### Uniformization theorem and Liouville action for punctured spheres

Location: E6-1 #4415
Speaker: 박진성 (KIAS)
Language: Korean
Abstract
In this talk, I will explain an approach of Poincare to prove the uniformization theorem for punctured spheres, and how it is related to the action functional in the Liouville theory.

### The Zilber-Pink Conjecture and the generalised Cosmetic Surgery Conjecture

Location: E6-1 #3433
Speaker: 전보광 (POSTECH)
Language: TBA
Abstract
In this talk, I will explain how one can generalise the cosmetic surgery conjecture under the assumption of another well-known conjecture in number theory, so called the Zilber-Pink conjecture.

### Quasi-isometric invariant of CAT(0) cube complexes

Location: E6-1 #4415
Speaker: 오상록 (KAIST)
Language: 한국어
Abstract
Bestvina, Kleiner and Sageev showed that every 2-quasiflat in a 2-dimensional CAT(0) cube complex is at finite Hausdorff distance from a finite union of 2-dimensional quarter-plane and Huang generalized it to $n$. Using this invariant, several quasi-isometric classification problems in right-angled Artin groups and graph braid groups are solved. In this talk, We discuss how this invariant works when we classify planar graph 2-braid groups up to quasi-isometries.

### The set of critical exponents of discrete groups acting on a regular tree

Location: E6-1 #4415
Speaker: 권상훈 (KIAS)
Language: 한국어
Abstract
We discuss the set of critical exponents of discrete groups acting on a regular tree. If the quotient graph is finite, then the critical exponent is an algebraic number. In general, given an arbitrary real number between 0 and the volume entropy of the regular tree, we discuss how we can construct a discrete group whose critical exponent realizes the number. We also study the minimal polynomials of Schottky free discrete groups of rank 2.

### Geometric inequalities and quasi-local mass for axially symmetric initial data in general relativity (1), (2)

Location: E6-1 #4415 on Sep. 19, and #1409 on Sep. 26
Speaker: 차예슬 Ye Sle Cha (Free University of Berlin)
Language: English
Abstract
In these two talks, I will introduce geometric problems related to mass in general relativity, such as a series of geometric inequalities, and conjectures regarding a notion of quasi-local mass proposed by Bartnik. The geometric inequalities we consider include the angular momentum-mass inequality for axially symmetric initial data for the Einstein equations. Note that the special cases treating maximal data have been proved by Dain et al. Here I will explain how to reduce the general (non-maximal) case to the known maximal case, and then discuss the solvability of the system of Elliptic PDEs arose in the process, for near maximal case. The second part of the talk will mainly provide an introduction to the static/stationary metric extension conjectures, related to Bartnik quasi-local mass. I will briefly discuss some known results for the static metric extension conjecture by Anderson, Anderson/Khuri, Miao et al., and show a local existence theorem for the solutions of axially symmetric, stationary vacuum Einstein equations.

### Dynamics of generalized beta-transformations

Location: E6-1 #4415
Speaker: Chenxi Wu (Rutgers University)
Language: English
Abstract
We study the dynamical properties of the topological generalized beta transformations, which generalizes the concept of generalized beta transformations defined by Gora. In particular, we generalize the result on admissible sequence for unimodular maps to the case of generalized beta maps, and also study the properties of the topological entropy and its Galois conjugates, generalizing some results by Tiozzo. This talk represents an ongoing collaboration with Diana Davis, Kathryn Lindsey and Harry Bray.

### Canonical metric on finite graphs

Location: E6-1 #1409
Speaker: Chenxi Wu (Rutgers University)
Language: English
Abstract
We prove a Kazhdan type theorem for the canonical metrics of finite graphs. Namely, we show that the canonical metric of finite normal coverings of the graph converges when the covering converges, and the limit depends only on the limit of the coverings. We also generalize the argument to higher dimensional simplicial complexes. The proof is mostly based on an analogous argument in the case of Riemann surfaces and Lück's approximation theorem for L^2 cohomology. This is joint work with Farbod Shokrieh.

