## Tuesday Seminar on Topology

Seminar information archive ～04/12｜Next seminar｜Future seminars 04/13～

Date, time & place | Tuesday 17:00 - 18:30 056Room #056 (Graduate School of Math. Sci. Bldg.) |
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Organizer(s) | KAWAZUMI Nariya, KITAYAMA Takahiro, SAKASAI Takuya |

**Seminar information archive**

### 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

### 2022/06/14

17:30-18:30 Online

Pre-registration required. See our seminar webpage.

Cartan calculi on the free loop spaces (JAPANESE)

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

Pre-registration required. See our seminar webpage.

**Katsuhiko Kuribayashi**(Shinshu University)Cartan calculi on the free loop spaces (JAPANESE)

[ Abstract ]

A typical example of a Cartan calculus is the Lie algebra representation of vector fields of a manifold on the derivation ring of the de Rham complex. In this talk, a `second stage' of the Cartan calculus is investigated. In a more general setting, the stage is formulated with a Lie algebra representation of the Andre-Quillen cohomology of a commutative differential graded algebra A on the endomorphism ring of the Hochschild homology of A in terms of the homotopy Cartan calculi in the sense of Fiorenza and Kowalzig. Moreover, the Lie algebra representation in the Cartan calculus is interpreted geometrically as a map from the rational homotopy group of the monoid of self-homotopy equivalences on a simply-connected space M to the derivation ring on the loop cohomology of M. An extension of the representation to the string cohomology and its geometric counterpart are also discussed together with the BV exactness which is a new rational homotopy invariant introduced in our work. This talk is based on joint work in progress with T. Naito, S. Wakatsuki and T. Yamaguchi.

[ Reference URL ]A typical example of a Cartan calculus is the Lie algebra representation of vector fields of a manifold on the derivation ring of the de Rham complex. In this talk, a `second stage' of the Cartan calculus is investigated. In a more general setting, the stage is formulated with a Lie algebra representation of the Andre-Quillen cohomology of a commutative differential graded algebra A on the endomorphism ring of the Hochschild homology of A in terms of the homotopy Cartan calculi in the sense of Fiorenza and Kowalzig. Moreover, the Lie algebra representation in the Cartan calculus is interpreted geometrically as a map from the rational homotopy group of the monoid of self-homotopy equivalences on a simply-connected space M to the derivation ring on the loop cohomology of M. An extension of the representation to the string cohomology and its geometric counterpart are also discussed together with the BV exactness which is a new rational homotopy invariant introduced in our work. This talk is based on joint work in progress with T. Naito, S. Wakatsuki and T. Yamaguchi.

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

### 2022/06/07

17:00-18:00 Online

Pre-registration required. See our seminar webpage.

Dynamical zeta functions for geodesic flows and the higher-dimensional Reidemeister torsion for Fuchsian groups (JAPANESE)

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

Pre-registration required. See our seminar webpage.

**早稲田大学**(Waseda University)Dynamical zeta functions for geodesic flows and the higher-dimensional Reidemeister torsion for Fuchsian groups (JAPANESE)

[ Abstract ]

We discuss a relation between a dynamical zeta function defined by the geodesic flow on a 2-dimensional hyperbolic orbifold and the asymptotic behavior of the Reidemeister torsion for the unit tangent bundle over the orbifold. The unit tangent bundle over a hyperbolic orbifold is a Seifert fibered space with a geometric structure given by the universal cover of PSL(2, R). This geometric structure induces an SL(2, R)-representation of the fundamental group. Here the asymptotic behavior of the Reidemeister torsion means the limit of the leading coefficient in the Reidemeister torsion for the unit tangent bundle over a hyperbolic orbifold and the SL(n, R)-representations induced by the SL(2, R)-one of its fundamental group. For a hyperbolic 3-manifold, we can derive the hyperbolic volume from the limit of the leading coefficient in the Reidemeister torsion with a dynamical zeta function according to previous works. For the unit tangent bundle over a 2-dimensional hyperbolic orbifold, which is not a hyperbolic 3-manifold, we can find the orbifold Euler characteristic of the orbifold in the limit of the leading coefficient in the Reidemeister torsion for the unit tangent bundle from the relation with the dynamical zeta function defined by the geodesic flow on the orbifold.

