## 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.) |
---|---|

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

**Seminar information archive**

### 2021/01/12

17:00-18:00 Online

Pre-registration required. See our seminar webpage.

Bounded cohomology of volume-preserving diffeomorphism groups (JAPANESE)

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

Pre-registration required. See our seminar webpage.

**Mitsuaki Kimura**(The University of Tokyo)Bounded cohomology of volume-preserving diffeomorphism groups (JAPANESE)

[ Abstract ]

Let M be a complete Riemannian manifold of finite volume. Brandenbursky and Marcinkowski proved that the third bounded cohomology of the volume-preserving diffeomorphism group of M is infinite dimensional when the fundamental group of M is "complicated enough". For example, if M is two-dimensional, the above condition is satisfied if the Euler characteristic is negative. Recently, we have extended this result in the following two directions.

(1) When M is two-dimensional and the Euler characteristic is greater than or equal to zero.

(2) When the volume of M is infinite.

In this talk, we will mainly discuss (1). The key idea is to use the fundamental group of the configuration space of M (i.e., the braid group), rather than the fundamental group of M. If time permits, we will also explain (2). For this extension, we introduce the notion of "norm controlled cohomology".

[ Reference URL ]Let M be a complete Riemannian manifold of finite volume. Brandenbursky and Marcinkowski proved that the third bounded cohomology of the volume-preserving diffeomorphism group of M is infinite dimensional when the fundamental group of M is "complicated enough". For example, if M is two-dimensional, the above condition is satisfied if the Euler characteristic is negative. Recently, we have extended this result in the following two directions.

(1) When M is two-dimensional and the Euler characteristic is greater than or equal to zero.

(2) When the volume of M is infinite.

In this talk, we will mainly discuss (1). The key idea is to use the fundamental group of the configuration space of M (i.e., the braid group), rather than the fundamental group of M. If time permits, we will also explain (2). For this extension, we introduce the notion of "norm controlled cohomology".

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

### 2020/12/15

17:00-18:00 Online

Pre-registration required. See our seminar webpage.

Braids, triangles and Lissajous curve (JAPANESE)

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

Pre-registration required. See our seminar webpage.

**Eiko Kin**(Osaka University)Braids, triangles and Lissajous curve (JAPANESE)

[ Abstract ]

The purpose of this talk is to introduce Lissajous 3-braids. Suppose we have a closed curve on the plane, and we consider the periodic motion of n points along the closed curve. If the motion is collision-free, then we get a braid obtained from the trajectory of the set of n points in question. In this talk, we consider 3-braids coming from the periodic motion of 3 points on Lissajous curves. We classify Lissajous 3-braids and present a parametrization in terms of natural numbers together with slopes. We also discuss some properties of pseudo-Anosov stretch factors for Lissajous 3-braids. The main tool is the shape sphere --- the configuration space of the oriented similarity classes of triangles. This is a joint work with Hiroaki Nakamura and Hiroyuki Ogawa.

[ Reference URL ]The purpose of this talk is to introduce Lissajous 3-braids. Suppose we have a closed curve on the plane, and we consider the periodic motion of n points along the closed curve. If the motion is collision-free, then we get a braid obtained from the trajectory of the set of n points in question. In this talk, we consider 3-braids coming from the periodic motion of 3 points on Lissajous curves. We classify Lissajous 3-braids and present a parametrization in terms of natural numbers together with slopes. We also discuss some properties of pseudo-Anosov stretch factors for Lissajous 3-braids. The main tool is the shape sphere --- the configuration space of the oriented similarity classes of triangles. This is a joint work with Hiroaki Nakamura and Hiroyuki Ogawa.

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

### 2020/12/08

17:30-18:30 Online

Pre-registration required. See our seminar webpage.

The intersection polynomials of a virtual knot (JAPANESE)

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

Pre-registration required. See our seminar webpage.

