## Tuesday Seminar on Topology

Seminar information archive ～11/02｜Next seminar｜Future seminars 11/03～

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**

### 2024/10/29

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.

Cartan calculus in string topology (JAPANESE)

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

Pre-registration required. See our seminar webpage.

**Takahito Naito**(Nippon Institute of Technology)Cartan calculus in string topology (JAPANESE)

[ Abstract ]

The homology of the free loop space of a closed oriented manifold (called the loop homology) has rich algebraic structures. In the theory of string topology due to Chas and Sullivan, it is well known that the loop homology has a structure of Gerstenhabar algebras with a multiplication called the loop product and a Lie bracket called the loop bracket. On the other hand, Kuribayashi, Wakatsuki, Yamaguchi and the speaker gave a Cartan calculus on the loop homology, which is a geometric description of a homotopy Cartan calculus in the sense of Fiorenza and Kowalzig on the Hochschild homology.

In this talk, we will investigate a relationship between the string topology operations and the Cartan calculus. Especially, we will show that the Cartan calculus can be described by using the loop product and the loop bracket with rational coefficients. As an application, the nilpotency of some loop homology classes are determined.

[ Reference URL ]The homology of the free loop space of a closed oriented manifold (called the loop homology) has rich algebraic structures. In the theory of string topology due to Chas and Sullivan, it is well known that the loop homology has a structure of Gerstenhabar algebras with a multiplication called the loop product and a Lie bracket called the loop bracket. On the other hand, Kuribayashi, Wakatsuki, Yamaguchi and the speaker gave a Cartan calculus on the loop homology, which is a geometric description of a homotopy Cartan calculus in the sense of Fiorenza and Kowalzig on the Hochschild homology.

In this talk, we will investigate a relationship between the string topology operations and the Cartan calculus. Especially, we will show that the Cartan calculus can be described by using the loop product and the loop bracket with rational coefficients. As an application, the nilpotency of some loop homology classes are determined.

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

### 2024/10/22

17:00-18:00 Online

Pre-registration required. See our seminar webpage.

On the generic existence of WKB spectral networks/Stokes graphs (JAPANESE)

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

Pre-registration required. See our seminar webpage.

**Tatsuki Kuwagaki**( Kyoto University)On the generic existence of WKB spectral networks/Stokes graphs (JAPANESE)

[ Abstract ]

The foliation determined by a quadratic differential on a Riemann surface is a classical object of study. In particular, considering leaves through zero points has been of interest in connection with WKB analysis, Teichmüller theory, and quantum field theory. WKB spectral network (or Stokes graph) is a higher-order-differential version of this notion. In this talk, I will discuss the proof of existence of WKB spectral network for a large class of differentials. If time permits, I will explain its relationship with Lagrangian intersection Floer theory.

[ Reference URL ]The foliation determined by a quadratic differential on a Riemann surface is a classical object of study. In particular, considering leaves through zero points has been of interest in connection with WKB analysis, Teichmüller theory, and quantum field theory. WKB spectral network (or Stokes graph) is a higher-order-differential version of this notion. In this talk, I will discuss the proof of existence of WKB spectral network for a large class of differentials. If time permits, I will explain its relationship with Lagrangian intersection Floer theory.

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

### 2024/10/17

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.

Slope inequalities for fibered complex surfaces (JAPANESE)

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

Pre-registration required. See our seminar webpage.

**Makoto Enokizono**(The University of Tokyo)Slope inequalities for fibered complex surfaces (JAPANESE)

[ Abstract ]

Slope inequalities of fibered surfaces are important in relation to the classification of algebraic surfaces and the complex structure of Lefschetz fibrations in four-dimensional topology. It is also known that many slope inequalities for semi-stable fibered surfaces can be derived from the intersection theory on the moduli space of stable curves. In this talk, after outlining the background of these studies, I will explain how various slope inequalities can be obtained for fibered surfaces that are not necessarily semi-stable by extending the discussion of the moduli space.