[ Reference URL ]We discuss a relation between a dynamical zeta function defined by the geodesic flow on a 2-dimensional hyperbolic orbifold and the asymptotic behavior of the Reidemeister torsion for the unit tangent bundle over the orbifold. The unit tangent bundle over a hyperbolic orbifold is a Seifert fibered space with a geometric structure given by the universal cover of PSL(2, R). This geometric structure induces an SL(2, R)-representation of the fundamental group. Here the asymptotic behavior of the Reidemeister torsion means the limit of the leading coefficient in the Reidemeister torsion for the unit tangent bundle over a hyperbolic orbifold and the SL(n, R)-representations induced by the SL(2, R)-one of its fundamental group. For a hyperbolic 3-manifold, we can derive the hyperbolic volume from the limit of the leading coefficient in the Reidemeister torsion with a dynamical zeta function according to previous works. For the unit tangent bundle over a 2-dimensional hyperbolic orbifold, which is not a hyperbolic 3-manifold, we can find the orbifold Euler characteristic of the orbifold in the limit of the leading coefficient in the Reidemeister torsion for the unit tangent bundle from the relation with the dynamical zeta function defined by the geodesic flow on the orbifold.

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

### 2022/05/31

17:00-18:00 Online

Pre-registration required. See our seminar webpage.

Stable Fukaya categories of Milnor fibers (JAPANESE)

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

Pre-registration required. See our seminar webpage.

**Kazushi Ueda**(The Univesity of Tokyo)Stable Fukaya categories of Milnor fibers (JAPANESE)

[ Abstract ]

We define the stable Fukaya category of a Liouville domain as the quotient of the wrapped Fukaya category by the full subcategory consisting of compact Lagrangians, and discuss the relation between the stable Fukaya categories of affine Fermat hypersurfaces and the Fukaya categories of projective hypersurfaces. We also discuss homological mirror symmetry for Milnor fibers of Brieskorn-Pham singularities along the way. This is a joint work in progress with Yanki Lekili.

[ Reference URL ]We define the stable Fukaya category of a Liouville domain as the quotient of the wrapped Fukaya category by the full subcategory consisting of compact Lagrangians, and discuss the relation between the stable Fukaya categories of affine Fermat hypersurfaces and the Fukaya categories of projective hypersurfaces. We also discuss homological mirror symmetry for Milnor fibers of Brieskorn-Pham singularities along the way. This is a joint work in progress with Yanki Lekili.

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

### 2022/05/24

17:00-18:00 Online

Pre-registration required. See our seminar webpage.

Polynomial functors associated with beaded open Jacobi diagrams (ENGLISH)

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

Pre-registration required. See our seminar webpage.

**Christine Vespa**(IRMA, Université de Strasbourg / JSPS)Polynomial functors associated with beaded open Jacobi diagrams (ENGLISH)

[ Abstract ]

The Kontsevich integral is a very powerful invariant of knots, taking values is the space of Jacobi diagrams. Using an extension of the Kontsevich integral to tangles in handlebodies, Habiro and Massuyeau construct a functor from the category of bottom tangles in handlebodies to the linear category A of Jacobi diagrams in handlebodies. The category A has a subcategory equivalent to the linearization of the opposite of the category of finitely generated free groups, denoted by $\textbf{gr}^{op}$. By restriction to this subcategory, morphisms in the linear category $\textbf{A}$ give rise to interesting contravariant functors on the category $\textbf{gr}$, encoding part of the composition structure of the category A.

In recent papers, Katada studies the functor given by the morphisms in the category A from 0. In particular, she obtains a family of polynomial functors on $\textbf{gr}^{op}$ which are outer functors, in the sense that inner automorphisms act trivially.

In this talk, I will explain these results and give extensions of Katada’s results concerning the functors given by the morphisms in the category A from any integer k. These functors give rise to families of polynomial functors on $\textbf{gr}^{op}$ which are no more outer functors. Our approach is based on an equivalence of categories given by Powell. Through this equivalence the previous polynomial functors correspond to functors given by beaded open Jacobi diagrams.

[ Reference URL ]The Kontsevich integral is a very powerful invariant of knots, taking values is the space of Jacobi diagrams. Using an extension of the Kontsevich integral to tangles in handlebodies, Habiro and Massuyeau construct a functor from the category of bottom tangles in handlebodies to the linear category A of Jacobi diagrams in handlebodies. The category A has a subcategory equivalent to the linearization of the opposite of the category of finitely generated free groups, denoted by $\textbf{gr}^{op}$. By restriction to this subcategory, morphisms in the linear category $\textbf{A}$ give rise to interesting contravariant functors on the category $\textbf{gr}$, encoding part of the composition structure of the category A.

In recent papers, Katada studies the functor given by the morphisms in the category A from 0. In particular, she obtains a family of polynomial functors on $\textbf{gr}^{op}$ which are outer functors, in the sense that inner automorphisms act trivially.