**Shin Satoh**(Kobe University)The intersection polynomials of a virtual knot (JAPANESE)

[ Abstract ]

We define two kinds of invariants of a virtual knot called the first and second intersection polynomials. The definition is based on the intersection number of a pair of curves on a closed surface. We study several properties of the polynomials. By introducing invariants of long virtual knots, we give connected sum formulae of the intersection polynomials, and prove that there are infinitely many connected sums of any two virtual knots as an application. Furthermore, by studying the behavior under a crossing change, we show that the intersection polynomials are finite type invariants of order two, and find an invariant of a flat virtual knot derived from the the intersection polynomials. This is a joint work with R. Higa, T. Nakamura, and Y. Nakanishi.

[ Reference URL ]We define two kinds of invariants of a virtual knot called the first and second intersection polynomials. The definition is based on the intersection number of a pair of curves on a closed surface. We study several properties of the polynomials. By introducing invariants of long virtual knots, we give connected sum formulae of the intersection polynomials, and prove that there are infinitely many connected sums of any two virtual knots as an application. Furthermore, by studying the behavior under a crossing change, we show that the intersection polynomials are finite type invariants of order two, and find an invariant of a flat virtual knot derived from the the intersection polynomials. This is a joint work with R. Higa, T. Nakamura, and Y. Nakanishi.

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

### 2020/12/01

17:00-18:00 Online

Pre-registration required. See our seminar webpage.

Goeritz groups of bridge decompositions (JAPANESE)

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

Pre-registration required. See our seminar webpage.

**Yuya Koda**(Hiroshima University)Goeritz groups of bridge decompositions (JAPANESE)

[ Abstract ]

For a bridge decomposition of a link in the 3-sphere, we define the Goeritz group to be the group of isotopy classes of orientation-preserving homeomorphisms of the 3-sphere that preserve each of the bridge sphere and link setwise. The Birman-Hilden theory tells us that this is a $\mathbb{Z} / 2 \mathbb{Z}$-quotient of a "hyperelliptic Goeritz group". In this talk, we discuss properties, mainly of dynamical nature, of this group using a measure of complexity called the distance of the decomposition. We then give an application to the asymptotic behavior of the minimal entropies for the original Goeritz groups of Heegaard splittings. This talk is based on a joint work with Susumu Hirose, Daiki Iguchi and Eiko Kin.

[ Reference URL ]For a bridge decomposition of a link in the 3-sphere, we define the Goeritz group to be the group of isotopy classes of orientation-preserving homeomorphisms of the 3-sphere that preserve each of the bridge sphere and link setwise. The Birman-Hilden theory tells us that this is a $\mathbb{Z} / 2 \mathbb{Z}$-quotient of a "hyperelliptic Goeritz group". In this talk, we discuss properties, mainly of dynamical nature, of this group using a measure of complexity called the distance of the decomposition. We then give an application to the asymptotic behavior of the minimal entropies for the original Goeritz groups of Heegaard splittings. This talk is based on a joint work with Susumu Hirose, Daiki Iguchi and Eiko Kin.

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

### 2020/11/24

17:30-18:30 Online

Pre-registration required. See our seminar webpage.

Intersection of Poincare holonomy varieties and Bers' simultaneous uniformization theorem (JAPANESE)

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

Pre-registration required. See our seminar webpage.

**Shinpei Baba**(Osaka University)Intersection of Poincare holonomy varieties and Bers' simultaneous uniformization theorem (JAPANESE)

[ Abstract ]

Given a marked compact Riemann surface X, the vector space of holomorphic quadratic differentials on X is identified with the space of CP

In this manner, different Riemann surfaces structures yield different half-dimensional smooth analytic subvarieties in the character variety. In this talk, we discuss some properties of their intersection. To do so, we utilize a cut-and-paste operation, called grafting, of CP

[ Reference URL ]Given a marked compact Riemann surface X, the vector space of holomorphic quadratic differentials on X is identified with the space of CP

^{1}-structures on X. Then, by the holonomy representations of CP^{1}-structures, this vector space properly embeds into the PSL(2, C)-character variety, the space of representations of the fundamental group of X into PSL(2,C).In this manner, different Riemann surfaces structures yield different half-dimensional smooth analytic subvarieties in the character variety. In this talk, we discuss some properties of their intersection. To do so, we utilize a cut-and-paste operation, called grafting, of CP

^{1}-structures.https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

### 2020/11/17

17:00-18:00 Online

Pre-registration required. See our seminar webpage.

