Tuesday Seminar on Topology

[Japanese]   [Past Programs]
17:00 -- 18:30 Graduate School of Mathematical Sciences, The University of Tokyo (with live streaming on Zoom)
17:00 -- 18:00 Online seminar on Zoom.

Last updated January 7, 2024
Information :@
Kazuo Habiro
Nariya Kawazumi
Takahiro Kitayama
Takuya Sakasai

Pre-registration is required.
Once you register, you can attend all our seminars until the end of March, 2024.
The zoom meeting will be opened 15 minutes before start.
Making audio and video recordings is prohibited.
October 10, 17:30-18:30 -- Room 056 with live streaming on Zoom

Masato Mimura (Tohoku University)

Invariant quasimorphisms and coarse geometry of scl

Abstract: The topic of this talk is completely independent from that of the intensive lecture (the Green--Tao theorem) from 9th to 13th, Oct. This talk is based on the series of the joint work with Morimichi Kawasaki, Mitsuaki Kimura, Takahiro Matsushita and Shuhei Maruyama. Quasimorphisms on a group are interesting objects, but for many naturally constructed groups the space of quasimorphisms tends to be either 'trivial' or infinite dimensional. We study the setting of a pair of a group and its normal subgroup, not of a single group, and invariant quasimorphisms. Then, we can obtain a non-zero finite dimensional vector space from this setting. The celebrated Bavard duality theorem is extended to this framework, and the resulting theorem yields some outcome on the coarse geometry of scl (stable commutator length). I will present an overview of the developments of this theory.

October 17, 17:00-18:00 -- Online on Zoom

Shunsuke Kano (MathCCS, Tohoku University)

Train track combinatorics and cluster algebras

Abstract: The concepts of train track was introduced by W. P. Thurston to study the measured foliations/laminations and the pseudo-Anosov mapping classes on a surface. In this talk, we translate some concepts of train tracks into the language of cluster algebras using the tropicalization of Goncharov--Shen's potential function. Using this, we translate a combinatorial property of a train track associated with a pseudo-Anosov mapping class into the combinatorial property in cluster algebras, called the sign stability which was introduced by Tsukasa Ishibashi and the speaker.

October 24, 17:00-18:00 -- Online on Zoom

Shin Hayashi (Aoyama Gakuin University)

Index theory for quarter-plane Toeplitz operators via extended symbols

Abstract: We consider index theory for some Toeplitz operators on a discrete quarter-plane. Index theory for such operators has been investigated by Simonenko, Douglas-Howe, Park and index formulas are obtained by Coburn-Douglas-Singer, Duducava. In this talk, we revisit Duducavafs idea and discuss an index formula for quarter-plane Toeplitz operators of two-variable rational matrix function symbols from a geometric viewpoint. By using Gohberg-Krein theory for matrix factorizations and analytic continuation, we see that the symbols of Fredholm quarter-plane Toeplitz operators defined originally on a two-dimensional torus can canonically be extended to some three-sphere, and show that their Fredholm indices coincides with the three-dimensional winding number of extended symbols. If time permits, we briefly mention a contact with a topic in condensed matter physics, called (higher-order) topological insulators.

October 31, 17:30-18:30 -- Room 056 with live streaming on Zoom

Naoki Chigira (Kumamoto University)

On Harada Conjecture II

Abstract: The Character table of finite group has a lot of information about the group. In this talk, we discuss about a conjecture of Koichiro Harada (so called Harada conjecture II) which is related to the product of all irreducible characters and the product of all conjugacy class sizes.

November 7, 17:00-18:30 -- Room 056 with live streaming on Zoom

Florent Schaffhauser (Heidelberg University)

Hodge numbers of moduli spaces of principal bundles on curves

Abstract: The Poincaré series of moduli stacks of semistable G-bundles on curves has been computed by Laumon and Rapoport. In this joint work with Melissa Liu, we show that the Hodge-Poincaré series of these moduli stacks can be computed in a similar way. As an application, we obtain a new proof of a joint result of the speaker with Erwan Brugallé, on the maximality on moduli spaces of vector bundles over real algebraic curves.

November 14, 17:00-18:30 -- Room 056 with live streaming on Zoom

Tomoo Yokoyama (Saitama University)

Dependency of the positive and negative long-time behaviors of flows on surfaces

Abstract: We discuss the dependence of a flow's positive and negative limit behaviors on a surface. In particular, I introduce the list of possible pairs of positive and negative limit behaviors that can and cannot occur. The idea of the dependence mechanism is illustrated using the dependence of the limit behavior of a toy model, a circle homeomorphism. We overview with as few prior knowledge assumptions as possible.

