## Tuesday Seminar on Topology

Seminar information archive ～10/04｜Next seminar｜Future seminars 10/05～

Date, time & place | Tuesday 17:00 - 18:30 056Room #056 (Graduate School of Math. Sci. Bldg.) |
---|---|

Organizer(s) | KAWAZUMI Nariya, KITAYAMA Takahiro, SAKASAI Takuya |

**Seminar information archive**

### 2023/07/04

17:00-18:30 Room #ハイブリッド開催/056 (Graduate School of Math. Sci. Bldg.)

Pre-registration required. See our seminar webpage.

Reciprocity of the Chern-Simons invariants of 3-manifolds (JAPANESE)

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

Pre-registration required. See our seminar webpage.

**Takefumi Nosaka**(Tokyo Institute of Technology)Reciprocity of the Chern-Simons invariants of 3-manifolds (JAPANESE)

[ Abstract ]

Given an oriented closed 3-manifold $M$ and a representation $\pi_1(M) \longrightarrow SL_2(\mathbb{C})$, we can define the Chern-Simons invariant and adjoint Reidemeister torsion. Recently, several physicists and topologists pose and study reciprocity conjectures of the torsions. Analogously, I pose reciprocity conjectures of the Chern-Simons invariants of 3-manifolds, and argue some supporting evidence on the conjectures. Especially, I show that the conjectures hold if Galois descent of a certain $K_3$-group is satisfied. In this talk, I will explain the backgrounds and the results in detail.

[ Reference URL ]Given an oriented closed 3-manifold $M$ and a representation $\pi_1(M) \longrightarrow SL_2(\mathbb{C})$, we can define the Chern-Simons invariant and adjoint Reidemeister torsion. Recently, several physicists and topologists pose and study reciprocity conjectures of the torsions. Analogously, I pose reciprocity conjectures of the Chern-Simons invariants of 3-manifolds, and argue some supporting evidence on the conjectures. Especially, I show that the conjectures hold if Galois descent of a certain $K_3$-group is satisfied. In this talk, I will explain the backgrounds and the results in detail.

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

### 2023/06/20

17:00-18:30 Room #ハイブリッド開催/056 (Graduate School of Math. Sci. Bldg.)

Pre-registration required. See our seminar webpage.

Moduli spaces of triangle chains (ENGLISH)

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

Pre-registration required. See our seminar webpage.

**Arnaud Maret**(Sorbonne Université)Moduli spaces of triangle chains (ENGLISH)

[ Abstract ]

In this talk, I will describe a moduli space of triangle chains in the hyperbolic plane with prescribed angles. We will relate this moduli space to a specific character variety of representations of surface groups into PSL(2,R). This identification provides action-angle coordinates for the Goldman symplectic form on the character variety. If time permits, I will explain why the mapping class group action on that particular character variety is ergodic.

[ Reference URL ]In this talk, I will describe a moduli space of triangle chains in the hyperbolic plane with prescribed angles. We will relate this moduli space to a specific character variety of representations of surface groups into PSL(2,R). This identification provides action-angle coordinates for the Goldman symplectic form on the character variety. If time permits, I will explain why the mapping class group action on that particular character variety is ergodic.

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

### 2023/06/13

17:00-18:00 Online

Pre-registration required. See our seminar webpage.

On a lower bound of the number of integers in Littlewood's conjecture (JAPANESE)

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

Pre-registration required. See our seminar webpage.

**Shunsuke Usuki**(Kyoto University)On a lower bound of the number of integers in Littlewood's conjecture (JAPANESE)

[ Abstract ]

Littlewood's conjecture is a famous and long-standing open problem on simultaneous Diophantine approximation. It is closely related to the action of diagonal matrices on ${\rm SL}(3,\mathbb{R})/{\rm SL}(3,\mathbb{Z})$, and M. Einsiedler, A. Katok and E. Lindenstrauss showed in 2000's that the exceptional set for Littlewood's conjecture has Hausdorff dimension zero by using some rigidity for invariant measures under the diagonal action. In this talk, I explain that we can obtain some quantitative result on the result of Einsiedler, Katok and Lindenstrauss by studying the empirical measures with respect to the diagonal action.

[ Reference URL ]Littlewood's conjecture is a famous and long-standing open problem on simultaneous Diophantine approximation. It is closely related to the action of diagonal matrices on ${\rm SL}(3,\mathbb{R})/{\rm SL}(3,\mathbb{Z})$, and M. Einsiedler, A. Katok and E. Lindenstrauss showed in 2000's that the exceptional set for Littlewood's conjecture has Hausdorff dimension zero by using some rigidity for invariant measures under the diagonal action. In this talk, I explain that we can obtain some quantitative result on the result of Einsiedler, Katok and Lindenstrauss by studying the empirical measures with respect to the diagonal action.

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

### 2023/06/06

17:30-18:30 Room #ハイブリッド開催/056 (Graduate School of Math. Sci. Bldg.)

Joint with Lie Groups and Representation Theory Seminar. See our seminar webpage.

Visible actions on reductive spherical homogeneous spaces and their invariant measures (JAPANESE)

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

Joint with Lie Groups and Representation Theory Seminar. See our seminar webpage.

