Time: Wednesday, 4:00 pm KST (unless otherwise noted). 

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Schedule for the semester (Fall 2026)

Sep 16: Zhiyan Zhao (Université Côte d'Azur) 

Sep 30: Reza Mohammadpour (Uppsala) 

Oct 14: Jiesong Zhang (KTH) 

Nov 4: Jaemin Park (Sookmyung Women's Univ.) 

Nov 25: Yubin Shin (NUS)

9.16: Zhiyan Zhao (Université Côte d'Azur)

Title: Local Rigidity of Isometries on Compact Riemannian Manifolds

Abstract: Rigidity phenomena play a central role in the interaction between geometry and dynamical systems. A fundamental question is whether a geometric action remains essentially unchanged under small perturbations. In this talk, we discuss the local rigidity of group actions by isometries on compact Riemannian manifolds. Using ideas inspired by Kolmogorov–Arnold–Moser (KAM) theory, we develop an iterative scheme showing that, under suitable assumptions, every sufficiently small smooth perturbation of such an action is smoothly conjugate to the original one. The approach combines geometric structures with techniques from dynamical systems and small divisor analysis. This is joint work with Laurent Stolovitch (Université Côte d'Azur).

9.30: Reza Mohammadpour (Uppsala) 

Title: Non-unique Equilibrium Measures for Matrix Cocycles 

Abstract: In this talk, we will discuss aspects of thermodynamic formalism for matrix cocycles, with a particular focus on the existence and uniqueness of equilibrium measures. Given a matrix cocycle $A$, we consider the family of potentials $\varphi_t(x)=t\log\|A(x)\|$, as $t$ ranges over the real numbers, and study how the equilibrium measures associated with these potentials depend on the parameter $t$. We present an example of a matrix cocycle for which the equilibrium measure is unique for every $t>-2$, while for $t\leq -2$ there are multiple equilibrium measures. This provides an example of a phase transition characterized by the loss of uniqueness of equilibrium measures. This is joint work with Anthony Quas.

10.14: Jiesong Zhang (KTH) 

Title: Invariant distributions of partially hyperbolic diffeomorphisms: excessive regularity and rigidity 

Abstract: Invariant distributions of Anosov and partially hyperbolic diffeomorphisms are generally only Hölder continuous. However, if they are sufficiently regular, the underlying systems can be highly rigid. In this talk, I will discuss this phenomenon, from classical results to recent progress, including joint work with Disheng Xu and with Martin Leguil and Disheng Xu.

11.4: Jaemin Park (Sookmyung Women's Univ.) 

Title: Random Cyclic Dynamics and Vietoris–Rips Complexes on the Circle 

Abstract: A finite sample of the circle carries a natural dynamical system: each point moves to the farthest sampled point within a prescribed clockwise distance. These finite cyclic systems, studied by Adamaszek, Adams and Motta, approximate rigid rotations for dense samples, yet their periodic-orbit structure can be quite different. The classification of Adamaszek and Adams, together with its periodic-orbit formulation, connects these dynamics to the homotopy type of the Vietoris–Rips complex of the sample. Building on this work, I will present exact finite-sample laws for periodic-orbit counts and homotopy types under independent uniform sampling, and a Poisson limit for critical orbit counts in shrinking parameter windows. This limit yields explicit limiting probabilities for the homotopy types near a topological transition. After introducing periodic orbits, winding fractions and the topological dictionary, I will explain the main proof ideas: circular spacings, orbit counting and estimates showing that rare candidate orbits survive the remaining sample points.

11.25: Yubin Shin (NUS)

Title: Equidistribution of expanding translates of smooth curves in homogeneous spaces under the action of a product of SO(n,1)'s

Abstract: Let G be a semisimple Lie group, Gamma be a lattice in G, and let A = {a_t} be a one-parameter diagonal subgroup. Given a point x in G/Gamma and a curve phi: I -> U^+ in the expanding horospherical subgroup of A, we consider the family of expanding translates a_t phi(I)x as t -> infinity. The equidistribution problem asks for conditions under which these translates eventually spread uniformly throughout G/Gamma with respect to the Haar probability measure.

Motivated by applications to Diophantine approximation and hyperbolic geometry, Shah initiated the study of this problem and obtained results in several important special cases involving SL_n(R) and a single copy of SO(n,1). Lei Yang later established equidistribution for analytic curves satisfying an algebraic condition when G = SO(n,1)^k. In this talk, I will discuss an extension of Lei Yang's result from analytic curves to smooth curves when G = SO(n_1,1) x ... x SO(n_k,1).