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.

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 C^\alpha for \alpha<r and impossible for \alpha>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 C^2, but possible for all smaller C^{1+\alpha}’s. Meanwhile, if one replaces Z by Z^d, the critical regularity is (1+1/d): there are such action for all C^{1+(1/d - \eps)}, and it is impossible for C^{1+ (1/d +\eps)}   (a result of Bertrand Deroin, Victor Kleptsyn, and Andres Navas, obtained in 2009).

The joint work with Victor Kleptsyn, that I will speak on, concerns critical regularities for actions of general nilpotent groups on one-dimensional manifolds: closed interval, circle, half-open interval. For all these groups and manifolds, we obtain the exact value of the critical regularity, describing it in purely algebraic terms (relative growth of some special subgroups).

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)

TBA

KIAS--Springer Lectures

March 7th (Fri) 2025, Part 1, 10 - 10:50 am, Part 2, 11 -  11:50 am

KIAS 1503 & Zoom https://kimsh.kr/vz

Richard Schwartz (Brown)

Le Retour de Pappus (The Return of Pappus)

Pappus's Theorem is a classic theorem in projective geometry, going back to antiquity. In this talk I will explain how Pappus's Theorem is related to Farey addition, representations of the modular group, and pleated surfaces contained in the rank 2 symmetric space associated to SL_3(R).    Some of these ideas go back to my 1993 paper, "Pappus's Theorem and the Modular Group", and some of the ideas are things I discovered when I returned to the topic this year.  I hope to explain  everything from scratch in the talk.

KIAS Mathematics Colloquium

March 12th (Wed) 2025, 4 pm

KIAS 1503 & Zoom https://kimsh.kr/vz

Richard Schwartz (Brown)

The Optimal Paper Moebius Band

A paper Moebius band is made  by twisting a 1 x L rectangular strip of paper in space and gluing together the length-1 sides. If L is large, this is easy to do. If L is small this is impossible. What is the cutoff value?  This question goes back at least to Wunderlich in 1962 and is most likely much older.  In 1977 B. Halpern and C. Weaver conjectured that L>sqrt(3) is a necessary and sufficient condition. In this talk I will explain my proof of the Halpern-Weaver Conjecture and I will also prove that a minimizing sequence of examples converges to a unique limiting shape, the equilateral triangle.

March 20 (Th), 11 am, KIAS 1423 & zoom https://kimsh.kr/vz

Homin Lee (Northwestern, KIAS)

Random dynamics on surfaces

In this talk, we will discuss about smooth random dynamical systems and group actions on surfaces. Random dynamical systems, especially understanding stationary measures, can play an important role when we consider a group action. For instance, when a group action on torus is given by toral automorphisms, using random dynamics, Benoist-Quint classified all orbit closures. 

In this talk, we will study on non-linear actions on surfaces using random dynamics. We will discuss about absolutely continuity of stationary measures, classification of orbit closure, and exact dimensionality of stationary measures. This talk will be mostly about the joint work with Aaron Brown, Davi Obata, and Yuping Ruan.