**Atsumu Sasaki**(Tokai University)Visible actions on reductive spherical homogeneous spaces and their invariant measures (JAPANESE)

[ Abstract ]

Toshiyuki Kobayashi has established propagation theorem of multiplicity-freeness property. This theorem enables us to give an unified explanation of multiplicity-freeness of multiplicity-free representations which have been found so far, and also to find new examples of multiplicity-free representations systematically. Kobayashi further has introduced the notion of visible actions on complex manifolds as a basic condition for propagation theorem of multiplicity-freeness property. This notion plays an important role to this theorem and also brings us to find various decomposition theorems of Lie groups and homogeneous spaces.

In this talk, we explain visible actions on reductive spherical homogeneous spaces. In particular, we see that for a visible action on reductive spherical homogeneous space our construction of a submanifold which meets every orbit (called a slice) is given by an explicit description of a Cartan decomposition for this space. As a corollary of this study, we characterize the invariant measure on a reductive spherical homogeneous space by giving an integral formula for a Cartan decomposition explicitly.

[ Reference URL ]Toshiyuki Kobayashi has established propagation theorem of multiplicity-freeness property. This theorem enables us to give an unified explanation of multiplicity-freeness of multiplicity-free representations which have been found so far, and also to find new examples of multiplicity-free representations systematically. Kobayashi further has introduced the notion of visible actions on complex manifolds as a basic condition for propagation theorem of multiplicity-freeness property. This notion plays an important role to this theorem and also brings us to find various decomposition theorems of Lie groups and homogeneous spaces.

In this talk, we explain visible actions on reductive spherical homogeneous spaces. In particular, we see that for a visible action on reductive spherical homogeneous space our construction of a submanifold which meets every orbit (called a slice) is given by an explicit description of a Cartan decomposition for this space. As a corollary of this study, we characterize the invariant measure on a reductive spherical homogeneous space by giving an integral formula for a Cartan decomposition explicitly.

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

### 2023/05/30

17:00-18:30 Room #ハイブリッド開催/056 (Graduate School of Math. Sci. Bldg.)

Pre-registration required. See our seminar webpage.

p-colorable subgroup of Thompson's group F (JAPANESE)

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

Pre-registration required. See our seminar webpage.

**Yuya Kodama**(Tokyo Metropolitan University)p-colorable subgroup of Thompson's group F (JAPANESE)

[ Abstract ]

Thompson's group F is a subgroup of Homeo([0, 1]). In 2017, Jones found a way to construct knots and links from elements in F. Moreover, any knot (or link) can be obtained in this way. So the next question is, which elements in F give the same knot (or link)? In this talk, I define a subgroup of F and show that every element (except the identity) gives a p-colorable knot (or link). When p=3, this gives a negative answer to a question by Aiello. This is a joint work with Akihiro Takano.

[ Reference URL ]Thompson's group F is a subgroup of Homeo([0, 1]). In 2017, Jones found a way to construct knots and links from elements in F. Moreover, any knot (or link) can be obtained in this way. So the next question is, which elements in F give the same knot (or link)? In this talk, I define a subgroup of F and show that every element (except the identity) gives a p-colorable knot (or link). When p=3, this gives a negative answer to a question by Aiello. This is a joint work with Akihiro Takano.

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

### 2023/05/16

17:00-18:30 Room #ハイブリッド開催/056 (Graduate School of Math. Sci. Bldg.)

Pre-registration required. See our seminar webpage.

Anderson self-duality of topological modular forms and heretoric string theory (JAPANESE)

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

Pre-registration required. See our seminar webpage.

**Mayuko Yamashita**(Kyoto University)Anderson self-duality of topological modular forms and heretoric string theory (JAPANESE)

[ Abstract ]

Topological Modular Forms (TMF) is an E-infinity ring spectrum which is conjectured by Stolz-Teichner to classify two-dimensional supersymmetric quantum field theories in physics. In the previous work with Y. Tachikawa, we proved the vanishing of anomalies in heterotic string theory mathematically by using TMF. In this talk, I explain our recent update on the previous work. Because of the vanishing result, we can consider a secondary transformation of spectra, which is shown to coincide with the Anderson self-duality morphism of TMF. This allows us to detect subtle torsion phenomena in TMF by differential-geometric ways.

[ Reference URL ]Topological Modular Forms (TMF) is an E-infinity ring spectrum which is conjectured by Stolz-Teichner to classify two-dimensional supersymmetric quantum field theories in physics. In the previous work with Y. Tachikawa, we proved the vanishing of anomalies in heterotic string theory mathematically by using TMF. In this talk, I explain our recent update on the previous work. Because of the vanishing result, we can consider a secondary transformation of spectra, which is shown to coincide with the Anderson self-duality morphism of TMF. This allows us to detect subtle torsion phenomena in TMF by differential-geometric ways.

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

### 2023/05/09

17:00-18:00 Online

Pre-registration required. See our seminar webpage.

Knots and frieze patterns (JAPANESE)

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

Pre-registration required. See our seminar webpage.

**Michihisa Wakui**(Kansai University)Knots and frieze patterns (JAPANESE)

[ Abstract ]

(joint work with Prof. Takeyoshi Kogiso (Josai University)) In the early 1970s, Conway and Coxeter introduced frieze patterns of positive integers arranged under the unimodular rule ad-bc=1, and showed that they are classified by triangulations of convex polygons. Currently, the frieze patterns by Conway and Coxeter are spotlighted in connection with cluster algebras which are introduced by Fomin and Zelevinsky in the early 2000s.