[ Reference URL ]Slope inequalities of fibered surfaces are important in relation to the classification of algebraic surfaces and the complex structure of Lefschetz fibrations in four-dimensional topology. It is also known that many slope inequalities for semi-stable fibered surfaces can be derived from the intersection theory on the moduli space of stable curves. In this talk, after outlining the background of these studies, I will explain how various slope inequalities can be obtained for fibered surfaces that are not necessarily semi-stable by extending the discussion of the moduli space.

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

### 2024/10/08

17:00-18:30 Room #ハイブリッド開催/123 (Graduate School of Math. Sci. Bldg.)

Pre-registration required. See our seminar webpage.

Dehn twists on 4-manifolds (JAPANESE)

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

Pre-registration required. See our seminar webpage.

**Hokuto Konno**(The University of Tokyo)Dehn twists on 4-manifolds (JAPANESE)

[ Abstract ]

Dehn twists on surfaces form a basic class of diffeomorphisms. On 4-manifolds, an analogue of Dehn twist can be defined by considering twists along Seifert fibered 3-manifolds. In this talk, I will explain how this type of diffeomorphism exhibits interesting properties from the perspective of differential topology, and occasionally from the viewpoint of symplectic geometry as well. The proof involves gauge theory for families. This talk includes joint work with Abhishek Mallick and Masaki Taniguchi, as well as joint work with Jianfeng Lin, Anubhav Mukherjee, and Juan Muñoz-Echániz.

[ Reference URL ]Dehn twists on surfaces form a basic class of diffeomorphisms. On 4-manifolds, an analogue of Dehn twist can be defined by considering twists along Seifert fibered 3-manifolds. In this talk, I will explain how this type of diffeomorphism exhibits interesting properties from the perspective of differential topology, and occasionally from the viewpoint of symplectic geometry as well. The proof involves gauge theory for families. This talk includes joint work with Abhishek Mallick and Masaki Taniguchi, as well as joint work with Jianfeng Lin, Anubhav Mukherjee, and Juan Muñoz-Echániz.

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

### 2024/07/23

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.

Shortest word problem in braid theory (JAPANESE)

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

Pre-registration required. See our seminar webpage.

**Keiko Kawamuro**(University of Iowa)Shortest word problem in braid theory (JAPANESE)

[ Abstract ]

Given a braid element in B_n, searching for a shortest braid word representative (using the band-generators) is called the Shortest Braid Problem. Up to braid index n = 4, this problem has been solved by Kang, Ko, and Lee in 1997. In this talk I will discuss recent development of this problem for braid index 5 or higher. I will also show diagrammatic computational technique of the Left Canonical Form of a given braid, that is a key to the three fundamental problems in braid theory; the Word Problem, the Conjugacy Problem and the Shortest Word Problem. This is joint work with Rebecca Sorsen and Michele Capovilla-Searle.

[ Reference URL ]Given a braid element in B_n, searching for a shortest braid word representative (using the band-generators) is called the Shortest Braid Problem. Up to braid index n = 4, this problem has been solved by Kang, Ko, and Lee in 1997. In this talk I will discuss recent development of this problem for braid index 5 or higher. I will also show diagrammatic computational technique of the Left Canonical Form of a given braid, that is a key to the three fundamental problems in braid theory; the Word Problem, the Conjugacy Problem and the Shortest Word Problem. This is joint work with Rebecca Sorsen and Michele Capovilla-Searle.

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

### 2024/07/09

17:00-18:00 Online

Pre-registration required. See our seminar webpage.

Knitted surfaces in the 4-ball and their chart description (JAPANESE)

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

Pre-registration required. See our seminar webpage.

**Inasa Nakamura**(Saga University)Knitted surfaces in the 4-ball and their chart description (JAPANESE)

[ Abstract ]

Knits (or BMW tangles) are tangles in a cylinder generated by generators of the BMW (Birman-Murakami-Wenzl) algebras, consisting of standard generators of the braid group and their inverses, and splices of crossings called pairs of hooks. We give a new construction of surfaces in $D^2 \times B^2$, called knitted surfaces (or BMW surfaces), that are described as the trace of deformations of knits, and we give the notion of charts for knitted surfaces, that are finite graphs in $B^2$. We show that a knitted surface has a chart description. Knitted surfaces and their chart description include 2-dimensional braids and their chart description. This is joint work with Jumpei Yasuda (Osaka University).

