room 8) Convex cocompact groups with three-dimensional limit sets)
14:30- 14:55 Minju Lee, On measures invariant under horospherical subgroups in rank one Lie groups
15:00-15:25 Jiyoung Han, On measures invariant under horospherical subgroups in rank one Lie groups
15:30-15:55 Taehyeong Kim, Large deviations and applications in homogeneous dynamics
Michael Bersudsky (SNU) Dynamics on the moduli space of k-lattices of n-space
The moduli space of homothety classes of k-lattices, namely rank-k discrete subgroups of R^n, may be viewed as a hybrid of the projective space of k-dimensional subspaces and the space of unimodular lattices. Correspondingly, matrix group actions on this space combine features of the contracting dynamics on the Grassmannian with the expanding dynamics on the space of lattices. This setting exhibits rich dynamics and gives rise to interesting orbit-closures for discrete subgroups of Lie groups. The dynamics of discrete subgroups of Lie groups on this space has attracted considerable attention in recent years, but a general understanding remains far from complete.
In this talk, I will discuss recent results on norm-ball averages along orbits of lattice subgroups. I will review recent joint work with Hao Xing on the SL(n,R) action, a joint work with Nimish Shah on actions of SO(n,1), and I will also discuss ongoing work with Seonhee Lim and Seongmin Kim on related problems for other Lie groups.
Dongryul Kim (SLMath) Classification of horospherical invariant infinite measures
In this talk, I will present the classification of horospherical invariant Radon measures for Anosov subgroups of arbitrary semisimple real algebraic groups. This generalizes the works of Burger and Roblin in rank one to higher ranks. At the same time, this extends the works of Furstenberg, Veech, and Dani, and a special case of Ratner's theorem for finite-volume homogeneous spaces to infinite-volume Anosov homogeneous spaces.
Especially, this resolves the open problems proposed by Landesberg--Lee--Lindenstrauss--Oh and by Oh. Our measure classification is in fact for a more general class of discrete subgroups, including relatively Anosov subgroups with respect to any parabolic subgroups, not necessarily minimal. Our method is rather geometric, not relying on continuous flows or ergodic theorems.
If time permits, I will also discuss the corresponding measure classification result for subgroups of mapping class groups. This is based on joint work with Inhyeok Choi.
Wooyeon Kim (KIAS) Representations of binary forms by quaternary quadratic forms
We discuss a local-global principle for representations of binary by quaternary quadratic forms. Our approach uses a recent measure rigidity result of Einsiedler and Lindenstrauss for higher rank diagonalizable actions, and the determinant method of Bombieri and Pila. This is joint work with Andreas Wieser and Pengyu Yang.
Minsung Kim (Postech) Rapid mixing for random walks on nilmanifolds
In chaotic systems, the mixing property is known for the fast decay of correlation. It is called rapid mixing if the correlation function decays super-polynomially. The mixing mechanism for hyperbolic systems and its compact group extensions were studied by Dolgopyat in a series of his papers in the late 90s'.
In this talk, we prove rapid mixing for almost all random walks generated by m≥2 translations on an arbitrary nilmanifold. For several classical classes of nilmanifolds, we show m = 2 suffices. This provides a partial answer to the question raised in the work of Dolgopyat ('02) about the prevalence of rapid mixing for random walks on homogeneous spaces. (This is joint work with Dmitry Dolgopyat and Spencer Durham.)
Taehyeong Kim (Brendeis Univ., USA) Large deviations and applications in homogeneous dynamics
The Birkhoff Ergodic Theorem can be viewed as a dynamical version of the law of large numbers. A natural question that arises is how rapidly the measure of the exceptional set decays, leading to the study of large deviations. In this talk, we present large deviation estimates for diagonalizable actions on homogeneous spaces and discuss an application to the effective uniqueness of the measure of maximal entropy. This is an ongoing joint work with Elon Lindenstrauss and Ron Mor.
Sanghoon Kwon (Catholic Kwandong Univ.) Equidistribution in homogeneous dynamics and exact rates over function fields
Equidistribution in homogeneous dynamics connects ergodic theory, geometry, and number theory by translating arithmetic distribution problems into orbit-distribution problems on homogeneous spaces. Classical examples include closed horocycles on the modular surface, unipotent orbits, diagonal flows, and their applications to Diophantine approximation and arithmetic counting. This talk begins with a brief survey of these classical equidistribution phenomena, with emphasis on expanding horocycle and horosphere translates and on the role of spectral gaps, mixing, and Eisenstein series in quantitative results. The second part presents recent results on exact equidistribution rates over function fields. In this setting, Bruhat–Tits trees and affine buildings replace the classical continuous geometry, allowing explicit transition formulas and exact discrepancy computations. Special attention will be given to the higher-rank case, where convergence rates depend on Cartan coordinates, chamber orientations, walls, and sectors. The talk aims to explain these formulas as positive-characteristic analogues of quantitative equidistribution for expanding horocycles and horospheres.
Minju Lee (KAIST) On measures invariant under horospherical subgroups in rank one Lie groups
Given a positive Laplace eigenfunction on a hyperbolic manifold, there exists a horospherically invariant measure corresponding to it, known as the Burger-Roblin measure. A question attributed to Babillot concerns whether every horospherically invariant measure arises from such a correspondence. In this talk, I will survey previous works where affirmative answers to this question were found and present a new result extending it to a broad class of subgroups in rank one Lie groups. This is joint work with Or Landesberg, Elon Lindenstrauss, and Hee Oh.
René Pfitscher (USCT, China) Integrability of Siegel transforms and an application
In this talk, we establish sharp algebraic criteria for the higher integrability of a natural generalization of the Siegel transform to the setting of rational representations of semisimple algebraic groups defined over the rationals, extending Siegel’s analytic work in the geometry of numbers. As an application, we derive an effective asymptotic formula for the number of rational approximations of bounded height to almost every real point on a rank-one flag variety at the Diophantine exponent. The argument combines the integrability criterion with effective equidistribution estimates for translated orbits of maximal compact subgroups, a result of independent interest.
Some other programs at the KMS meeting
22nd of June
13:30-14:55 Special Session on Geometric Topology (Medium Conference Room 8)
15:10-15:30 Opening ceremony and group photo session
15:30-16:30 Plenary lecture 1, June E Huh
16:50-18:20 Special Session on Geometric Topology (Medium Conference Room 8)
Seoul National University, Gwanak Campus, Building 27, Room 220,
1 Gwanak-ro, Gwanak-gu, Seoul, 08826, Korea
Naver Map: https://naver.me/x1uSY5w5