## Tuesday Seminar on Topology

Seminar information archive ～04/13｜Next seminar｜Future seminars 04/14～

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**

### 2024/04/09

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.

Topological stability theorem and Gromov-Hausdorff convergence (JAPANESE)

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

Pre-registration required. See our seminar webpage.

**Shouhei Honda**(The University of Tokyo)Topological stability theorem and Gromov-Hausdorff convergence (JAPANESE)

[ Abstract ]

Gromov-Hausdorff distance defines a distance on the set of all isometry classes of compact metric spaces. It is natural to ask about topological relationships between two compact metric spaces whose Gromov-Hausdorff distance is small. Cheeger-Colding provided a striking result about this question, under a (lower) curvature bound on Ricci curvature. In this talk we will improve this result sharply. This is a joint work with Yuanlin Peng (Tohoku University). If time permits, along this direction, we will also discuss a recent work about a topological stability result to flat tori via harmonic maps, where this is a joint work with Christian Ketterer (University of Freiburg), Ilaria Mondello (Université de Paris Est Créteil), Chiara Rigoni (University of Vienna) and Raquel Perales (CIMAT).

[ Reference URL ]Gromov-Hausdorff distance defines a distance on the set of all isometry classes of compact metric spaces. It is natural to ask about topological relationships between two compact metric spaces whose Gromov-Hausdorff distance is small. Cheeger-Colding provided a striking result about this question, under a (lower) curvature bound on Ricci curvature. In this talk we will improve this result sharply. This is a joint work with Yuanlin Peng (Tohoku University). If time permits, along this direction, we will also discuss a recent work about a topological stability result to flat tori via harmonic maps, where this is a joint work with Christian Ketterer (University of Freiburg), Ilaria Mondello (Université de Paris Est Créteil), Chiara Rigoni (University of Vienna) and Raquel Perales (CIMAT).

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

### 2024/02/13

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.

Measures on the moduli space of curves and super volumes (ENGLISH)

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

Pre-registration required. See our seminar webpage.

**Paul Norbury**(The University of Melbourne)Measures on the moduli space of curves and super volumes (ENGLISH)

[ Abstract ]

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

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

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

### 2024/01/23

17:00-18:00 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 rational cohomology of spin hyperelliptic mapping class groups (JAPANESE)

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

Pre-registration required. See our seminar webpage.

**Gefei Wang**(The University of Tokyo)On the rational cohomology of spin hyperelliptic mapping class groups (JAPANESE)

[ Abstract ]

Let $G$ be the subgroup $S_{n−q} \times S_q$ of the $n$-th symmetric group $S_n$ for $n-q \ge q$. In this talk, we study the $G$-invariant part of the rational cohomology group of the pure braid group $P_n$. The invariant part $H^*(P_n)^G$ includes the rational cohomology of a spin hyperelliptic mapping class group of genus $g$ as a subalgebra when $n=2g+2$. Based on the study of Lehrer-Solomon, we prove that they are independent of n and q in degree $* \le q-1$. We also give a formula to calculate the dimension of $H^* (P_n)^G$ and calculate it in all degree for $q \le 3$.

[ Reference URL ]Let $G$ be the subgroup $S_{n−q} \times S_q$ of the $n$-th symmetric group $S_n$ for $n-q \ge q$. In this talk, we study the $G$-invariant part of the rational cohomology group of the pure braid group $P_n$. The invariant part $H^*(P_n)^G$ includes the rational cohomology of a spin hyperelliptic mapping class group of genus $g$ as a subalgebra when $n=2g+2$. Based on the study of Lehrer-Solomon, we prove that they are independent of n and q in degree $* \le q-1$. We also give a formula to calculate the dimension of $H^* (P_n)^G$ and calculate it in all degree for $q \le 3$.

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

### 2024/01/16

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

Pre-registration required. See our seminar webpage.

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

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

Pre-registration required. See our seminar webpage.

**Jin Miyazawa**(The University of Tokyo)A gauge theoretic invariant of embedded surfaces in 4-manifolds and exotic P

^{2}-knots (JAPANESE)
[ Abstract ]

When two embeddings of surfaces on a 4-dimensional manifold are given, if they are topologically isotopic but not smoothly isotopic, we call them a pair of exotic surfaces. While there is a great deal of study of exotic surfaces in 4-manifolds, studies of closed exotic surfaces in S

[ Reference URL ]When two embeddings of surfaces on a 4-dimensional manifold are given, if they are topologically isotopic but not smoothly isotopic, we call them a pair of exotic surfaces. While there is a great deal of study of exotic surfaces in 4-manifolds, studies of closed exotic surfaces in S

^{4}are limited. In particular, the existence of orientable exotic surfaces in S^{4}remains unknown to date. There are some examples of non-orientable exotic surfaces in S^{4}, including the initial example given by Finashin-Kreck-Viro in 1988, but all such cases have genus greater than or equal to 5. The difficulty in detecting exotic surfaces in S^{4}is to prove that two embeddings of surfaces are not smoothly isotopic. All examples of exotic non-orientable surfaces in S^{4}have been detected by proving the 4-manifolds obtained by the double branched covers are exotic. If we attempt to apply this technique to low-genus non-orientable surfaces in S^{4}, we have to discover exotic small 4-manifolds, which is known to be difficult. In this seminar, we construct an invariant for embedded surfaces in 4-manifolds using Real Seiberg-Witten theory. As an application, we give an infinite family of exotic embeddings into S^{4}for the real projective plane.https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

### 2024/01/09

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

Pre-registration required. See our seminar webpage.