[ Reference URL ]Knits (or BMW tangles) are tangles in a cylinder generated by generators of the BMW (Birman-Murakami-Wenzl) algebras, consisting of standard generators of the braid group and their inverses, and splices of crossings called pairs of hooks. We give a new construction of surfaces in $D^2 \times B^2$, called knitted surfaces (or BMW surfaces), that are described as the trace of deformations of knits, and we give the notion of charts for knitted surfaces, that are finite graphs in $B^2$. We show that a knitted surface has a chart description. Knitted surfaces and their chart description include 2-dimensional braids and their chart description. This is joint work with Jumpei Yasuda (Osaka University).

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

### 2024/07/02

17:00-18:30 Room #ハイブリッド開催/117 (Graduate School of Math. Sci. Bldg.)

Pre-registration required. See our seminar webpage.

The second quandle homology group of the knot $n$-quandle (JAPANESE)

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

Pre-registration required. See our seminar webpage.

**Kokoro Tanaka**(Tokyo Gakugei University)The second quandle homology group of the knot $n$-quandle (JAPANESE)

[ Abstract ]

We compute the second quandle homology group of the knot $n$-quandle for each integer $n>1$, where the knot $n$-quandle is a certain quotient of the knot quandle (of an oriented classical knot in the $3$-sphere). Although the second quandle homology group of the knot quandle can only detect the unknot, it turns out that that of its 3-quandle can detect the unknot, the trefoil and the cinqfoil. This is a joint work with Yuta Taniguchi.

[ Reference URL ]We compute the second quandle homology group of the knot $n$-quandle for each integer $n>1$, where the knot $n$-quandle is a certain quotient of the knot quandle (of an oriented classical knot in the $3$-sphere). Although the second quandle homology group of the knot quandle can only detect the unknot, it turns out that that of its 3-quandle can detect the unknot, the trefoil and the cinqfoil. This is a joint work with Yuta Taniguchi.

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

### 2024/06/25

17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Joint with RIKEN iTHEMS. Pre-registration required. See our seminar webpage.

Liouville symmetry groups and pseudo-isotopies (ENGLISH)

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

Joint with RIKEN iTHEMS. Pre-registration required. See our seminar webpage.

**Emmy Murphy**(University of Toronto)Liouville symmetry groups and pseudo-isotopies (ENGLISH)

[ Abstract ]

Even though $\mathbb{C}^n$ is the most basic symplectic manifold, when $n>2$ its compactly supported symplectomorphism group remains mysterious. For instance, we do not know if it is connected. To understand it better, one can define various subgroups of the symplectomorphism group, and a number of Serre fibrations between them. This leads us to the Liouville pseudo-isotopy group of a contact manifold, important for relating (for instance) compactly supported symplectomorphisms of $\mathbb{C}^n$, and contactomorphisms of the sphere at infinity. After explaining this background, the talk will focus on a new result: that the pseudo-isotopy group is connected, under a Liouville-vs-Weinstein hypothesis.

[ Reference URL ]Even though $\mathbb{C}^n$ is the most basic symplectic manifold, when $n>2$ its compactly supported symplectomorphism group remains mysterious. For instance, we do not know if it is connected. To understand it better, one can define various subgroups of the symplectomorphism group, and a number of Serre fibrations between them. This leads us to the Liouville pseudo-isotopy group of a contact manifold, important for relating (for instance) compactly supported symplectomorphisms of $\mathbb{C}^n$, and contactomorphisms of the sphere at infinity. After explaining this background, the talk will focus on a new result: that the pseudo-isotopy group is connected, under a Liouville-vs-Weinstein hypothesis.

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

### 2024/06/20

17:00-18:30 Room #002 (Graduate School of Math. Sci. Bldg.)