Working with Takeyoshi Kogiso in Josai University the speaker study on relationship between rational links and Conway-Coxeter friezes through ancestor triangles of rational numbers introduced by Shuji Yamada in Kyoto Sangyo University, and show that rational links are characterized by Conway-Coxeter friezes of zigzag type. At nearly the same time Morier-Genoud and Ovsienko also introduce the concept of q-deformation of rational numbers based on continued fraction expansions, and derive closely related results to our research. In this seminar we will talk about an outline of these results.

[ Reference URL ](joint work with Prof. Takeyoshi Kogiso (Josai University)) In the early 1970s, Conway and Coxeter introduced frieze patterns of positive integers arranged under the unimodular rule ad-bc=1, and showed that they are classified by triangulations of convex polygons. Currently, the frieze patterns by Conway and Coxeter are spotlighted in connection with cluster algebras which are introduced by Fomin and Zelevinsky in the early 2000s.

Working with Takeyoshi Kogiso in Josai University the speaker study on relationship between rational links and Conway-Coxeter friezes through ancestor triangles of rational numbers introduced by Shuji Yamada in Kyoto Sangyo University, and show that rational links are characterized by Conway-Coxeter friezes of zigzag type. At nearly the same time Morier-Genoud and Ovsienko also introduce the concept of q-deformation of rational numbers based on continued fraction expansions, and derive closely related results to our research. In this seminar we will talk about an outline of these results.

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

### 2023/04/25

17:00-18:00 Online

Pre-registration required. See our seminar webpage.

Harmonic measures and rigidity of surface group actions on the circle (JAPANESE)

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

Pre-registration required. See our seminar webpage.

**Hiraku Nozawa**(Ritsumeikan University)Harmonic measures and rigidity of surface group actions on the circle (JAPANESE)

[ Abstract ]

We study rigidity properties of surface group actions on the circle via harmonic measures on the suspension bundles, which are measures invariant under the heat diffusion along leaves. We will explain a curvature estimate and a Gauss-Bonnet formula for an S^1-connection obtained by taking the average of the flat connection on the suspension bundle with respect to a harmonic measure. As consequences, we give a precise description of the harmonic measure on suspension foliations with maximal Euler number and an alternative proof of semiconjugacy rigidity theorems of Matsumoto and Burger-Iozzi-Wienhard for actions with maximal Euler number. This is joint work with Masanori Adachi and Yoshifumi Matsuda.

[ Reference URL ]We study rigidity properties of surface group actions on the circle via harmonic measures on the suspension bundles, which are measures invariant under the heat diffusion along leaves. We will explain a curvature estimate and a Gauss-Bonnet formula for an S^1-connection obtained by taking the average of the flat connection on the suspension bundle with respect to a harmonic measure. As consequences, we give a precise description of the harmonic measure on suspension foliations with maximal Euler number and an alternative proof of semiconjugacy rigidity theorems of Matsumoto and Burger-Iozzi-Wienhard for actions with maximal Euler number. This is joint work with Masanori Adachi and Yoshifumi Matsuda.

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

### 2023/04/18

17:00-18:00 Online

Pre-registration required. See our seminar webpage.

A crossed homomorphism on a big mapping class group (JAPANESE)

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

Pre-registration required. See our seminar webpage.

**Shuhei Maruyama**(Chuo University)A crossed homomorphism on a big mapping class group (JAPANESE)

[ Abstract ]

Big mapping class groups are mapping class groups of surfaces of infinite type. Calegari and Chen determined the second (co)homology group of the mapping class group of the sphere minus a Cantor set. They also raised related questions: one of the questions asks an explicit form of certain crossed homomorphisms on the big mapping class group. In this talk, we provide a construction of crossed homomorphisms via group actions on the circle, which answers the question of Calegari and Chen.

[ Reference URL ]Big mapping class groups are mapping class groups of surfaces of infinite type. Calegari and Chen determined the second (co)homology group of the mapping class group of the sphere minus a Cantor set. They also raised related questions: one of the questions asks an explicit form of certain crossed homomorphisms on the big mapping class group. In this talk, we provide a construction of crossed homomorphisms via group actions on the circle, which answers the question of Calegari and Chen.

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

### 2023/04/11

17:00-18:30 Room #ハイブリッド開催/056 (Graduate School of Math. Sci. Bldg.)

Pre-registration required. See our seminar webpage.

On the stable cohomology of the (IA-)automorphism groups of free groups (JAPANESE)

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

Pre-registration required. See our seminar webpage.

**Kazuo Habiro**(The Univesity of Tokyo)On the stable cohomology of the (IA-)automorphism groups of free groups (JAPANESE)

[ Abstract ]

By combining Borel's stability and vanishing theorem for the stable cohomology of GL(n,Z) with coefficients in algebraic GL(n,Z)-representations and the Hochschild-Serre spectral sequence, we compute the twisted first cohomology of the automorphism group Aut(F_n) of the free group F_n of rank n. This method is used also in the study of the stable rational cohomology of the IA-automorphism group IA_n of F_n. We propose a conjectural algebraic structure of the stable rational cohomology of IA_n, and consider some relations to known results and conjectures. We also consider a conjectural structure of the stable rational cohomology of the Torelli groups of surfaces. This is a joint work with Mai Katada.

