February 24, 25 (Mon, Tue), 11 am, KIAS 1423 & zoom https://kimsh.kr/vz
Maximiliano Escayola (USACH)
Critical regularities of nilpotent groups acting on one-manifolds (Parts 1 & 2)
In the context of group actions, there is a general setting of a critical regularity: some actions are possible in class Cα for α<r and impossible for α>r. For instance, the classical Denjoy theorem and Denjoy/Herman examples state that an action of ℤ on the circle with a Cantor minimal set is impossible in C2, but possible for all smaller C1+α’s. Meanwhile, if one replaces ℤ by ℤd, the critical regularity is (1+1/d): there are such action for all C1+(1/d - ε), and it is impossible for C1+ (1/d +ε) (a result of Bertrand Deroin, Victor Kleptsyn, and Andres Navas, obtained in 2009).
February 26th, 2024 (Wed), 9 am KST (Joint seminar with KIAS HCMC Topology Seminar)
Zoom only (link) | Meeting ID: 894 7323 8682 | Passcode: kias
Roberta Shapiro (University of Michigan)
Geometry, topology, and combinatorics of fine curve graph variants
The fine curve graph of a surface S is a graph whose vertices are essential simple closed curves in S and whose edges connect curves that are disjoint. This contrasts past work on the curve graph, which is similar to the above but considers everything up to isotopy. In this talk, we will explore existing and new results on both classical and fine curve graphs and their variants. We will reach into the fields of geometry, topology, and combinatorics to gain new perspectives on these graphs and the surfaces used to construct them. We will further prove a sampling of results about finitary curve graphs, whose vertices are essential simple closed curves in S and edges connect curves that intersect at finitely many points.
January 17th (Fri) 2025, 11 am
KIAS 1423 & Zoom https://kimsh.kr/vz
Inhyeok Choi (KIAS / Cornell)
Free discrete subgroups of Homeo(S) and the fine curve graph.
Recently, Bowden-Hensel-Webb proposed the notion of the fine curve graph for the study of Homeo(S) as an analogue of the curve graph for the mapping class group Mod(S). In the case of Mod(S), the orbit map from a subgroup of Mod(S) to the curve graph may be seriously distorted. In this talk, I will describe an analogous example for Homeo(S) acting on the fine curve graph, namely, a free discrete subgroup of Homeo(S) without global fixed points, whose one free factor is a loxodromic and the other one free factor fixes the basepoint on the fine curve graph. If time allows, I will explain why the (metric) WPD of point-pushing pseudo-Anosov maps help understand this subgroup. This project is in progress.
February 12 (Wed), 11 am, KIAS 1423 & zoom https://kimsh.kr/vz
Clarence Kineider (MPI Leipzig)
Obstructions for Anosov subgroups of SO(p,q)
Anosov representations form a class of representations from Gromov-hyperbolic groups into various Lie groups. This class of representations enjoy many nice properties: they are discrete and faithful, stable under small perturbations... However little is known about which kind of hyperbolic groups can admit Anosov representations in a given Lie group. I will give examples of topological obstructions when the Lie group is SO(p,q), that are in some cases strong enough to force the hyperbolic group to be either free or a surface group.
February 13 (Th), 11 am, KIAS 1503 (note the room!) & zoom https://kimsh.kr/vz
Jineon Baek (Yonsei)
Optimality of Gerver's Sofa
We resolve the \textit{moving sofa problem}, posed by Moser in 1966, which asks for the maximum area of a connected planar shape that can move around the right-angled corner of a \textit{L}-shaped hallway with unit width. We confirm the conjecture made by Gerver in 1994 that his construction, known as Gerver's sofa, with 18 curve sections attains the maximum area 2.2195...
February 19 (Wed), 11 am, KIAS 1423 & zoom https://kimsh.kr/vz
David Xu
Convex-cocompact representations into the isometry group of the infinite-dimensional hyperbolic space
Similarly to Euclidean spaces, there is an infinite-dimensional analog for the (algebraic) hyperbolic spaces. This space enjoys all the "geometric" properties of its finite-dimensional siblings. However, the topological aspects of this space and its group of isometries are more involved than in finite dimension. In particular, even the notion of discrete isometry groups needs to be specified in this context. In this talk, I will present the infinite-dimensional hyperbolic space and describe some of its properties, emphasizing some differences with finite dimension. Then, I will discuss about a generalization of the classic stability result of convex-cocompact representations of finitely generated groups in hyperbolic spaces.
February 21st, 2024 (Fri), 10 am KST (Joint seminar with KIAS HCMC Topology Seminar)
Zoom only (link) | Meeting ID: 894 7323 8682 | Passcode: kias
Inyoung Ryu (Texas A&M University)
Connected components of spaces of type-preserving representations
We investigate the spaces of representations of surface groups into PSL(2, R). For a closed surface, by the classic result of Goldman, the Euler class together with the Milnor-Wood inequality provide a complete classification of the connected components of the spaces of the representations. However, describing the connected components becomes more subtle when considering the space of type-preserving representations for punctured surfaces. In this talk, I will present a recent joint work with Tian Yang that addresses this problem.