Joint with RIKEN iTHEMS. Pre-registration required. See our seminar webpage.

Rigidity and Flexibility of Iosmetric Embeddings (ENGLISH)

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

Joint with RIKEN iTHEMS. Pre-registration required. See our seminar webpage.

**Dominik Inauen**(University of Leipzig)Rigidity and Flexibility of Iosmetric Embeddings (ENGLISH)

[ Abstract ]

The problem of embedding abstract Riemannian manifolds isometrically (i.e. preserving the lengths) into Euclidean space stems from the conceptually fundamental question of whether abstract Riemannian manifolds and submanifolds of Euclidean space are the same. As it turns out, such embeddings have a drastically different behaviour at low regularity (i.e. $C^1$) than at high regularity (i.e. $C^2$). For example, by the famous Nash--Kuiper theorem it is possible to find $C^1$ isometric embeddings of the standard $2$-sphere into arbitrarily small balls in $\mathbb{R}^3$, and yet, in the $C^2$ category there is (up to translation and rotation) just one isometric embedding, namely the standard inclusion. Analoguous to the Onsager conjecture in fluid dynamics, one might ask if there is a sharp regularity threshold in the Hölder scale which distinguishes these flexible and rigid behaviours. In my talk I will review some known results and argue why the Hölder exponent 1/2 can be seen as a critical exponent in the problem.

[ Reference URL ]The problem of embedding abstract Riemannian manifolds isometrically (i.e. preserving the lengths) into Euclidean space stems from the conceptually fundamental question of whether abstract Riemannian manifolds and submanifolds of Euclidean space are the same. As it turns out, such embeddings have a drastically different behaviour at low regularity (i.e. $C^1$) than at high regularity (i.e. $C^2$). For example, by the famous Nash--Kuiper theorem it is possible to find $C^1$ isometric embeddings of the standard $2$-sphere into arbitrarily small balls in $\mathbb{R}^3$, and yet, in the $C^2$ category there is (up to translation and rotation) just one isometric embedding, namely the standard inclusion. Analoguous to the Onsager conjecture in fluid dynamics, one might ask if there is a sharp regularity threshold in the Hölder scale which distinguishes these flexible and rigid behaviours. In my talk I will review some known results and argue why the Hölder exponent 1/2 can be seen as a critical exponent in the problem.

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

### 2024/06/11

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.

A topological proof of Wolpert's formula of the Weil-Petersson symplectic form in terms of the Fenchel-Nielsen coordinates (JAPANESE)

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

Pre-registration required. See our seminar webpage.

**Nariya Kawazumi**(The University of Tokyo)A topological proof of Wolpert's formula of the Weil-Petersson symplectic form in terms of the Fenchel-Nielsen coordinates (JAPANESE)

[ Abstract ]

Wolpert explicitly described the Weil-Petersson symplectic form on the Teichmüller space in terms of the Fenchel-Nielsen coordinate system, which comes from a pants decomposition of a surface. By introducing a natural cell-decomposition associated with the decomposition, we give a topological proof of Wolpert's formula, where the symplectic form localizes near the simple closed curves defining the decomposition.

[ Reference URL ]Wolpert explicitly described the Weil-Petersson symplectic form on the Teichmüller space in terms of the Fenchel-Nielsen coordinate system, which comes from a pants decomposition of a surface. By introducing a natural cell-decomposition associated with the decomposition, we give a topological proof of Wolpert's formula, where the symplectic form localizes near the simple closed curves defining the decomposition.

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

### 2024/06/04

17:00-18:00 Online

Pre-registration required. See our seminar webpage.

The trapezoidal conjecture for the links of braid index 3 (JAPANESE)

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

Pre-registration required. See our seminar webpage.

**Katsumi Ishikawa**(RIMS, Kyoto University)The trapezoidal conjecture for the links of braid index 3 (JAPANESE)

[ Abstract ]

The trapezoidal conjecture is a classical famous conjecture posed by Fox, which states that the coefficient sequence of the Alexander polynomial of any alternating link is trapezoidal. In this talk, we show this conjecture for any alternating links of braid index 3. Although the result holds for any choice of the orientation, we shall mainly discuss the case of the closures of alternating 3-braids with parallel orientations.