Stabilizer subgroups of Thompson's group F in Thompson knot theory (JAPANESE)

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

Pre-registration required. See our seminar webpage.

**Akihiro Takano**(The University of Tokyo)Stabilizer subgroups of Thompson's group F in Thompson knot theory (JAPANESE)

[ Abstract ]

Thompson knot theory, introduced by Vaughan Jones, is a study of knot theory using Thompson's group F.

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

[ Reference URL ]Thompson knot theory, introduced by Vaughan Jones, is a study of knot theory using Thompson's group F.

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

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

### 2023/12/19

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.

Topological quantum computing, tensor networks and operator algebras (JAPANESE)

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

Pre-registration required. See our seminar webpage.

**Yasuyuki Kawahigashi**(The University of Tokyo)Topological quantum computing, tensor networks and operator algebras (JAPANESE)

[ Abstract ]

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

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

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

### 2023/12/14

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.

Torus orbit closures in the flag variety (JAPANESE)

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

Pre-registration required. See our seminar webpage.

**Mikiya Masuda**(Osaka City University)Torus orbit closures in the flag variety (JAPANESE)

[ Abstract ]

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

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

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

### 2023/12/12

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.

Multivariable knot polynomials from braided Hopf algebras with automorphisms (ENGLISH)

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

Pre-registration required. See our seminar webpage.

**Stavros Garoufalidis**(Southern University of Science and Technology)Multivariable knot polynomials from braided Hopf algebras with automorphisms (ENGLISH)

[ Abstract ]

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

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

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

### 2023/12/05

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 Euler class for flat S

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

Pre-registration required. See our seminar webpage.

**Teruaki Kitano**(Soka University)On the Euler class for flat S

^{1}-bundles, C^{∞}vs C^{ω}(JAPANESE)
[ Abstract ]

We describe low dimensional homology groups of the real analytic, orientation preserving diffeomorphism group of S

[ Reference URL ]We describe low dimensional homology groups of the real analytic, orientation preserving diffeomorphism group of S

^{1}in terms of BΓ_{1}by applying a theorem of Thurston. It is an open problem whether some power of the rational Euler class vanishes for real analytic flat S^{1}bundles. In this talk we discuss that if it does, then the homology group should contain many torsion classes that vanish in the smooth case. Along this line we can give a new proof for the non-triviality of any power of the rational Euler class in the smooth case. If time permits, we will mention some attempts to study a Mather-Thurston map in the analytic case. This talk is based on a joint work with Shigeyuki Morita and Yoshihiko Mitsumatsu.https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

### 2023/11/28

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.

An analogue of the Johnson-Morita theory for the handlebody group (ENGLISH)

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

Pre-registration required. See our seminar webpage.

**Gwénaël Massuyeau**(Université de Bourgogne)An analogue of the Johnson-Morita theory for the handlebody group (ENGLISH)

[ Abstract ]

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

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

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

### 2023/11/21

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

Pre-registration required. See our seminar webpage.

Shadows, divides and hyperbolic volumes (JAPANESE)

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

Pre-registration required. See our seminar webpage.

**Yuya Koda**(Keio University)Shadows, divides and hyperbolic volumes (JAPANESE)

[ Abstract ]

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

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

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

### 2023/11/14

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.

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

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

Pre-registration required. See our seminar webpage.

**Tomoo Yokoyama**(Saitama University)Dependency of the positive and negative long-time behaviors of flows on surfaces (JAPANESE)

[ Abstract ]

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

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

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

### 2023/11/07

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.

Hodge numbers of moduli spaces of principal bundles on curves (ENGLISH)

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

Pre-registration required. See our seminar webpage.

**Florent Schaffhauser**(Heidelberg University)Hodge numbers of moduli spaces of principal bundles on curves (ENGLISH)

[ Abstract ]

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

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

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

### 2023/10/31

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

Pre-registration required. See our seminar webpage.

On Harada Conjecture II (JAPANESE)

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

Pre-registration required. See our seminar webpage.

**Naoki Chigira**(Kumamoto University)On Harada Conjecture II (JAPANESE)

[ Abstract ]

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

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

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

### 2023/10/24

17:00-18:00 Online

Pre-registration required. See our seminar webpage.

Index theory for quarter-plane Toeplitz operators via extended symbols (JAPANESE)

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

Pre-registration required. See our seminar webpage.

**Shin Hayashi**(Aoyama Gakuin University)Index theory for quarter-plane Toeplitz operators via extended symbols (JAPANESE)

[ Abstract ]

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

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

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

### 2023/10/17

17:00-18:00 Online

Pre-registration required. See our seminar webpage.

Train track combinatorics and cluster algebras (JAPANESE)

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

Pre-registration required. See our seminar webpage.

**Shunsuke Kano**(MathCCS, Tohoku University)Train track combinatorics and cluster algebras (JAPANESE)

[ Abstract ]

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

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

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

### 2023/10/10

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

Pre-registration required. See our seminar webpage.

Invariant quasimorphisms and coarse geometry of scl (JAPANESE)

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

Pre-registration required. See our seminar webpage.

**Masato Mimura**(Tohoku University)Invariant quasimorphisms and coarse geometry of scl (JAPANESE)

[ Abstract ]

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

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

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

### 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