In this talk, I will explain these results and give extensions of Katada’s results concerning the functors given by the morphisms in the category A from any integer k. These functors give rise to families of polynomial functors on $\textbf{gr}^{op}$ which are no more outer functors. Our approach is based on an equivalence of categories given by Powell. Through this equivalence the previous polynomial functors correspond to functors given by beaded open Jacobi diagrams.

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

### 2022/05/17

17:00-18:00 Online

Pre-registration required. See our seminar webpage.

Contribution of simple loops to the configuration space integral (JAPANESE)

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

Pre-registration required. See our seminar webpage.

**Tatsuro Shimizu**(Tokyo Denki University)Contribution of simple loops to the configuration space integral (JAPANESE)

[ Abstract ]

For a manifold with a representation of the fundamental group, the configuration space integral associated with a graph (Feynman diagram) gives a real number. An appropriate linear combination of graphs gives an invariant of the manifold with the representation. In this talk, we discuss the contribution of simple loops to the configuration space integral. Hutchings, Lee and Kitayama give geometric descriptions of the Reidemeister torsion by using circle valued Morse functions. By using these descriptions and a Morse theoretical description of the configuration space integral, we have equations among the Reidemeister torsion and the contributions of simple loops in some cases. In this talk, we extend the equations for some other cases and give a computational example of the configuration space integrals by using Morse function.

[ Reference URL ]For a manifold with a representation of the fundamental group, the configuration space integral associated with a graph (Feynman diagram) gives a real number. An appropriate linear combination of graphs gives an invariant of the manifold with the representation. In this talk, we discuss the contribution of simple loops to the configuration space integral. Hutchings, Lee and Kitayama give geometric descriptions of the Reidemeister torsion by using circle valued Morse functions. By using these descriptions and a Morse theoretical description of the configuration space integral, we have equations among the Reidemeister torsion and the contributions of simple loops in some cases. In this talk, we extend the equations for some other cases and give a computational example of the configuration space integrals by using Morse function.

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

### 2022/05/10

17:00-18:00 Online

Pre-registration required. See our seminar webpage.

Nielsen realization, knots, and Seiberg-Witten (Floer) homotopy theory (JAPANESE)

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

Pre-registration required. See our seminar webpage.

**Hokuto Konno**(The Univesity of Tokyo)Nielsen realization, knots, and Seiberg-Witten (Floer) homotopy theory (JAPANESE)

[ Abstract ]

I will discuss two different kinds of applications of Seiberg-Witten (Floer) homotopy theory involving involutions. The first application is about the Nielsen realization problem, which asks whether a given finite subgroup of the mapping class group of a manifold lifts to a subgroup of the diffeomorphism group. Although every finite subgroup is known to lift in dimension 2, there are manifolds of dimension greater than 2 for which the Nielsen realization fails. However, only few examples have been known in dimension 4. I will show that "4-dimensional Dehn twists" yield a large class of new examples. The second application is about 4-dimensional invariants of knots. I will introduce a version of "Floer K-theory for knots", and will explain that this framework gives the first comparison result for the smooth and topological versions of a certain knot invariant, called stabilizing number. Although the above two topics (Nielsen realization and knots) may seem to have different flavors, they are derived from a common idea. The first one is proved using a constraint on smooth involutions on a closed 4-manifold from Seiberg-Witten homotopy theory by Yuya Kato, and the second one is derived from a generalization of Kato's result to 4-manifolds with boundary using Seiberg-Witten Floer homotopy theory. This talk is partially based on joint work with Jin Miyazawa and Masaki Taniguchi.

[ Reference URL ]I will discuss two different kinds of applications of Seiberg-Witten (Floer) homotopy theory involving involutions. The first application is about the Nielsen realization problem, which asks whether a given finite subgroup of the mapping class group of a manifold lifts to a subgroup of the diffeomorphism group. Although every finite subgroup is known to lift in dimension 2, there are manifolds of dimension greater than 2 for which the Nielsen realization fails. However, only few examples have been known in dimension 4. I will show that "4-dimensional Dehn twists" yield a large class of new examples. The second application is about 4-dimensional invariants of knots. I will introduce a version of "Floer K-theory for knots", and will explain that this framework gives the first comparison result for the smooth and topological versions of a certain knot invariant, called stabilizing number. Although the above two topics (Nielsen realization and knots) may seem to have different flavors, they are derived from a common idea. The first one is proved using a constraint on smooth involutions on a closed 4-manifold from Seiberg-Witten homotopy theory by Yuya Kato, and the second one is derived from a generalization of Kato's result to 4-manifolds with boundary using Seiberg-Witten Floer homotopy theory. This talk is partially based on joint work with Jin Miyazawa and Masaki Taniguchi.