[ Reference URL ]The trapezoidal conjecture is a classical famous conjecture posed by Fox, which states that the coefficient sequence of the Alexander polynomial of any alternating link is trapezoidal. In this talk, we show this conjecture for any alternating links of braid index 3. Although the result holds for any choice of the orientation, we shall mainly discuss the case of the closures of alternating 3-braids with parallel orientations.

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

### 2024/05/28

17:00-18:00 Online

Pre-registration required. See our seminar webpage.

Knot invariants and their Harer-Zagier transform (ENGLISH)

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

Pre-registration required. See our seminar webpage.

**Andreani Petrou**(Okinawa Institute of Science and Technology)Knot invariants and their Harer-Zagier transform (ENGLISH)

[ Abstract ]

The Harer-Zagier (HZ) transform is a discrete Laplace transform that can be applied to knot polynomials, mapping them into a rational function of two variables $\lambda$ and $q$. The HZ transform of the HOMFLY-PT polynomial has a simple form, as it can be written as a sum of factorised terms. For some special families of knots, it can be fully factorised and it is completely determined by a set of exponents. There is an interesting relation between such exponents and Khovanov homology. Moreover, we conjecture that there is an 1-1 correspondence with such factorisability and a relation between the HOMFLY-PT and Kauffman polynomials. Furthermore, we suggest that by fixing the variable $\lambda= q^n$ for some "magical" exponent $n$, the HZ transform of any knot can obtain a factorised form in terms of cyclotomic polynomials. Finally, the zeros of the HZ transform show an interesting behaviour, which shall be discussed.

[ Reference URL ]The Harer-Zagier (HZ) transform is a discrete Laplace transform that can be applied to knot polynomials, mapping them into a rational function of two variables $\lambda$ and $q$. The HZ transform of the HOMFLY-PT polynomial has a simple form, as it can be written as a sum of factorised terms. For some special families of knots, it can be fully factorised and it is completely determined by a set of exponents. There is an interesting relation between such exponents and Khovanov homology. Moreover, we conjecture that there is an 1-1 correspondence with such factorisability and a relation between the HOMFLY-PT and Kauffman polynomials. Furthermore, we suggest that by fixing the variable $\lambda= q^n$ for some "magical" exponent $n$, the HZ transform of any knot can obtain a factorised form in terms of cyclotomic polynomials. Finally, the zeros of the HZ transform show an interesting behaviour, which shall be discussed.

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

### 2024/05/21

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.

γ-supports and sheaves (JAPANESE)

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

Pre-registration required. See our seminar webpage.

**Yuichi Ike**(Institute of Mathematics for Industry, Kyushu University)γ-supports and sheaves (JAPANESE)

[ Abstract ]

The space of smooth compact exact Lagrangians of a cotangent bundle carries the spectral metric γ, and we consider its completion. With an element of the completion, Viterbo associated a closed subset called γ-support. In this talk, I will explain how we can use sheaf-theoretic methods to explore the completion and γ-supports. I will show that we can associate a sheaf with an element of the completion, and its (reduced) microsupport is equal to the γ-support through the correspondence. With this equality, I will also show several properties of γ-supports. This is joint work with Tomohiro Asano (RIMS), Stéphane Guillermou (Nantes Université), Vincent Humilière (Sorbonne Université), and Claude Viterbo (Université Paris-Saclay).

[ Reference URL ]The space of smooth compact exact Lagrangians of a cotangent bundle carries the spectral metric γ, and we consider its completion. With an element of the completion, Viterbo associated a closed subset called γ-support. In this talk, I will explain how we can use sheaf-theoretic methods to explore the completion and γ-supports. I will show that we can associate a sheaf with an element of the completion, and its (reduced) microsupport is equal to the γ-support through the correspondence. With this equality, I will also show several properties of γ-supports. This is joint work with Tomohiro Asano (RIMS), Stéphane Guillermou (Nantes Université), Vincent Humilière (Sorbonne Université), and Claude Viterbo (Université Paris-Saclay).

