## Virtual Seminar on

## Geometry and Topology

### https://visgat.cayley.kr

## How to join

**Zoom Meeting ID** 739-035-2844

**Password** Zero minus the Euler characteristic of a closed oriented surface of genus 4000

**Time **Wed 4 pm or Thursday 9 am (or so) in KST (Korea Standard Time)

## 2021

**July ****14**** (W) 4 pm Korea**

InSung Park (ICERM)

*Julia sets with conformal dimension one (continued)*

In this talk, we discuss ideas used to prove the main theorem of the last talk: the equivalence between Julia sets having conformal dimension one and crochet-type Julia sets. The main technique that we use is the conformal energy of maps between graphs, introduced by D. Thurston. The motivation of the conformal energy is to provide a "positive" characterization of post-critically finite rational maps; a "negative" characterization was given by W. Thruston in the 1980s. We will start with historical backgrounds on the characterization problem and see how the conformal energy can be used to estimate the conformal dimensions of Julia sets.

**July 7 (W) 4 pm Korea**

InSung Park (ICERM)

*Julia sets with conformal dimension one*

Complex dynamics is the study of dynamical systems defined by iterating rational maps on the Riemann sphere. For a post-critically finite rational map f, the Julia set $J_f$ is a fractal defined as the repeller of the dynamics of f. As a fractal embedded in the Riemann sphere, the Julia set of a post-critically finite rational map has conformal dimension between 1 and 2. The Julia set has conformal dimension 2 if and only if it is the whole sphere. However, the other extreme case, when conformal dimension=1, contains diverse Julia sets, including Julia sets of post-critically finite polynomials and Newton maps. In this talk, we show that a Julia set $J_f$ has conformal dimension one if and only if there is an f-invariant graph that has topological entropy zero. In the spirit of Sullivanâ€™s dictionary, we also compare this result with Carrasco and Mackay's work on Gromov hyperbolic groups whose boundaries have conformal dimension one.

**April 21 (W) 4 pm Korea **

Yi Liu (BICMR)

*Finite-volume hyperbolic 3-manifolds are almost determined by their finite quotient groups*

In this talk, I will outline a proof for showing that the profinite completion of the fundamental group determines finite-volume hyperbolic 3-manifolds up to finitely many possibilities. As one of the main steps, I will explain why the Thurston norm on the first (real) cohomology is determined by the completion.

**May 13 (Th) 4 pm Korea**

Masato Mimura (Tohoku U)

*The Green--Tao theorem for number fields*

Joint work with colleagues in Tohoku University: Wataru Kai, Shin-ichiro Seki, Akihiro Munemasa and Kiyoto Yoshino. Ben Green and Terence Tao have proved that the set of rational primes contains arbitrarily long arithmetic progressions. We establish generalizations of this theorem in the setting of (arbitrary) number fields. No serious background on number theory is assumed for this talk. For the preprint, see https://arxiv.org/abs/2012.15669

**May 27 (Th) 10 am Korea (= May 26 9 pm EST)**

Tarik Aougab (Haverford College)

*Simple length rigidity for covers*

Suppose X and Y are finite covers of a fixed hyperbolic surface S. We first show that if for all closed curves gamma on S, gamma admits a simple lift to X if and only if it does to Y, then X and Y are equivalent covers. Using similar ideas, we address the question of when two covers of a fixed hyperbolic surface are isometric when their unmarked simple length spectra agree. We outline some sufficient criteria on the covers for this and generate families of examples. This represents joint work with Max Lahn, Marissa Loving, and Nick Miller.

**July 7, 4 - 5 pm (Korea)**

InSung Park (ICERM)