[ Reference URL ]By combining Borel's stability and vanishing theorem for the stable cohomology of GL(n,Z) with coefficients in algebraic GL(n,Z)-representations and the Hochschild-Serre spectral sequence, we compute the twisted first cohomology of the automorphism group Aut(F_n) of the free group F_n of rank n. This method is used also in the study of the stable rational cohomology of the IA-automorphism group IA_n of F_n. We propose a conjectural algebraic structure of the stable rational cohomology of IA_n, and consider some relations to known results and conjectures. We also consider a conjectural structure of the stable rational cohomology of the Torelli groups of surfaces. This is a joint work with Mai Katada.

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

### 2023/01/17

17:00-18:00 Online

Pre-registration required. See our seminar webpage.

Integral structures in the local algebra of a singularity (ENGLISH)

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

Pre-registration required. See our seminar webpage.

**Chenghan Zha**(The Univesity of Tokyo)Integral structures in the local algebra of a singularity (ENGLISH)

[ Abstract ]

We compute the image of the Milnor lattice of an ADE singularity under a period map. We also prove that the Milnor lattice can be identified with an appropriate relative K-group defined through the Berglund-Huebsch dual of the corresponding singularity. Furthermore, we figure out the image of the Milnor lattice of the singularity of an invertible polynomial of chain type using the basis of middle homology constructed by Otani-Takahashi. We calculated the Seifert form of the basis as well.

[ Reference URL ]We compute the image of the Milnor lattice of an ADE singularity under a period map. We also prove that the Milnor lattice can be identified with an appropriate relative K-group defined through the Berglund-Huebsch dual of the corresponding singularity. Furthermore, we figure out the image of the Milnor lattice of the singularity of an invertible polynomial of chain type using the basis of middle homology constructed by Otani-Takahashi. We calculated the Seifert form of the basis as well.

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

### 2023/01/10

17:00-18:00 Online

Pre-registration required. See our seminar webpage.

Some calculations of an earthquake map in the cross ratio coordinates and the earthquake theorem of cluster algebras of finite type (JAPANESE)

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

Pre-registration required. See our seminar webpage.

**Takeru Asaka**(The Univesity of Tokyo)Some calculations of an earthquake map in the cross ratio coordinates and the earthquake theorem of cluster algebras of finite type (JAPANESE)

[ Abstract ]

Thurston defined an earthquake, which cuts the Poincaré half-plane model and shifts it. Though it is a discontinuous bijective map, it can be extended to a homeomorphism of a circumference. Also, if an earthquake is equivalent relative to a Fuchsian group, the homeomorphism is equivalent, too. Moreover, Thurston proved the earthquake theorem saying that there uniquely exists an earthquake for any orient-preserving homeomorphism of a circumference, and Bonsante-Krasnov-Schlenker extended it to the case of marked surfaces. We calculate some earthquake maps by the cross ratio coordinates. The cross ratio coordinates are deeply related by the cluster algebra (Fock-Goncharov). We prove the earthquake theorem of cluster algebras of finite type. It is a joint work with Tsukasa Ishibashi and Shunsuke Kano.

[ Reference URL ]Thurston defined an earthquake, which cuts the Poincaré half-plane model and shifts it. Though it is a discontinuous bijective map, it can be extended to a homeomorphism of a circumference. Also, if an earthquake is equivalent relative to a Fuchsian group, the homeomorphism is equivalent, too. Moreover, Thurston proved the earthquake theorem saying that there uniquely exists an earthquake for any orient-preserving homeomorphism of a circumference, and Bonsante-Krasnov-Schlenker extended it to the case of marked surfaces. We calculate some earthquake maps by the cross ratio coordinates. The cross ratio coordinates are deeply related by the cluster algebra (Fock-Goncharov). We prove the earthquake theorem of cluster algebras of finite type. It is a joint work with Tsukasa Ishibashi and Shunsuke Kano.

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

### 2022/12/13

17:30-18:30 Online

Pre-registration required. See our seminar webpage.

Spectral convergence in geometric quantization on K3 surfaces (JAPANESE)

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

Pre-registration required. See our seminar webpage.

**Kota Hattori**(Keio University)Spectral convergence in geometric quantization on K3 surfaces (JAPANESE)

[ Abstract ]

In this talk I will explain the geometric quantization on K3 surfaces from the viewpoint of the spectral convergence. We take a special Lagrangian fibrations on the K3 surfaces and a family of hyper-Kähler structures tending to large complex structure limit and show a spectral convergence of the d-bar-Laplacians on the prequantum line bundle to the spectral structure related to the set of Bohr-Sommerfeld fibers.

[ Reference URL ]In this talk I will explain the geometric quantization on K3 surfaces from the viewpoint of the spectral convergence. We take a special Lagrangian fibrations on the K3 surfaces and a family of hyper-Kähler structures tending to large complex structure limit and show a spectral convergence of the d-bar-Laplacians on the prequantum line bundle to the spectral structure related to the set of Bohr-Sommerfeld fibers.

