Virtual Seminar on
Geometry and Topology
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)
September 29 (W) 4 pm Korea
Kiho Park (KIAS)
September 1, 2, 3 (W, Th, F) 4 pm Korea
The question of whether two non-abelian free groups of different finite ranks satisfy the same collection of first order sentences was originally formulated by Tarski in the 1950s and remained open until 2001, when it was answered in the affirmative by Z. Sela in a celebrated series of papers relying heavily on geometric group theory techniques (an independent solution is due to Kharlampovich and Myasnikov). In this series of talks I will present some of the core ideas in Sela's proof. In particular, I will give an outline of the shortening argument and two of its applications: the construction of Makanin-Razborov diagrams for the analysis of solution sets of equations over free groups and the existence of formal solutions witnessing the validity of sentences. I will assume a certain degree of familiarity with Bass-Serre theory.
July 14 (W) 4 pm Korea
InSung Park (ICERM)
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 Sullivan’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)