Lefschetz fibration on the Milnor fibers of simple elliptic and cusp singularities (JAPANESE)

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

Pre-registration required. See our seminar webpage.

**Yoshihiko Mitsumatsu**(Chuo University)Lefschetz fibration on the Milnor fibers of simple elliptic and cusp singularities (JAPANESE)

[ Abstract ]

In this talk a joint work with Naohiko Kasuya(Kyoto Sangyo U.), Hiroki Kodama(Tohoku U.), and Atsuhide Mori(Osaka Dental U.) is reported. The main result is the following.

There exist a Lefschetz fibration of the Milnor fiber of T_{pqr}-singularity (1/p + 1/q + 1/r ≦ 1) to the unit disk with regular fiber diffeomorphic to T^2.

An outline of the construction will be explained, through which, the space of 2-jets of (R^4, 0) to (R^2, 0) is analysed. This is motivated by F. Presas' suggestion that the speaker's construction of regular Poisson structures(=leafwise symplectic foliations) on S^5 might be interpreted by ``leafwise Lefschetz fibration''. These Lefschetz fibrations give a way to look at K3 surfaces through an extended class of Arnol'd's strange duality. These applications are introduced as well.

[ Reference URL ]In this talk a joint work with Naohiko Kasuya(Kyoto Sangyo U.), Hiroki Kodama(Tohoku U.), and Atsuhide Mori(Osaka Dental U.) is reported. The main result is the following.

There exist a Lefschetz fibration of the Milnor fiber of T_{pqr}-singularity (1/p + 1/q + 1/r ≦ 1) to the unit disk with regular fiber diffeomorphic to T^2.

An outline of the construction will be explained, through which, the space of 2-jets of (R^4, 0) to (R^2, 0) is analysed. This is motivated by F. Presas' suggestion that the speaker's construction of regular Poisson structures(=leafwise symplectic foliations) on S^5 might be interpreted by ``leafwise Lefschetz fibration''. These Lefschetz fibrations give a way to look at K3 surfaces through an extended class of Arnol'd's strange duality. These applications are introduced as well.

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

### 2020/10/27

17:00-18:00 Online

Pre-registration required. See our seminar webpage.

Vassiliev derivatives of Khovanov homology and its application (JAPANESE)

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

Pre-registration required. See our seminar webpage.

**Jun Yoshida**(The University of Tokyo)Vassiliev derivatives of Khovanov homology and its application (JAPANESE)

[ Abstract ]

Khovanov homology is a categorification of the Jones polynomial. It is known that Khovanov homology also arises from a categorical representation of braid groups, so we can regard it as a kind of quantum knot invariant. However, in contrast to the case of classical quantum invariants, its relation to Vassiliev invariants remains unclear. In this talk, aiming at the problem, we discuss a categorified version of Vassiliev skein relation on Khovanov homology. Namely, we extend Khovanov homology to singular links so that extended ones can be seen as "derivatives" in view of Vassiliev theory. As an application, we compute first derivatives to determine Khovanov homologies of twist knots. This talk is based on papers arXiv:2005.12664 (joint work with N.Ito) and arXiv:2007.15867.