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

### 2022/12/06

17:00-18:00 Online

Pre-registration required. See our seminar webpage.

Torsion in the abelianization of the Johnson kernel (ENGLISH)

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

Pre-registration required. See our seminar webpage.

**Quentin Faes**(The Univesity of Tokyo)Torsion in the abelianization of the Johnson kernel (ENGLISH)

[ Abstract ]

The Johnson kernel is the subgroup of the mapping class group of a closed oriented surface that is generated by Dehn twists along separating simple closed curves, and is also the second term of the so-called Johnson filtration of the mapping class group. The rational abelianization of this group is known, but it was recently proved by Nozaki, Sato and Suzuki, that the abelianization has torsion. They used the LMO homomorphism. In this talk, I will explain a purely two-dimensional proof of this result, which provides a lower bound for the cardinality of the torsion part of the abelianization. These results are also valid for the case of an open surface. This is joint work with Gwénaël Massuyeau.

[ Reference URL ]The Johnson kernel is the subgroup of the mapping class group of a closed oriented surface that is generated by Dehn twists along separating simple closed curves, and is also the second term of the so-called Johnson filtration of the mapping class group. The rational abelianization of this group is known, but it was recently proved by Nozaki, Sato and Suzuki, that the abelianization has torsion. They used the LMO homomorphism. In this talk, I will explain a purely two-dimensional proof of this result, which provides a lower bound for the cardinality of the torsion part of the abelianization. These results are also valid for the case of an open surface. This is joint work with Gwénaël Massuyeau.

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

### 2022/11/29

17:00-18:00 Online

Pre-registration required. See our seminar webpage.

GKM graph with legs and graph equivariant cohomology (JAPANESE)

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

Pre-registration required. See our seminar webpage.

**Shintaro Kuroki**(Okayama University of Science)GKM graph with legs and graph equivariant cohomology (JAPANESE)

[ Abstract ]

A GKM (Goresky-Kottiwicz-MacPherson) graph is a regular graph labeled on edges with some conditions. This notion has been introduced by Guillemin-Zara in 2001 to study the geometry of a nice class of manifolds with torus actions, called a GKM manifold, by a combinatorial way. In particular, we can define a ring on a GKM graph called a graph equivariant cohomology, and it is often isomorphic to the equivariant cohomology of a GKM manifold. In this talk, we introduce the GKM graph with legs (i.e., non-compact edges) related to non-compact manifolds with torus actions that may not satisfy the usual GKM conditions, and study the graph equivariant cohomology which is related to the GKM graph with legs. The talk is mainly based on the joint work with Grigory Solomadin (arXiv:2207.11380) and partially on the joint work with Vikraman Uma (arXiv:2106.11598).

[ Reference URL ]A GKM (Goresky-Kottiwicz-MacPherson) graph is a regular graph labeled on edges with some conditions. This notion has been introduced by Guillemin-Zara in 2001 to study the geometry of a nice class of manifolds with torus actions, called a GKM manifold, by a combinatorial way. In particular, we can define a ring on a GKM graph called a graph equivariant cohomology, and it is often isomorphic to the equivariant cohomology of a GKM manifold. In this talk, we introduce the GKM graph with legs (i.e., non-compact edges) related to non-compact manifolds with torus actions that may not satisfy the usual GKM conditions, and study the graph equivariant cohomology which is related to the GKM graph with legs. The talk is mainly based on the joint work with Grigory Solomadin (arXiv:2207.11380) and partially on the joint work with Vikraman Uma (arXiv:2106.11598).

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

### 2022/11/22

17:00-18:00 Online

Pre-registration required. See our seminar webpage.

Epimorphism between knot groups and isomorphisms on character varieties (JAPANESE)

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

Pre-registration required. See our seminar webpage.

**Teruaki Kitano**(Soka University)Epimorphism between knot groups and isomorphisms on character varieties (JAPANESE)

[ Abstract ]

A partial order on the set of prime knots is given by the existence of an epimorphism between the fundamental groups of the knot complements. In this talk we will survey some basic properties of this order, and discuss some results and questions in connection with the SL(2,C)-character variety. In particular we study to what extend the SL(2,C)-character variety to determine the knot. This talk will be based on joint works with Michel Boileau(Univ. Aix-Marseille), Steven Sivek(Imperial College London), and Raphael Zentner(Univ. Regensburg).

[ Reference URL ]A partial order on the set of prime knots is given by the existence of an epimorphism between the fundamental groups of the knot complements. In this talk we will survey some basic properties of this order, and discuss some results and questions in connection with the SL(2,C)-character variety. In particular we study to what extend the SL(2,C)-character variety to determine the knot. This talk will be based on joint works with Michel Boileau(Univ. Aix-Marseille), Steven Sivek(Imperial College London), and Raphael Zentner(Univ. Regensburg).

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

### 2022/11/15

17:00-18:00 Online

Pre-registration required. See our seminar webpage.

Stable cohomology of mapping class groups with some particular twisted contravariant coefficients (ENGLISH)

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

Pre-registration required. See our seminar webpage.

