**
PS-SNU Geometry/Topology Online Seminar**

**Time:** 09:00 - 10:30 (France Time) / 16:00 - 17:30 (Korea Time)

**Date:** 2nd or 3rd Tuesday of each month, starting from February 2024

**Organizers:** Bruno Duchesne (Paris-Saclay University) and Gye-Seon Lee (Seoul National University)

The seminar consists of two 40-minute talks, with a 10-minute break in between.

To join the seminar, please email the organizers to be added to the mailing list.

The seminar is hosted via Zoom, and access information will be sent to the mailing list.

**Schedule**

__14 May 2024__

**Jiyoung Han** (Korea Institute for Advanced Study)
**Title**: Dynamics on Homogeneous Spaces and Their Applications to Counting Problems
**Abstract**: The Oppenheim conjecture, now known as Margulis' theorem, asserts the density of integral vectors under a non-degenerate indefinite quadratic form Q in ℝ. Margulis achieved this using homogeneous dynamics, focusing on the dynamical properties of a unipotent flow {u_{t}}_{t ∈ ℝ} at a point g_{Q}Γ relative to Q in the homogeneous space G/Γ, where G = SL_{d}(ℝ) and Γ = SL_{d}(ℤ). Ratner's trilogy simplified and generalized Margulis' theorem, which is a contribution of Dani and Margulis. This talk provides an overview of Ratner's theorems, the idea of proofs behind Margulis' and Dani's theorems, and recent advancements in quantifying these results.

**TBA** (TBA)
**Title**: TBA
**Abstract**: TBA

__18 June 2024__

**Seul Bee Lee** (IBS Center for Geometry and Physics)
**Title**: Diophantine approximation by rational numbers of certain parity types
**Abstract**: In the study of Diophantine approximation, a natural question is which rationals p/q minimize |qx-p| with a bounded condition over q. We call such rationals the best approximations. The regular continued fraction provides an algorithm for generating the best approximations. From a broader perspective, we are interested in the best approximations with congruence conditions on their numerators and denominators. In this talk, we will characterize the set of best approximations with parity conditions. Subsequently, we will explore a symbolic sequence of real numbers associated with a specific triangle group in the upper half-plane to give an algorithm detecting best approximations with parity conditions.

**TBA** (TBA)
**Title**: TBA
**Abstract**: TBA

__13 February 2024__

**David Xu** (Paris-Saclay University)
**Title**: Deformations of convex-cocompact representations into the isometry group of the infinite-dimensional hyperbolic space
**Abstract**: Convex-cocompact representations of hyperbolic groups into the groups of isometries of hyperbolic spaces are a generalisation of quasi-Fuchsian representations in PSL(2,C). A classical result due to Marden and Thurston states that convex-cocompact representations into PO(n,1) form an open subset of the space of representations (this is called stability of convex-cocmpact representations). In this talk, we will discuss the case of representations into the group of isometries of the infinite-dimensional hyperbolic space, where this stability property remains true. This allows the use of bending to produce convex-cocompact representations by deformation.

**Balthazar Fléchelles** (Institut des Hautes Études Scientifiques)
**Title**: Cubulated hyperbolic groups admit Anosov representations
**Abstract**: Anosov representations of hyperbolic groups are a generalization of convex-cocompact representations that satisfy very interesting dynamical and geometric properties. Unfortunately, it is not always easy to build Anosov representations for a given hyperbolic group. In this work in collaboration with Sami Douba, Theodore Weisman and Feng Zhu, we prove using Vinberg theoretic tools that any hyperbolic group acting properly discontinuously and cocompactly on a CAT(0) cube complex (such a group is said to be cubulated and hyperbolic) admits explicit Anosov representations. Among these groups, some were not previously known to admit Anosov representations. In fact, random groups (of density < 1/6) are cubulated and hyperbolic, so our result applies to a large variety of groups.

__12 March 2024__

**Hongtaek Jung** (Seoul National University)
**Title**: Volumes of Hitchin-Riemann moduli spaces are infinite
**Abstract**: Let S be a closed orientable surface of genus at least 2 and let Mod(S) be the mapping class group. Because Mod(S) acts properly on the Hitchin component Hit_{d}(S), we can form the Hitchiin-Riemann moduli space M_{d}(S)= Hit_{d}(S)/Mod(S), generalizing the classical Riemann moduli space. We show, on the contrary to the Riemann moduli space, that the total Atiyah-Bott-Goldman volume of M_{d}(S) is infinite provided d > 1.

**Yusen Long** (Paris-Saclay University)
**Title**: Fine curve graph and its large scale geometry
**Abstract**: In last two decades, curve graph has been an important tool for studying combinatorics properties of mapping class groups of finite-type surfaces. Recently, to study diffeomorphism group of surfaces, Bowden, Hensel and Webb proposed an analogue called fine curve graph. In this talk, we will present some geometric properties of this graph at large scale and its topological properties at infinity.

__9 April 2024__

**KyeongRo Kim** (Seoul National University)
**Title**: laminar groups and Kleinian groups
**Abstract**: Thurston showed the universal circle theorem as a first step of the geometrization conjecture of tautly foliated three manifolds. The theorem says that the fundamental group of a closed three manifold, slithering over the circle, acts on the circle preserving a pair of laminations. In this talk, I talk about the converse of the universal circle theorem in terms of laminar groups. Also, I will overview recent results about the laminar groups and discuss related open problems. This is based on works joint with Harry Hyungryul Baik and Hongtaek Jung.

**Blandine Galiay** (Institut des Hautes Études Scientifiques)
**Title**: Divisible convex sets in flag manifolds and rigidity
**Abstract**: Divisible convex sets have been widely studied since the 1960s. They are proper domains of the projective space that admit a cocompact action of a discrete subgroup of the linear projective group. The best-known examples are symmetric spaces embedded in the projective space, but there are also many nonsymmetric examples. It is natural to seek to generalize this theory, by replacing the projective space by a flag variety G/P, where G is a real semisimple non-compact Lie group and P a parabolic of G. A question of W. van Limbeek and A. Zimmer is then: are there examples of divisible convex sets in G/P that are nonsymmetric? In a number of cases, it has been proved that there are not. In this talk, we will focus on some particular classes of flag varieties in which rigidity can indeed be observed.