# Abstracts

Mini-course

Marco Linton (ICMAT)

Research Talks

Tattybubu Arap Kyzy (Pusan National University)

Jiyoung Han (KIAS)

Susumu Hirose (Tokyo University of Science)

On generating mapping class groups by pseudo-Anosov elements

Wajnryb proved that the mapping class group of a closed oriented surface is generated by two elements. We proved that the mapping class group is generated by two pseudo-Anosov elements. In particular, if the genus is greater than or equal to nine, we can take the generators to two conjugate pseudo-Anosov elements with arbitrarily large dilatations. This is a joint work with Naoyuki Monden (Okayama University, Japan.)

Jan Kim (EIMS at Ewha Womans University)

Minkyu Kim (KIAS)

A categorified eigenring action on Jacobi diagrams in handlebodies

For an algebra $R$ and a multiplicative submodule $J$, the eigenring, denoted by $E_{R}(J)$, is a canonical subquotient algebra of $R$ by $J$. In the case where $J$ is a two-sided ideal, $E_{R}(J)$ coincides with the quotient algebra $R/J$. The formal resemblance of linear categories to algebras allows us to extend this construction to that of a subquotient linear category. This talk presents an example that arises in the context of quantum topology, demonstrating this general framework. To be precise, we consider a linear category $A$ generated by Jacobi diagrams in handlebodies. We then relate the Lie operad to a “categorified” eigenring of $A$. The category $A$ was introduced by Habiro and Massuyeau in order to extend the Kontsevich integral.

Sang-hyun Kim (KIAS)

Sungwoon Kim (Jeju National University)

Taehyeong Kim (KIAS)

Equidistribution of divergent geodesics via entropy methods

Duke’s Theorem (1988) is concerned with the equidistribution of closed geodesics on the modular surface, and Einsiedler, Lindenstrauss, Michel and Venkatesh (2012) gave an ergodic theoretic proof of Duke's Theorem using maximal entropy methods. Along these lines, David and Shapira (2018) established an equidistribution result of certain divergent geodesics on the modular surface under non-escape of mass assumption. In this talk, we will relax this non-escape of mass assumption and give its applications to continued fractions of rationals. This is joint work with Ofir David, Ron Mor, and Uri Shapira.

Mitsuaki Kimura (Osaka Dental University)

Erika Kuno (Osaka University)

Automorphisms of fine curve graphs for nonorientale surfaces

The fine curve graph of a surface was introduced by Bowden, Hensel, and Webb as a graph consisting of the actual essential simple closed curves on the surface. Long, Margalit, Pham, Verberne, and Yao proved that the automorphism group of the fine curve graph of a closed orientable surface of genus g ≥ 2 is isomorphic to the homeomorphism group of the surface. We generalized their result to closed nonorientable surfaces of genus g ≥ 4. This is a joint work with Mitsuaki Kimura.

Sanghoon Kwak (KIAS)

Donsung Lee (Seoul National University)

Sangjin Lee (KIAS)

A Higher-Dimensional Generalization of the Pseudo-Anosov Mapping Class

In this talk, I will present an example of applying concepts from Geometric Group Theory and surface mapping class group theory to symplectic topology. I will begin with a brief introduction to symplectic topology and then illustrate a specific application: a higher-dimensional generalization of the pseudo-Anosov mapping class in the symplectic setting.

Seonhee Lim (Seoul National University)

Makoto Sakuma (Hiroshima University, OCAMI)

Two-bridge knot groups and problems arising from their study

Two-bridge knot groups are particularly elegant examples of one-relator groups. In a series of joint works with Donghi Lee conducted between 2011 and 2016, we fully resolved several problems related to two-bridge knot groups, which had been on my mind since around 1997 and formally proposed in 1999. These solutions found applications in McShane’s identity, Bowditch-Tan-Wong-Zhang’s end invariants, and epimorphisms between two-bridge knot groups. These studies further inspired Yuya Koda and me to introduce the concept of homotopy motion groups of surfaces in 3-manifolds. In this talk, I will briefly review the joint work with Donghi Lee on two-bridge knot groups

and discuss how this work led to the development of homotopy motion groups. I will then present some open questions for further exploration.

Carl-Fredrik Nyberg-Brodda (KIAS)

Donggyun Seo (Seoul National University)