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

### 2024/05/14

17:00-18:00 Online

Pre-registration required. See our seminar webpage.

Exotic 4-manifolds with signature zero (JAPANESE)

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

Pre-registration required. See our seminar webpage.

**Noriyuki Hamada**(Institute of Mathematics for Industry, Kyushu University)Exotic 4-manifolds with signature zero (JAPANESE)

[ Abstract ]

We will talk about our novel examples of symplectic 4-manifolds, which are homeomorphic but not diffeomorphic to the standard simply-connected closed 4-manifolds with signature zero. In particular, they provide such examples with the smallest Euler characteristics known to date. Our method employs the time-honored approach of reverse-engineering, while the key new ingredients are the model manifolds that we build from scratch as Lefschetz fibrations. Notably, our method greatly simplifies pi_1 calculations, typically the most intricate aspect in existing literature.

This is joint work with Inanc Baykur (University of Massachusetts Amherst).

[ Reference URL ]We will talk about our novel examples of symplectic 4-manifolds, which are homeomorphic but not diffeomorphic to the standard simply-connected closed 4-manifolds with signature zero. In particular, they provide such examples with the smallest Euler characteristics known to date. Our method employs the time-honored approach of reverse-engineering, while the key new ingredients are the model manifolds that we build from scratch as Lefschetz fibrations. Notably, our method greatly simplifies pi_1 calculations, typically the most intricate aspect in existing literature.

This is joint work with Inanc Baykur (University of Massachusetts Amherst).

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

### 2024/05/07

17:00-18:00 Online

Pre-registration required. See our seminar webpage.

The Thurston spine and the Systole function of Teichmüller space (ENGLISH)

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

Pre-registration required. See our seminar webpage.

**Ingrid Irmer**(Southern University of Science and Technology)The Thurston spine and the Systole function of Teichmüller space (ENGLISH)

[ Abstract ]

The systole function $f_{sys}$ on Teichm\"uller space $\mathcal{T}_{g}$ of a closed genus $g$ surface is a piecewise-smooth map $\mathcal{T}_{g}\rightarrow \mathbb{R}$ whose value at any point is the length of the shortest geodesic on the corresponding hyperbolic surface. It is known that $f_{sys}$ gives a mapping class group-equivariant handle decomposition of $\mathcal{T}_{g}$ via an analogue of Morse Theory. This talk explains the relationship between this handle decomposition and the Thurston spine of $\mathcal{T}_{g}$.

[ Reference URL ]The systole function $f_{sys}$ on Teichm\"uller space $\mathcal{T}_{g}$ of a closed genus $g$ surface is a piecewise-smooth map $\mathcal{T}_{g}\rightarrow \mathbb{R}$ whose value at any point is the length of the shortest geodesic on the corresponding hyperbolic surface. It is known that $f_{sys}$ gives a mapping class group-equivariant handle decomposition of $\mathcal{T}_{g}$ via an analogue of Morse Theory. This talk explains the relationship between this handle decomposition and the Thurston spine of $\mathcal{T}_{g}$.

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

### 2024/04/23

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.

Pochette surgery on 4-manifolds and the Ozsváth--Szabó $d$-invariants of Brieskorn homology 3-spheres (JAPANESE)

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

Pre-registration required. See our seminar webpage.

**Tatsumasa Suzuki**(Meiji University)Pochette surgery on 4-manifolds and the Ozsváth--Szabó $d$-invariants of Brieskorn homology 3-spheres (JAPANESE)

[ Abstract ]

This talk consists of the following two research contents:

I. The boundary sum of $S^1 \times D^3$ and $D^2 \times S^2$ is called a pochette. The pochette surgery, which is a generalization of Gluck surgery and a special case of torus surgery, was discovered by Zjuñici Iwase and Yukio Matsumoto in 2004. For a pochette $P$ embedded in a 4-manifold $X$, a pochette surgery on $X$ is the operation of removing the interior of $P$ and gluing $P$ by a diffeomorphism of the boundary of $P$. In this talk, we focus on the fact that pochette surgery is a surgery with a cord and the 2-sphere $S^2$, and attempt to classify the diffeomorphism type of pochette surgery on the 4-sphere $S^4$.