**Arthur Soulié**(IBS Center for Geometry and Physics, POSTECH)Stable cohomology of mapping class groups with some particular twisted contravariant coefficients (ENGLISH)

[ Abstract ]

The twisted cohomology of mapping class groups of compact orientable surfaces (with one boundary) is very difficult to compute generally speaking. In this talk, I will describe the computation of the stable cohomology algebra of these mapping class groups with twisted coefficients given by the first homology of the unit tangent bundle of the surface. This type of computation is out of the scope of the traditional framework for homological stability. Indeed, these twisted coefficients define a contravariant functor over the classical category associated to mapping class groups to study homological stability, rather than a covariant one. I will also present the computation of the stable cohomology algebras with with twisted coefficients given by the exterior powers and tensor powers of the first homology of the unit tangent bundle of the surface. All this represents a joint work with Nariya Kawazumi.

[ Reference URL ]The twisted cohomology of mapping class groups of compact orientable surfaces (with one boundary) is very difficult to compute generally speaking. In this talk, I will describe the computation of the stable cohomology algebra of these mapping class groups with twisted coefficients given by the first homology of the unit tangent bundle of the surface. This type of computation is out of the scope of the traditional framework for homological stability. Indeed, these twisted coefficients define a contravariant functor over the classical category associated to mapping class groups to study homological stability, rather than a covariant one. I will also present the computation of the stable cohomology algebras with with twisted coefficients given by the exterior powers and tensor powers of the first homology of the unit tangent bundle of the surface. All this represents a joint work with Nariya Kawazumi.

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

### 2022/11/08

17:00-18:00 Online

Pre-registration required. See our seminar webpage.

Milnor fibers of hyperplane arrangements (JAPANESE)

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

Pre-registration required. See our seminar webpage.

**Masahiko Yoshinaga**(Osaka University)Milnor fibers of hyperplane arrangements (JAPANESE)

[ Abstract ]

A (central) hyperplane arrangement is a union of finitely many hyperplanes in a linear space. There are many relationships between the intersection lattice of the arrangement and geometry of related spaces. In this talk, we focus on the Milnor fiber of a hyperplane arrangement. The first Betti number of the Milnor fiber is expected to be determined by the combinatorial structure of the intersection lattice, however, it is still open. We discuss two results on the problem. The first (discouraging) one is concerning the dimension of (-1)-eigenspace of the monodromy action on the first cohomology group. We show that it is related to 2-torsions in the first homology of double covering of the (projectivized) complement (j.w. Ishibashi and Sugawara). The second (encouraging) one is related to the Aomoto complex, which is defined in purely combinatorial way. We show that a q-analogue of Aomoto complex determines all nontrivial monodromy eigenspaces of the Milnor fiber.

[ Reference URL ]A (central) hyperplane arrangement is a union of finitely many hyperplanes in a linear space. There are many relationships between the intersection lattice of the arrangement and geometry of related spaces. In this talk, we focus on the Milnor fiber of a hyperplane arrangement. The first Betti number of the Milnor fiber is expected to be determined by the combinatorial structure of the intersection lattice, however, it is still open. We discuss two results on the problem. The first (discouraging) one is concerning the dimension of (-1)-eigenspace of the monodromy action on the first cohomology group. We show that it is related to 2-torsions in the first homology of double covering of the (projectivized) complement (j.w. Ishibashi and Sugawara). The second (encouraging) one is related to the Aomoto complex, which is defined in purely combinatorial way. We show that a q-analogue of Aomoto complex determines all nontrivial monodromy eigenspaces of the Milnor fiber.

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

### 2022/11/01

17:00-18:00 Online

Pre-registration required. See our seminar webpage.

An obstruction problem associated with finite path-integral (JAPANESE)

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

Pre-registration required. See our seminar webpage.

**Minkyu Kim**(The Univesity of Tokyo)An obstruction problem associated with finite path-integral (JAPANESE)

[ Abstract ]

Finite path-integral introduced by Dijkgraaf and Witten in 1990 is a mathematical methodology to construct an Atiyah-Segal type TQFT from finite gauge theory. In three dimensions, it is generalized to Hopf algebra gauge theory of Meusburger, and the corresponding TQFT is known as Turaev-Viro model. On the one hand, the bicommutative Hopf algebra gauge theory is covered by homological algebra. In this talk, we will explain an obstruction problem associated with a refined finite path-integral construction of TQFT's from homological algebra. This talk is based on our study of a folklore claim in condensed matter physics that gapped lattice quantum system, e.g. toric code, is approximated by topological field theories in low temperature.

[ Reference URL ]Finite path-integral introduced by Dijkgraaf and Witten in 1990 is a mathematical methodology to construct an Atiyah-Segal type TQFT from finite gauge theory. In three dimensions, it is generalized to Hopf algebra gauge theory of Meusburger, and the corresponding TQFT is known as Turaev-Viro model. On the one hand, the bicommutative Hopf algebra gauge theory is covered by homological algebra. In this talk, we will explain an obstruction problem associated with a refined finite path-integral construction of TQFT's from homological algebra. This talk is based on our study of a folklore claim in condensed matter physics that gapped lattice quantum system, e.g. toric code, is approximated by topological field theories in low temperature.

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

### 2022/10/25

17:00-18:00 Online

Pre-registration required. See our seminar webpage.

Stabilized convex symplectic manifolds are Weinstein (JAPANESE)

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

Pre-registration required. See our seminar webpage.

