Tuesday Seminar on Topology (October, 2021 -- January, 2022)

[Japanese] [Past Programs]

This is an online seminar on Zoom.

Last updated March 23, 2022
Information :　
Nariya Kawazumi
Takahiro Kitayama
Takuya Sakasai

October 5, 17:00-18:00 -- Online on Zoom. Pre-registration required.

Hiroshi Goda (Tokyo University of Agriculture and Technology)

Twisted Alexander polynomials, chirality, and local deformations of hyperbolic 3-cone-manifolds

Abstract: We discuss a relationship between the chirality of knots and higher dimensional twisted Alexander polynomials associated with holonomy representations of hyperbolic 3-cone-manifolds. In particular, we provide a new necessary condition for a knot, that appears in a hyperbolic 3-cone-manifold of finite volume as a singular set, to be amphicheiral. Moreover, we can detect the chirality of hyperbolic twist knots, according to our criterion, using low-dimensional irreducible representations. (This is a joint work with Takayuki Morifuji.)

October 12, 17:00-18:00 -- Online on Zoom. Pre-registration required.

Nobuo Iida (The Univesity of Tokyo)

Seiberg-Witten Floer homotopy and contact structures

Abstract: Seiberg-Witten theory has been an efficient tool to study 4-dimensional symplectic and 3-dimensional contact geometry. In this talk, we introduce new homotopical invariants related to these structures using Seiberg-Witten theory and explain their properties and applications. These invariants have two main origins:
1. Kronheimer-Mrowka's invariant for 4-manifold with contact boundary, whose construction is based on Seiberg-Witten equation on 4-manifolds with conical end.
2. Bauer-Furuta and Manolescu's homotopical method called finite dimensional approximation in Seiberg-Witten theory.
This talk includes joint works with Masaki Taniguchi(RIKEN) and Anubhav Mukherjee(Georgia tech).

October 19, 17:00-18:00 -- Online on Zoom. Pre-registration required.

Yoshihiko Shinomiya (Shizuoka University)

Period matrices of some hyperelliptic Riemann surfaces

Abstract: In this talk, we give new examples of period matrices of hyperelliptic Riemann surfaces. For generic genus, there were few examples of period matrices. The period matrix of a Riemann surface depends only on the choice of symplectic basis of the first homology group. It is difficult to find a symplectic basis in general. We construct hyperelliptic Riemann surfaces of generic genus from some rectangles and find their symplectic bases. Moreover, we give their algebraic equations. The algebraic equations are of the form w²=z(z²-1)(z²-a₁²)(z²-a₂²) … (z²-a_g-1²) (1< a₁ < a₂ < … < a_g-1). From them, we can calculate period matrices of our Riemann surfaces. We also show that all algebraic curves of this types of equations are obtained by our construction.

October 26, 17:00-18:00 -- Online on Zoom. Pre-registration required.

Naohiko Kasuya (Hokkaido University)

On the strongly pseudoconcave boundary of a compact complex surface

Abstract: On the strongly pseudoconvex (resp. pseudoconcave) boundary of a complex surface, the complextangency defines a positive (resp. negative) contact structure. Bogomolov and De Oliveira proved that the boundary contact structure of a strongly pseudoconvex surface is Stein fillable. Therefore, for a closed contact 3-manifold, Stein fillability and holomorphic fillability are equivalent. Then what about the boundary of a strongly pseudoconcave surface? We prove that any closed negative contact 3-manifold can be realized as the boundary of a strongly pseudoconcave surface. The proof is done by establishing holomorphic handle attaching method to the strongly pseudoconcave boundary of a complex surface, based on Eliashberg's handlebody construction of Stein manifolds. This is a joint work with Daniele Zuddas (University of Trieste).

November 2, 17:00-18:00 -- Online on Zoom. Pre-registration required.

Takefumi Nosaka (Tokyo Institute of Technology)

Meta-nilpotent knot invariants and symplectic automorphism groups of free nilpotent groups

Abstract: There are many developments of fibered knots and homology cylinders from topological and algebraic viewpoints. In a converse sense, we discuss meta-nilpotent localization of knot groups, which can deal with any knot like fibered knots. The monodoromy can be regarded as a symplectic automorphism of free nilpotent group, and the conjugacy classes in the outer automorphism groups produce knot invariants in terms of Johnson homomorphisms. In this talk, I show the construction of the monodoromies, and some results on the knot invariants. I also talk approaches from Fox pairings.

November 9, 17:00-18:00 -- Online on Zoom. Pre-registration required.

Shuhei Maruyama (Nagoya University)

The spaces of non-descendible quasimorphisms and bounded characteristic classes

Abstract: A quasimorphism is a real-valued function on a group which is a homomorphism up to bounded error. In this talk, we discuss the (non-)descendibility of quasimorphisms. In particular, we consider the space of non-descendible quasimorphisms on universal covering groups and explain its relation to the space of bounded characteristic classes of foliated bundles. This talk is based on a joint work with Morimichi Kawasaki.