### The free product structure of diffeomorphism groups

Location: E6-1 #1409
Speaker: Thomas Koberda (University of Virginia)
Language: English
Abstract
I will discuss some aspects of the algebraic structure of finitely generated groups of diffeomorphisms of compact one-manifolds. In particular, we show that if G is not virtually metabelian then (G x Z)*Z cannot act faithfully by C^2 diffeomorphisms on a compact one-manifold. Among the consequences of this result is a completion of the classification of right-angled Artin groups which admit faithful C^{\infty} actions on the circle, a program initiated together with H. Baik and S. Kim. This represents joint work with S. Kim.

### Towers of regular self-covers and linear endomorphisms of tori

Location: E6-1 #1409
Speaker: Wouter van Limbeek (University of Michigan)
Language: English
Abstract
Let M be a closed manifold that admits a nontrivial cover diffeomorphic to itself. Which manifolds have such a self-similar structure? Examples are provided by tori, in which case the covering is homotopic to a linear endomorphism. Under the assumption that all iterates of the covering of M are regular, we show that any self-cover is induced by a linear endomorphism of a torus on a quotient of the fundamental group. Under further hypotheses we show that a finite cover of M is a principal torus bundle. We use this to give an application to holomorphic self-covers of Kaehler manifolds.

### Towers of regular self-covers and linear endomorphisms of tori

Location: E6-1 #1409
Speaker: Wouter van Limbeek (University of Michigan)
Language: English
Abstract
Let M be a closed manifold that admits a nontrivial cover diffeomorphic to itself. Which manifolds have such a self-similar structure? Examples are provided by tori, in which case the covering is homotopic to a linear endomorphism. Under the assumption that all iterates of the covering of M are regular, we show that any self-cover is induced by a linear endomorphism of a torus on a quotient of the fundamental group. Under further hypotheses we show that a finite cover of M is a principal torus bundle. We use this to give an application to holomorphic self-covers of Kaehler manifolds.

### Is a typical biPerron algebraic unit a pseudo-Anosov dilatation?

Speaker: 백형렬 (KAIST)
Language: Korean
Abstract
We give a statistical answer to a variaion of this problem.

### Translation length on extension graphs

Speaker: 신현식 (KAIST)
Language: Korean
Abstract
We report a recent result on the translation length on extension graphs of RAAGs.

### Geometric convergence of Kleinian groups

Speaker: 백형렬 (KAIST)
Language: Korean
Abstract
We introduce the geometric topology on the space of Kleinian groups, and discuss how to understand it with a simple example.

### Bipolar filtration of topologically slice knots

Speaker: 김민훈 (KIAS)
Language: Korean
Abstract
We show that the bipolar filtration of the smooth concordance group of topologically slice knots introduced by Cochran, Harvey and Horn has nontrivial graded quotients at every stage. To detect a nontrivial element in the quotient, the proof uses Cheeger-Gromov $L^2$ $\rho$-invariants and infinitely many Heegaard Floer correction term invariants simultaneously. This is joint work with Jae Choon Cha.

### Random walks in weakly hyperbolic groups V

Speaker: TBA
Language: Korean
Abstract
Survey of the paper by Joseph Maher and Giulio Tiozzo V

### Random walks in weakly hyperbolic groups IV

Speaker: 정홍택 (KAIST)
Language: Korean
Abstract
Survey of the paper by Joseph Maher and Giulio Tiozzo IV

### Random walks in weakly hyperbolic groups III

Speaker: 정홍택 (KAIST)
Language: Korean
Abstract
Survey of the paper by Joseph Maher and Giulio Tiozzo III

### Random walks in weakly hyperbolic groups II

Speaker: 정성구 (KAIST)
Language: Korean
Abstract
Survey of the paper by Joseph Maher and Giulio Tiozzo II

### Collar lemma for Hitchin representations

Speaker: 이계선 (University of Heidelberg)
Language: English
Abstract
There is a classical result first due to Keen known as the collar lemma for hyperbolic surfaces. A consequence of the collar lemma is that if two closed curves A and B on a closed orientable hyperbolizable surface have non-zero geometric intersection number, then there is an explicit lower bound for the length of A in terms of the length of B, which holds for any hyperbolic structure on the surface. By slightly weakening this lower bound, we generalize this statement to hold for all Hitchin representations. This is a joint work with Tengren Zhang.

### Random walks in weakly hyperbolic groups I

Speaker: 오상록 (KAIST)
Language: Korean
Abstract
Survey of the paper by Joseph Maher and Giulio Tiozzo I