**Noboru Ogawa**(Tokai University)Stabilized convex symplectic manifolds are Weinstein (JAPANESE)

[ Abstract ]

There are two important classes of convexity in symplectic geometry: Liouville and Weinstein structures. Basic objects such as cotangent bundles and Stein manifolds have these structures. In 90s, Eliashberg and Gromov formulated them as symplectic counterparts of Stein manifolds, since then, they have played a significant role in the study of symplectic topology. By definition, a Weinstein structure is a Liouville structure, but the converse is not true in general; McDuff gave the first example which is a Liouville manifold without any Weinstein structures. The purpose of this talk is to present the recent advances on the difference of both structures, up to homotopy. In particular, I will show that the stabilization of the McDuff’s example admits a flexible Weinstein structure. The main part is based on a joint work with Yakov Eliashberg (Stanford University) and Toru Yoshiyasu (Kyoto University of Education). If time permits, I would like to discuss some open questions and progress.

[ Reference URL ]There are two important classes of convexity in symplectic geometry: Liouville and Weinstein structures. Basic objects such as cotangent bundles and Stein manifolds have these structures. In 90s, Eliashberg and Gromov formulated them as symplectic counterparts of Stein manifolds, since then, they have played a significant role in the study of symplectic topology. By definition, a Weinstein structure is a Liouville structure, but the converse is not true in general; McDuff gave the first example which is a Liouville manifold without any Weinstein structures. The purpose of this talk is to present the recent advances on the difference of both structures, up to homotopy. In particular, I will show that the stabilization of the McDuff’s example admits a flexible Weinstein structure. The main part is based on a joint work with Yakov Eliashberg (Stanford University) and Toru Yoshiyasu (Kyoto University of Education). If time permits, I would like to discuss some open questions and progress.

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

### 2022/10/11

17:00-18:00 Online

Pre-registration required. See our seminar webpage.

Magnitude homology of graphs (JAPANESE)

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

Pre-registration required. See our seminar webpage.

**Yasuhiko Asao**(Fukuoka University)Magnitude homology of graphs (JAPANESE)

[ Abstract ]

Magnitude is introduced by Leinster in 00’s as an ``Euler characteristic of metric spaces”. It is defined for the metric structure itself rather than the topology induced from the metric. Magnitude homology is a categorification of magnitude in a sense that ordinary homology categorifies the Euler characteristic. The speaker’s interest is in geometric meaning of this theory. In this talk, after an introduction to basic ideas, I will explain that magnitude truly extends the Euler characteristic. From this perspective, magnitude homology can be seen as one of the categorification of the Euler characteristic, and the path homology (Grigor’yan—Muranov—Lin—S-T. Yau et.al) appears as a part of another one. These structures are aggregated in a spectral sequence obtained from the classifying space of "filtered set enriched categories" which includes ordinary small categories and metric spaces.

[ Reference URL ]Magnitude is introduced by Leinster in 00’s as an ``Euler characteristic of metric spaces”. It is defined for the metric structure itself rather than the topology induced from the metric. Magnitude homology is a categorification of magnitude in a sense that ordinary homology categorifies the Euler characteristic. The speaker’s interest is in geometric meaning of this theory. In this talk, after an introduction to basic ideas, I will explain that magnitude truly extends the Euler characteristic. From this perspective, magnitude homology can be seen as one of the categorification of the Euler characteristic, and the path homology (Grigor’yan—Muranov—Lin—S-T. Yau et.al) appears as a part of another one. These structures are aggregated in a spectral sequence obtained from the classifying space of "filtered set enriched categories" which includes ordinary small categories and metric spaces.

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

### 2022/10/04

17:00-18:30 Online

Pre-registration required. See our seminar webpage.

Orientable rho-Q-manifolds and their modular classes (JAPANESE)

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

Pre-registration required. See our seminar webpage.

**Shuichi Harako**(The Univesity of Tokyo)Orientable rho-Q-manifolds and their modular classes (JAPANESE)

[ Abstract ]

A rho-commutative algebra, or an almost commutative algebra, is a graded algebra whose commutativity is given by a function called a commutation factor. It is one generalization of a commutative algebra or a superalgebra. We obtain a rho-Lie algebra, or an epsilon-Lie algebra, by a similar generalization of a Lie algebra. On the other hand, we have the modular class of an orientable Q-manifold. Here, a Q-manifold is a supermanifold with an odd vector field whose Lie bracket with itself vanishes, and its orientability is described in terms of the Berezinian bundle. In this talk, we introduce the concept of a rho-manifold, which is a graded manifold whose functional algebra is a rho-commutative algebra, then we show that we can define Q-structures, Berezinian bundle, volume forms, and modular classes of a rho-manifold with some examples.

[ Reference URL ]A rho-commutative algebra, or an almost commutative algebra, is a graded algebra whose commutativity is given by a function called a commutation factor. It is one generalization of a commutative algebra or a superalgebra. We obtain a rho-Lie algebra, or an epsilon-Lie algebra, by a similar generalization of a Lie algebra. On the other hand, we have the modular class of an orientable Q-manifold. Here, a Q-manifold is a supermanifold with an odd vector field whose Lie bracket with itself vanishes, and its orientability is described in terms of the Berezinian bundle. In this talk, we introduce the concept of a rho-manifold, which is a graded manifold whose functional algebra is a rho-commutative algebra, then we show that we can define Q-structures, Berezinian bundle, volume forms, and modular classes of a rho-manifold with some examples.

