Shucheng Yu (University of Science and Technology of China), 11 am -12 pm, Jan 09, 2026, Rm 129-301
Title: Extreme events for some unipotent actions on the space of lattices
Abstract: In this talk we study the extreme value law for cusp excursions of certain unipotent actions on the space of lattices. We analyze this problem in terms of the hitting time and impact statistics for the unipotent action with respect to a shrinking surface of section, following the strategy of Pollicott and Marklof.
We will also discuss the asymptotics of the corresponding limiting distribution, which in a special case agrees with the directional statistics of Euclidean lattices. This is joint work with Jens Marklof and Andreas Strömbergsson.
Michael Bersudsky (SNU), Lecture series, Nov-Dec, 2025
Title: Uniform distribution in homogeneous spaces, o-minimal structures and homogeneous dynamics
Yubin Shin (OSU), Nov 13, 2025
Title: Equidistribution of expanding translates of smooth curves in homogeneous spaces under the action of a product of SO(n,1)'s
Abstract: We study the limiting distributions of expanding translates of a compact segment of a smooth curve under a diagonal subgroup of G = SO(n₁,1) × ⋯ × SO(nₖ,1), where G acts on a finite volume homogeneous space L/Γ as a subgroup. We show that the expanding translates of the curve become equidistributed in the orbit closure of G, provided that Lebesgue almost every point on the curve avoids a certain countable collection of algebraic obstructions. In this talk, we will discuss the characterization of this collection of obstructions, compare this to previous results and discuss difficulties that arise due to the smooth curve condition.
Aaron Brown, Oct 20, 2025
Title: Regularity and Lyapunov rigidity of stationary measures
Abstract: For non-linear random walks on the 2-torus, we study the existence and regularity of stationary measures. For random walks close to certain (Zariski dense) affine random walks, we show the only non-atomic (ergodic) stationary measure is absolutely continuous. One can then study the top Lyapunov exponent relative to this stationary measure. Entropy considerations given an inequality between the top Lyapunov exponent for the non-linear and affine random walks; equality holds only when the non-linear random walk is smoothly conjugate to an affine random walk.
Taehyeong Kim Sep 17, 2025
Title: On the rate of convergence of continued fraction statistics of random rationals
Abstract: The statistical behavior of continued fraction expansions for typical real numbers is a classic subject, such as the Lévy-Khintchine Theorem. A natural question arises when we restrict our attention to the set of rational numbers. In 2018, David and Shapira showed that the continued fraction statistics of random rationals with a fixed denominator converge to the Gauss–Kuzmin distribution as the denominator grows. In this talk, we will present our recent result establishing a polynomial rate of convergence for these statistics. Our approach relies on an equidistribution result for divergent orbits of the geodesic flow on SL(2,R)/SL(2,Z), combined with entropy bounds for invariant measures that spend significant time in the cusp. This is joint work with Ofir David, Ron Mor, and Uri Shapira.
Wooyeon Kim, Aug 12, 2025
Title: MOMENTS OF MARGULIS FUNCTIONS AND INDEFINITE TERNARY QUADRATIC FORMS
Abstract: In this paper, we prove a quantitative version of the Oppenheim conjecture for indefinite ternary quadratic forms: for any indefinite irrational ternary quadratic form Q that is not extremely well approxiable by rational forms, and for a ă b the number of integral vectors of norm at most T satisfying a ă Qpvq ă b is asymptotically equivalent to `CQpb ́ aq ` IQpa, bq ̆Tas T tends to infinity, where the constant CQ ą 0 depends only on Q, and the term IQpa, bqTaccounts for the contribution from rational isotropic lines and degenerate planes. The main technical ingredient is a uniform bound for the λ-moment of the Margulis α-function along expanding translates of a unipotent orbit in SL3pRq{ SL3pZq, for some λ ą 1. To establish this, we introduce a new height function rα on the space of lattices, which captures the failure of the classical Margulis inequality. This moment bound implies equidistribution of such translates with respect to a class of unbounded test functions, including the Siegel transform.