November 21, 17:30-18:30 -- Room 056 with live streaming on Zoom

Yuya Koda (Keio University)

Shadows, divides and hyperbolic volumes

Abstract: In 2008, Costantino and D.Thurston revealed that the combinatorial structure of the Stein factorizations of stable maps from 3-manifolds into the real plane can be used to describe the hyperbolic structures of the complement of the set of definite fold points, which is a link. The key was that the Stein factorizations can naturally be embedded into 4-manifolds, and nice ideal polyhedral decompositions become visible on their boundaries. In this talk, we consider divides, which are the images of a proper and generic immersions of compact 1-manifolds into the 2-disk. Due to A'Campo's theory, each divide is associated with a link in the 3-sphere. By embedding a polyhedron induced from a given divide into the 4-ball as was done to Stein factorization, we can read off the ideal polyhedral decompositions on the boundary. We then show that the complement of the link of the divide can be obtained by Dehn filling a hyperbolic 3-manifold that admits a decomposition into several ideal regular hyperbolic polyhedra, where the number of each polyhedron is determined by types of the double points of the divide. This immediately gives an upper bound of the hyperbolic volume of the links of divides, which is shown to be asymptotically sharp. As in the case of Stein factorizations, an idea from the theory of Turaev's shadows plays an important role here. This talk is based on joint work with Ryoga Furutani (Hiroshima University).

November 28, 17:00-18:30 -- Room 056 with live streaming on Zoom

Gwénaël Massuyeau (Université de Bourgogne)

An analogue of the Johnson-Morita theory for the handlebody group

Abstract: The Johnson-Morita theory provides an approach for the mapping class group of a surface by considering its actions on the successive nilpotent quotients of the fundamental group of the surface. In this talk, after an outline of the original theory, we will present an analogue of the Johnson-Morita theory for the handlebody group, i.e. the mapping class group of a handlebody. This is joint work with Kazuo Habiro; as we shall explain if time allows, our motivation is to recover the "tree reduction" of a certain functor on the category of bottom tangles in handlebodies that we introduced (a few years ago) using the Kontsevich integral.

December 5, 17:00-18:30 -- Room 056 with live streaming on Zoom

Teruaki Kitano (Soka University)

On the Euler class for flat S1-bundles, C vs Cω

Abstract: We describe low dimensional homology groups of the real analytic, orientation preserving diffeomorphism group of S1 in terms of BΓ1 by applying a theorem of Thurston. It is an open problem whether some power of the rational Euler class vanishes for real analytic flat S1 bundles. In this talk we discuss that if it does, then the homology group should contain many torsion classes that vanish in the smooth case. Along this line we can give a new proof for the non-triviality of any power of the rational Euler class in the smooth case. If time permits, we will mention some attempts to study a Mather-Thurston map in the analytic case. This talk is based on a joint work with Shigeyuki Morita and Yoshihiko Mitsumatsu.

December 12, 17:00-18:30 -- Room 056 with live streaming on Zoom

Stavros Garoufalidis (Southern University of Science and Technology)

Multivariable knot polynomials from braided Hopf algebras with automorphisms

Abstract: We will discuss a unified approach to define multivariable polynomial invariants of knots that include the colored Jones polynomials, the ADO polynomials and the invariants defined using the theory of quantum groups. Our construction uses braided Hopf algebras with automorphisms. We will give examples of 2-variable invariants, and discuss their structural properties. Joint work with Rinat Kashaev.

December 14 (Thu), 17:00-18:30 -- Room 056 with live streaming on Zoom

Mikiya Masuda (Osaka City University)

Torus orbit closures in the flag variety

Abstract: The study of torus orbit closures in the flag variety was initiated by Gelfand-Serganova and Klyachko in 1980fs but has not been studied much since then. Recently, I have studied its geometry and topology jointly with Eunjeong Lee, Seonjeong Park, Jongbaek Song in connection with combinatorics of polytopes, Coxeter matroids, and polygonal triangulations. In this talk I will report on the development of this subject.