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

### 2022/07/12

17:00-18:00 Online

Pre-registration required. See our seminar webpage.

Cable knots and involutive Heegaard Floer homology (ENGLISH)

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

Pre-registration required. See our seminar webpage.

**Sungkyung Kang**(Center for Geometry and Physics, Institute of Basic Science)Cable knots and involutive Heegaard Floer homology (ENGLISH)

[ Abstract ]

Heegaard Floer homology (and its variants) carries an intrinsic symmetry, which conjecturally corresponds to the Pin(2)-equivariance in Seiberg-Witten Floer homology. By exploiting the symmetry, we prove that (odd,1)-cables of the figure-eight knots are linearly independent in the concordance group of rationally slice knots, and present a first example of rationally slice knots of complexity 1 which are not slice. Furthermore, we establish an explicit connection between involutive knot Floer theory and involutive bordered Floer theory of knot complements, and use it to prove a similar result for iterated cables of figure-eight knots. A part of this talk is based on a joint work with J. Hom, M. Stoffregen, and J. Park.

[ Reference URL ]Heegaard Floer homology (and its variants) carries an intrinsic symmetry, which conjecturally corresponds to the Pin(2)-equivariance in Seiberg-Witten Floer homology. By exploiting the symmetry, we prove that (odd,1)-cables of the figure-eight knots are linearly independent in the concordance group of rationally slice knots, and present a first example of rationally slice knots of complexity 1 which are not slice. Furthermore, we establish an explicit connection between involutive knot Floer theory and involutive bordered Floer theory of knot complements, and use it to prove a similar result for iterated cables of figure-eight knots. A part of this talk is based on a joint work with J. Hom, M. Stoffregen, and J. Park.

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

### 2022/07/05

17:00-18:00 Online

Pre-registration required. See our seminar webpage.

Non-existence of Lyapunov exponents for homoclinic bifurcations of surface diffeomorphisms (JAPANESE)

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

Pre-registration required. See our seminar webpage.

**Yushi Nakano**(Tokai University)Non-existence of Lyapunov exponents for homoclinic bifurcations of surface diffeomorphisms (JAPANESE)

[ Abstract ]

Lyapunov exponent is widely used in natural science including mathematics, such as a tool to find chaotic signal or a foundation of non-uniformly hyperbolic systems theory. However, its existence (outside of the supports of invariant probability measures) is seldom discussed. In this talk, I consider the problem of whether the Lyapunov irregular set, i.e. the set of points at which Lyapunov exponent fails to exist, has positive Lebesgue measure. I will show that surface diffeomorphisms with a robust homoclinic tangency given by Colli and Vargas, as well as other several known nonhyperbolic dynamics, has the Lyapunov irregular set of positive Lebesgue measure. This is a joint work with S. Kiriki, X. Li and T. Soma.

[ Reference URL ]Lyapunov exponent is widely used in natural science including mathematics, such as a tool to find chaotic signal or a foundation of non-uniformly hyperbolic systems theory. However, its existence (outside of the supports of invariant probability measures) is seldom discussed. In this talk, I consider the problem of whether the Lyapunov irregular set, i.e. the set of points at which Lyapunov exponent fails to exist, has positive Lebesgue measure. I will show that surface diffeomorphisms with a robust homoclinic tangency given by Colli and Vargas, as well as other several known nonhyperbolic dynamics, has the Lyapunov irregular set of positive Lebesgue measure. This is a joint work with S. Kiriki, X. Li and T. Soma.

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

### 2022/06/21

17:00-18:00 Online

Pre-registration required. See our seminar webpage.

Cosmetic surgeries on knots in the 3-sphere (JAPANESE)

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

Pre-registration required. See our seminar webpage.

**Kazuhiro Ichihara**(Nihon University)Cosmetic surgeries on knots in the 3-sphere (JAPANESE)

[ Abstract ]

A pair of Dehn surgeries on a knot is called purely (resp. chirally) cosmetic if the obtained manifolds are orientation-preservingly (resp. -reversingly) homeomorphic. It is conjectured that if a knot in the 3-sphere admits purely (resp. chirally) cosmetic surgeries, then the knot is a trivial knot (resp. a torus knot or an amphicheiral knot). In this talk, after giving a brief survey on the studies on these conjectures, I will explain recent progresses on the conjectures. This is based on joint works with Tetsuya Ito (Kyoto University), In Dae Jong (Kindai University), and Toshio Saito (Joetsu University of Education).

[ Reference URL ]A pair of Dehn surgeries on a knot is called purely (resp. chirally) cosmetic if the obtained manifolds are orientation-preservingly (resp. -reversingly) homeomorphic. It is conjectured that if a knot in the 3-sphere admits purely (resp. chirally) cosmetic surgeries, then the knot is a trivial knot (resp. a torus knot or an amphicheiral knot). In this talk, after giving a brief survey on the studies on these conjectures, I will explain recent progresses on the conjectures. This is based on joint works with Tetsuya Ito (Kyoto University), In Dae Jong (Kindai University), and Toshio Saito (Joetsu University of Education).

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html