Jiyoung Han, Aug 4, 2025
Title: Toward a random quantification of the Oppenheim conjecture for symplectic forms
Abstract: Since the celebrated theorem of Margulis, solving Oppneheim conjecture-typed problems has been a central theme of homogeneous dynamics. In this talk, we introduce the Oppenheim conjecture and its random quantitative formulation. If time permits, we will briefly sketch the proofs, especially how Ratner’s orbit closure theorem enters in the qualitative result, and how Rogers’ higher moment formulas influence the dimension-rank condition in the random quantitative version.
8. Amir Mohammadi, Jul 22, 2025
Title: Effective equidistribution in rank 2 homogeneous spaces and values of quadratic forms
Abstract: We establish effective equidistribution theorems, with a polynomial error rate, for orbits of unipotent subgroups in quotients of quasi-split, almost simple Linear algebraic groups of absolute rank 2. As an application, inspired by the results of Eskin, Margulis and Mozes, we establish quantitative results regarding the distribution of values of an indefinite ternary quadratic form at integer points, giving in particular an effective and quantitative proof of the Oppenheim Conjecture.
9. Miri Son, July 09, 2025
Title: Classification of SL(n,R)-actions on closed manifolds
Abstract: Recently, Fisher and Melnick classified SL(n,R)-actions on n-dimensional manifolds for n≥3. We generalize this result by classifying actions on m-dimensional manifolds for 3≤n≤m≤2n-3. This work is motivated by the Zimmer program and is central to it, as Lie group actions restrict to their lattice actions.
10. Insung Park, June 20, 2025
Title: Zelditch's trace formula and effective equidistribution of closed geodesics in hyperbolic surfaces
Abstract: In the early 1990s, Zelditch adapted the Selberg trace formula to prove an effective version of Bowen's equidistribution theorem for closed geodesics on hyperbolic surfaces. Building on his approach, in joint work with Junehyuk Jung and Peter Zenz, we refined Zelditch's idea to achieve the optimal error term in the equidistribution of closed geodesics on compact hyperbolic surfaces. In this talk, we begin by reviewing the basic ideas of trace formulas, followed by a discussion of the new contributions. No prior knowledge of trace formulas is required.
11. Heejong Lee, May 30, 2025
Title: p-adic Galois representations and modularity
Abstract: Galois representations naturally arise from arithmetic geometry and automorphic representations. For the first half of the talk, I will give a brief overview of p-adic Galois representations and its relationship with automorphic representations, including one example that any high-schooler can understand. For the second half of the talk, I will introduce a result about (families of) p-adic local Galois representations and how to prove it using algebro-geometric techniques, which also have a slight dynamical flavor. This is based on a joint work with Daniel Le, Bao Le Hung, Brandon Levin, and Stefano Morra.
12. Junghoon Lee, May 27, 2025
Title: A crash course on Berkovich spaces
13. Yuchan Lee, Mar 27, 2025
Title: An asymptotic formula for the number of integral matrices with a fixed characteristic polynomial via orbital integrals
Abstract: For an arbitrarily given irreducible polynomial χ(x) in Z{x} of degree n, let N (X, T ) be the number of n × n matrices over Z whose characteristic polynomial is χ(x), bounded by a positive number T with respect to a certain norm. We provide an asymptotic formula for N (X, T ) as T → ∞ in terms of the orbital integrals of gln. This generalizes the work of A. Eskin, S. Mozes, and N. Shah (1996) which assumed that Z{x}/(χ(x)) is the ring of integers. In addition, we will provide an asymptotic formula for N (X, T ), using the orbital integrals of gln, when Q is generalized to a totally real number field k and when n is a prime number. Here we need a mild restriction on splitness of χ(x) over kv at p-adic places v of k for p ≤ n when k[x]/(χ(x)) is unramified Galois over k. This is a joint work with Seoungsu Jeon.
14. Dongryul Kim, Feb 19, 2025
Title: Horocycles in hyperbolic 3-manifolds with Sierpiński limit sets
Abstract: Let M be a geometrically finite hyperbolic 3-manifold whose limit set is a round Sierpiński carpet, i.e. M is geometrically finite and acylindrical with a compact, totally geodesic convex core boundary. In this paper, we classify orbit closures of the 1-dimensional horocycle flow on the frame bundle of M. As a result, the closure of a horocycle in M is a properly immersed submanifold. This extends the work of McMullen-Mohammadi-Oh where M is further assumed to be convex cocompact.
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