November 16, 17:00-18:00 -- Online on Zoom. Pre-registration required.

Wataru Yuasa (RIMS, Kyoto University)

Skein and cluster algebras of marked surfaces without punctures for sl(3)

Abstract: We consider a skein algebra consisting of sl(3)-webs with the boundary skein relations for a marked surface without punctures. We construct a quantum cluster algebra coming from the moduli space of decorated SL(3)-local systems of the surface inside the skew-field of fractions of the skein algebra. In this talk, we introduce the sticking trick and the cutting trick for sl(3)-webs. The sticking trick expands the boundary-localized skein algebra into the cluster algebra. The cutting trick gives Laurent expressions of "elevation-preserving" webs with positive coefficients in certain clusters. We can also apply these tricks in the case of sp(4). This talk is based on joint works with Tsukasa Ishibashi.

November 30, 17:00-18:00 -- Online on Zoom. Pre-registration required.

Masatoshi Sato (Tokyo Denki University)

A non-commutative Reidemeister-Turaev torsion of homology cylinders

Abstract: The Reidemeister-Turaev torsion of homology cylinders takes values in the integral group ring of the first homology of a surface. We lift it to a torsion valued in the K₁-group of the completed rational group ring of the fundamental group of the surface. We show that it induces a finite type invariant of homology cylinders, and describe the induced map on the graded quotient of the Y-filtration of homology cylinders via the 1-loop part of the LMO functor and the Enomoto-Satoh trace. This talk is based on joint work with Yuta Nozaki and Masaaki Suzuki.

December 7, 17:00-18:00 -- Online on Zoom. Pre-registration required.

Taketo Sano (The Univesity of Tokyo)

A Bar-Natan homotopy type

Abstract: In year 2000, Khovanov introduced a categorification of the Jones polynomial, now known as Khovanov homology. In 2014, Lipshitz and Sarkar introduced a spatial refinement of Khovanov homology, called Khovanov homotopy type, which is a finite CW spectrum whose reduced cellular cohomology recovers Khovanov homology. On the algebraic level, there are several deformations of Khovanov homology, such as Lee homology and Bar-Natan homology. These variants are also important in that they give knot invariants such as Rasmussen’s $s$-invariant. Whether these variants admit spatial refinements have been open.

In 2021, the speaker constructed a spatial refinement of Bar-Natan homology and determined its stable homotopy type. The construction follows that of Lipshitz and Sarkar, which is based on the construction proposed by Cohen, Segal and Jones using the concept of flow categories. Also, we adopt techniques called “Morse moves in flow categories” introduced by Lobb et.al. to determine the stable homotopy type. Spacialy (or homotopically) refining the $s$-invariant is left as a future work.

December 21, 17:30-18:30 [Joint with Lie Groups and Representation Theory Seminar] -- Online on Zoom. Pre-registration is not required.

Hiroki Shimakura (Tohoku University)

Classification of holomorphic vertex operator algebras of central charge 24

Abstract: Holomorphic vertex operator algebras are imporant in vertex operator algebra theory. For example, the famous moonshine vertex operator algebra is holomorphic. One of the fundamental problems is to classify holomorphic vertex operator algebras. It is known that holomorphic vertex operator algebras of central charge 8 and 16 are lattice vertex operator algebras. I will talk about recent progress on the classification of holomorphic vertex operator algebras of central charge 24.

January 11, 17:00-18:00 [Joint with Lie Groups and Representation Theory Seminar] -- Online on Zoom. Pre-registration is not required.

Keiichi Maeta (The Univesity of Tokyo)

On the existence problem of Compact Clifford-Klein forms of indecomposable pseudo-Riemannian symmetric spaces with signature (n,2)

Abstract: For a homogeneous space G/H and its discontinuous group Γ⊂ G, the double coset space Γ\G/H is called a Clifford-Klein form of G/H. In the study of Clifford-Klein forms, the classification of homogeneous spaces which admit compact Clifford-Klein forms is one of the important open problems, which was introduced by Toshiyuki Kobayashi in 1980s. We consider this problem for indecomposable and reducible pseudo-Riemannian symmetric spaces with signature (n,2). We show the non-existence of compact Clifford-Klein forms for some series of symmetric spaces, and construct new compact Clifford-Klein forms of countably infinite five-dimensional pseudo-Riemannian symmetric spaces with signature (3,2).

January 25, 17:00-18:00 -- Online on Zoom. Pre-registration required.

Xiaobing Sheng (The Univesity of Tokyo)

Some obstructions on subgroups of the Brin-Thompson group 2V

Abstract: Motivated by Burillo, Cleary and Röver's summary of the obstruction for subgroups of Thompson's group V, we investigate the higher dimensional version, the group 2V and found out that they have similar obstructions on torsion subgroups and certain Baumslag-Solitar groups.