December 19, 17:00-18:30 -- Room 056 with live streaming on Zoom

Yasuyuki Kawahigashi (The University of Tokyo)

Topological quantum computing, tensor networks and operator algebras

Abstract: Modular tensor categories have caught much attention in connection to topological quantum computing based on anyons recently. Condensed matter physicists recently try to understand structures of modular tensor categories appearing in two-dimensional topological order using tensor networks. We present understanding of their tools in terms of operator algebras. For example, 4-tensors they use are exactly bi-unitary connections in the Jones theory of subfactors and their sequence of finite dimensional Hilbert spaces on which their gapped Hamiltonians act is given by the so-called higher relative commutants of a subfactor. No knowledge on operator algebras are assumed.

January 9, 17:00-18:00 -- Room 056 with live streaming on Zoom

Akihiro Takano (The University of Tokyo)

Stabilizer subgroups of Thompson's group F in Thompson knot theory

Abstract: Thompson knot theory, introduced by Vaughan Jones, is a study of knot theory using Thompson's group F. More specifically, he defined a method of constructing a knot from an element of F, and proved that any knot can be realized in his way. This fact is called Alexanderfs theorem, which is an analogy of the braid group. In this talk, we consider Thompson knot theory in terms of a relation between subgroups of F and knots obtained from their elements. In particular, we focus on stabilizer subgroups of F with respect to the natural action on the unit interval. This talk is based on joint work with Yuya Kodama (Tokyo Metropolitan University).

January 16, 17:00-18:00 -- Room 056 with live streaming on Zoom

Jin Miyazawa (The University of Tokyo)

A gauge theoretic invariant of embedded surfaces in 4-manifolds and exotic P2-knots

Abstract: When two embeddings of surfaces on a 4-dimensional manifold are given, if they are topologically isotopic but not smoothly isotopic, we call them a pair of exotic surfaces. While there is a great deal of study of exotic surfaces in 4-manifolds, studies of closed exotic surfaces in S4 are limited. In particular, the existence of orientable exotic surfaces in S4 remains unknown to date. There are some examples of non-orientable exotic surfaces in S4, including the initial example given by Finashin-Kreck-Viro in 1988, but all such cases have genus greater than or equal to 5. The difficulty in detecting exotic surfaces in S4 is to prove that two embeddings of surfaces are not smoothly isotopic. All examples of exotic non-orientable surfaces in S4 have been detected by proving the 4-manifolds obtained by the double branched covers are exotic. If we attempt to apply this technique to low-genus non-orientable surfaces in S4, we have to discover exotic small 4-manifolds, which is known to be difficult. In this seminar, we construct an invariant for embedded surfaces in 4-manifolds using Real Seiberg-Witten theory. As an application, we give an infinite family of exotic embeddings into S4 for the real projective plane.

January 23, 17:00-18:00 -- Room 056 with live streaming on Zoom

Gefei Wang (The University of Tokyo)

On the rational cohomology of spin hyperelliptic mapping class groups

Abstract: Let G be the subgroup Sn|q × Sq of the n-th symmetric group Sn for n-q ≥ q. In this talk, we study the G-invariant part of the rational cohomology group of the pure braid group Pn. The invariant part H*(Pn)G includes the rational cohomology of a spin hyperelliptic mapping class group of genus g as a subalgebra when n=2g+2. Based on the study of Lehrer-Solomon, we prove that they are independent of n and q in degree * ≤ q-1. We also give a formula to calculate the dimension of H*(Pn)G and calculate it in all degree for q ≤ 3.

February 13, 17:00-18:30 -- Room 056 with live streaming on Zoom

Paul Norbury (The University of Melbourne)

Measures on the moduli space of curves and super volumes

Abstract: In this lecture I will define a family of finite measures on the moduli space of smooth curves with marked points. The measures are defined via a construction analogous to that of the Weil-Petersson metric using the extra data of a spin structure. In fact, the measures arise naturally out of the super Weil-Petersson metric defined over the moduli space of super curves. The total measure can be identified with the volume of the moduli space of super curves. It can be calculated in many examples, and conjecturally satisfies a recursion analogous to Mirzakhani's recursion relations between Weil-Petersson volumes of moduli spaces of hyperbolic surfaces. This conjecture has been verified in many cases, including the so-called Neveu-Schwarz case where it coincides with the recursion of Stanford and Witten. The general case produces deformations of the Neveu-Schwarz volume polynomials, satisfying the same Mirzakhani-like recursion relations.