We discuss a construction of a (singular) foliation on 3-manifold with a left-orderable fundamental group. We discuss this in the context of L-space conjecture.

Let $L$ be a knot or link, embedded in $S^3$. We call $L$ is fibered if $S^3 - L$ admits a fiber bundle structure over $S^1$. Murasugi, Harer, Stalling and more mathematicians have piled up a lot of tools to detect and construct a fibered knot&link, and Thurston gave a nice description of fiberedness of given 3-manifold by so-called “Thurston norm”. In this talk, I will introduce a classical question, “What can be a shape of Thurston unit norm ball?” and share some of my results, which will appear in the upcoming paper. This is a joint work with Hyungryul Baik and Philippe Tranchida.

Recent developments in the theory of random walks on (partially) hyperbolic spaces led to a detailed description of random isometries of such spaces. In this talk, I will explain the relationship between random mapping class and frequently contracting geodesics in Teichmuller space. If time allows, I will also describe an analogous phenomenon regarding $\mathrm{Out}(F_n)$ and Outer space.

Profinite groups were first designed in the area of Galois theory. In 2017, H. Wilton and P. Zalesskii showed that some geometric properties can be detected by the profinite completion of its fundamental group. After this, many other geometric properties were also known as having profinite rigidity, by M. R. Bridson, D. B. McReynolds, A. W. Reid, etc. In this talk, we will discuss their result and briefly introduce my work, the construction of group action on an asymptotic cone. Using this action, we deduce that any torsion-free, residually finite, non-elementary hyperbolic group can be embedded into the isometry group of the $2^{\aleph_0}$- universal tree.

A quandle is an algebraic structure closely related to knot theory. A quandle is defined on a set with a binary operation satisfying some conditions for Reidemeister moves. The first part of this talk surveys quandle theory and its applications to knot theory. In the second part, we introduce recent results on the relationship between quandle 2-cocycle extensions and group 2-cocycle extensions.

In 1976, Cappell and Shaneson constructed an infinite family of smooth homotopy 4-spheres, called Cappell-Shaneson homotopy 4-spheres. Cappell-Shaneson homotopy 4-spheres are the most notable potential counterexamples to the smooth 4-dimensional Poincare conjecture and they are related to other important conjectures including Gluck, Schoenflies, the slice-ribbon conjectures. In this talk, I would like to give a survey on Cappell-Shaneson homotopy 4-spheres.

Let $O$ be a compact reflection $n$-orbifold whose underlying space is homeomorphic to a truncation $n$-polytope, i.e. a polytope obtained from an $n$-simplex by successively truncating vertices. In this talk, I will give a complete description of the deformation space of convex projective structures on the orbifold $O$ of dimension at least 4. Joint work with Suhyoung Choi and Ludovic Marquis.

The unknotting number of knots is a difficult quantity to compute, and even its behavior under basic satelliting operations is not understood. We establish a lower bound on the unknotting number of cable knots and iterated cable knots purely in terms of the winding number of the pattern. This is joint work with Jennifer Hom and Tye Lidman.

Gromov found a class of finitely presented groups that share properties with lattices of hyperbolic spaces. We now call them word-hyperbolic groups. In this talk, we will see some basic properties of word-hyperbolic groups , shown by Gromov, and application to mapping class groups and RAAGs. This is joint work with Baik and Shin.

Billera—Lee '81 and Stanley '80 proved McMullen's g-conjecture, which provides us a comprehensive combinatorial characterization of a simplicial (or simple) polytope in terms of the face numbers. Stanley’s proof for the necessity part of this conjecture employed toric geometry, in particular he used the relationship between the face numbers of lattice polytopes and the Betti numbers of rationally smooth toric varieties. In this talk, we will examine the primary idea of this relationship and see how it might be applied to singular toric varieties.

The homology cobordism group of 3-dimensional homology cylinders can be considered as an enlargement of both the mapping class group of a surface and the concordance group of string links. In this talk, we consider extending their invariants to homology cylinders. Especially, all of Johnson homomorphisms and Morita homomorphisms of a mapping class group of a surface, Milnor invariants and Orr invariants of (string) links are related to lower central series of a free group. The invariants also give rise to filtrations. We compare and extend those. Also we get relations of the filtratins to automorphism groups of free nilpotent groups, and free Lie algebras.

Let $\varphi \colon \Lambda \to \Gamma$ be a regular cover of simplicial graphs without isolated vertices. If $f$ is an automorphism of $A_\Gamma$, the right-angled Artin-Tits group associated to $\Gamma$, does there exist an automorphism $F$ of $A_\Lambda$ which is a lift of $f$? We will establish criterion for such a lift to exist and then study the group $\textrm{LAut}(\varphi)$ consisting of all the liftable automorphisms of $A_\Gamma$ and the group $\textrm{FD}(\varphi)$ of all the lifts of the identity, This is a joint work with Sangrok Oh and Donggyun Seo.

The curve graph of a surface of finite type is a fundamental object in the study of its mapping class group both from the metric and the combinatorial point of view. I will discuss joint work with Valentina Disarlo and Thomas Koberda where we conduct a thorough study of curve graphs from the model theoretic point of view, with particular emphasis in the problem of interpretability between different curve graphs and other geometric complexes.