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

### 2022/04/26

17:00-18:00 Online

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

On the existence of discrete series for homogeneous spaces (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.

**Yoshiki Oshima**(The Univesity of Tokyo)On the existence of discrete series for homogeneous spaces (JAPANESE)

[ Abstract ]

When a Lie group $G$ acts transitively on a manifold $X$, an irreducible subrepresentation of $L^2(X)$ is called a discrete series representation of $X$. One may ask which homogeneous space $X$ has a discrete series representation. For reductive symmetric spaces, it is known that the existence of discrete series is equivalent to a rank condition by works of Flensted-Jensen, T.Matsuki, and T.Oshima. The problem for general reductive homogeneous spaces was considered by T.Kobayashi and a sufficient condition for the existence of discrete series was obtained by using his theory of admissible restriction. In this talk, we would like to see another sufficient condition for general homogeneous spaces and also the case of their line bundles in terms of the orbit method.

[ Reference URL ]When a Lie group $G$ acts transitively on a manifold $X$, an irreducible subrepresentation of $L^2(X)$ is called a discrete series representation of $X$. One may ask which homogeneous space $X$ has a discrete series representation. For reductive symmetric spaces, it is known that the existence of discrete series is equivalent to a rank condition by works of Flensted-Jensen, T.Matsuki, and T.Oshima. The problem for general reductive homogeneous spaces was considered by T.Kobayashi and a sufficient condition for the existence of discrete series was obtained by using his theory of admissible restriction. In this talk, we would like to see another sufficient condition for general homogeneous spaces and also the case of their line bundles in terms of the orbit method.

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

### 2022/04/19

17:30-18:30 Online

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

On the classification and construction of conformal symmetry breaking operators for anti-de Sitter spaces (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.

**Toshihisa Kubo**(Ryukoku University)On the classification and construction of conformal symmetry breaking operators for anti-de Sitter spaces (JAPANESE)

[ Abstract ]

Let $X$ be a smooth manifold and $Y$ a smooth submanifold of $X$. Take $G' \subset G$ to be a pair of Lie groups that act on $Y \subset X$, respectively. Consider a $G'$-intertwining differential operator $\mathcal{D}$ from the space of smooth sections for a $G$-equivariant vector bundle over $X$ to that for a $G'$-equivariant vector bundle over $Y$. Toshiyuki Kobayashi called such a differential operator $\mathcal{D}$ a

In [Kobayashi-K-Pevzner, Lecture Notes in Math. 2170 (2016)], we explicitly constructed and classified all the differential SBOs from the space of differential $i$-forms $\mathcal{E}^i(S^n)$ over the standard Riemann sphere $S^n$ to that of differential $j$-forms $\mathcal{E}^j(S^{n-1})$ over the totally geodesic hypersphere $S^{n-1}$. In this talk, by extending the results in a Riemannian setting, we discuss about the classification and construction of differential SBOs in a pseudo-Riemannian setting such as anti-de Sitter spaces and hyperbolic spaces. This is a joint work with Toshiyuki Kobayashi and Michael Pevzner.

[ Reference URL ]Let $X$ be a smooth manifold and $Y$ a smooth submanifold of $X$. Take $G' \subset G$ to be a pair of Lie groups that act on $Y \subset X$, respectively. Consider a $G'$-intertwining differential operator $\mathcal{D}$ from the space of smooth sections for a $G$-equivariant vector bundle over $X$ to that for a $G'$-equivariant vector bundle over $Y$. Toshiyuki Kobayashi called such a differential operator $\mathcal{D}$ a

**differential symmetry breaking operator**(differential SBO for short) ([T.~Kobayashi, Differential Geom. Appl. (2014)]).In [Kobayashi-K-Pevzner, Lecture Notes in Math. 2170 (2016)], we explicitly constructed and classified all the differential SBOs from the space of differential $i$-forms $\mathcal{E}^i(S^n)$ over the standard Riemann sphere $S^n$ to that of differential $j$-forms $\mathcal{E}^j(S^{n-1})$ over the totally geodesic hypersphere $S^{n-1}$. In this talk, by extending the results in a Riemannian setting, we discuss about the classification and construction of differential SBOs in a pseudo-Riemannian setting such as anti-de Sitter spaces and hyperbolic spaces. This is a joint work with Toshiyuki Kobayashi and Michael Pevzner.

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