II. In 2003, Peter Ozsváth and Zoltán Szabó introduced a homology cobordism invariant for homology 3-spheres called a $d$-invariant. In this talk, we present new computable examples by refining the Karakurt--Şavk formula for any Brieskorn homology 3-sphere $\Sigma(p,q,r)$ with $p$ is odd and $pq+pr-qr=1$. Furthermore, by refining the Can--Karakurt formula for the $d$-invariant of any $\Sigma(p,q,r)$, we also introduce the relationship with the $d$-invariant of $\Sigma(p,q,r)$ and those of lens spaces.

This talk includes contents of joint work with Motoo Tange (University of Tsukuba).

[ Reference URL ]This talk consists of the following two research contents:

I. The boundary sum of $S^1 \times D^3$ and $D^2 \times S^2$ is called a pochette. The pochette surgery, which is a generalization of Gluck surgery and a special case of torus surgery, was discovered by Zjuñici Iwase and Yukio Matsumoto in 2004. For a pochette $P$ embedded in a 4-manifold $X$, a pochette surgery on $X$ is the operation of removing the interior of $P$ and gluing $P$ by a diffeomorphism of the boundary of $P$. In this talk, we focus on the fact that pochette surgery is a surgery with a cord and the 2-sphere $S^2$, and attempt to classify the diffeomorphism type of pochette surgery on the 4-sphere $S^4$.

II. In 2003, Peter Ozsváth and Zoltán Szabó introduced a homology cobordism invariant for homology 3-spheres called a $d$-invariant. In this talk, we present new computable examples by refining the Karakurt--Şavk formula for any Brieskorn homology 3-sphere $\Sigma(p,q,r)$ with $p$ is odd and $pq+pr-qr=1$. Furthermore, by refining the Can--Karakurt formula for the $d$-invariant of any $\Sigma(p,q,r)$, we also introduce the relationship with the $d$-invariant of $\Sigma(p,q,r)$ and those of lens spaces.

This talk includes contents of joint work with Motoo Tange (University of Tsukuba).

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

### 2024/04/16

17:00-18:00 Online

Pre-registration required. See our seminar webpage.

Skein algebras and quantum tori in view of pants decompositions (JAPANESE)

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

Pre-registration required. See our seminar webpage.

**Hiroaki Karuo**(Gakushuin University)Skein algebras and quantum tori in view of pants decompositions (JAPANESE)

[ Abstract ]

To understand the algebraic structures of skein algebras and their generalizations, we usually try to embed these algebras into quantum tori using ideal triangulations of a surface and the splitting map. However, such a construction does not work for the skein algebras of closed surfaces and the Roger--Yang skein algebras of punctured surfaces.

In the talk, we define filtrations on these algebras using pants decompositions and embed the associated graded algebras into quantum tori. As a consequence, Roger--Yang skein algebras are quantizations of decorated Teichmuller spaces. This talk is based on a joint work with Wade Bloomquist (Morningside University) and Thang Le (Georgia Institute of Technology).

[ Reference URL ]To understand the algebraic structures of skein algebras and their generalizations, we usually try to embed these algebras into quantum tori using ideal triangulations of a surface and the splitting map. However, such a construction does not work for the skein algebras of closed surfaces and the Roger--Yang skein algebras of punctured surfaces.

In the talk, we define filtrations on these algebras using pants decompositions and embed the associated graded algebras into quantum tori. As a consequence, Roger--Yang skein algebras are quantizations of decorated Teichmuller spaces. This talk is based on a joint work with Wade Bloomquist (Morningside University) and Thang Le (Georgia Institute of Technology).

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

### 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