[ Reference URL ]Khovanov homology is a categorification of the Jones polynomial. It is known that Khovanov homology also arises from a categorical representation of braid groups, so we can regard it as a kind of quantum knot invariant. However, in contrast to the case of classical quantum invariants, its relation to Vassiliev invariants remains unclear. In this talk, aiming at the problem, we discuss a categorified version of Vassiliev skein relation on Khovanov homology. Namely, we extend Khovanov homology to singular links so that extended ones can be seen as "derivatives" in view of Vassiliev theory. As an application, we compute first derivatives to determine Khovanov homologies of twist knots. This talk is based on papers arXiv:2005.12664 (joint work with N.Ito) and arXiv:2007.15867.

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

### 2020/10/20

17:00-18:00 Online

Pre-registration required. See our seminar webpage.

Poincaré duality for free loop spaces (ENGLISH)

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

Pre-registration required. See our seminar webpage.

**Alexandru Oancea**(Sorbonne Université)Poincaré duality for free loop spaces (ENGLISH)

[ Abstract ]

A certain number of dualities between homological and cohomological invariants of free loop spaces have been observed over the years, having the flavour of Poincaré duality but nevertheless holding in an infinite dimensional setting. The goal of the talk will be to explain these through a new duality theorem, whose proof uses symplectic methods. The talk will report on joint work with Kai Cieliebak and Nancy Hingston.

[ Reference URL ]A certain number of dualities between homological and cohomological invariants of free loop spaces have been observed over the years, having the flavour of Poincaré duality but nevertheless holding in an infinite dimensional setting. The goal of the talk will be to explain these through a new duality theorem, whose proof uses symplectic methods. The talk will report on joint work with Kai Cieliebak and Nancy Hingston.

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

### 2020/10/06

17:30-18:30 Online

Pre-registration required. See our seminar webpage.

The Atiyah-Patodi-Singer index of manifolds with boundary and domain-wall fermions (JAPANESE)

https://zoom.us/meeting/register/tJcqdO6pqz0pGNbwpZOpG-o2h4xJwmpma3zL

Pre-registration required. See our seminar webpage.

**Shinichiroh Matsuo**(Nagoya University)The Atiyah-Patodi-Singer index of manifolds with boundary and domain-wall fermions (JAPANESE)

[ Abstract ]

We introduce a mathematician-friendly formulation of the physicist-friendly derivation of the Atiyah-Patodi-Singer index.

In a previous work, motivated by the study of lattice gauge theory, we derived a formula expressing the Atiyah-Patodi-Singer index in terms of the eta invariant of “domain-wall fermion Dirac operators” when the base manifold is a flat 4-dimensional torus. Now we generalise this formula to any even dimensional closed Riemannian manifolds, and prove it mathematically rigorously. Our proof uses a Witten localisation argument combined with a devised embedding into a cylinder of one dimension higher. Our viewpoint sheds some new light on the interplay among the Atiyah-Patodi-Singer boundary condition, domain-wall fermions, and edge modes.

This talk is based on a joint paper arXiv:1910.01987, to appear in CMP, with H. Fukaya, M. Furuta, T. Onogi, S. Yamaguchi, and M. Yamashita.

[ Reference URL ]We introduce a mathematician-friendly formulation of the physicist-friendly derivation of the Atiyah-Patodi-Singer index.

In a previous work, motivated by the study of lattice gauge theory, we derived a formula expressing the Atiyah-Patodi-Singer index in terms of the eta invariant of “domain-wall fermion Dirac operators” when the base manifold is a flat 4-dimensional torus. Now we generalise this formula to any even dimensional closed Riemannian manifolds, and prove it mathematically rigorously. Our proof uses a Witten localisation argument combined with a devised embedding into a cylinder of one dimension higher. Our viewpoint sheds some new light on the interplay among the Atiyah-Patodi-Singer boundary condition, domain-wall fermions, and edge modes.

This talk is based on a joint paper arXiv:1910.01987, to appear in CMP, with H. Fukaya, M. Furuta, T. Onogi, S. Yamaguchi, and M. Yamashita.

https://zoom.us/meeting/register/tJcqdO6pqz0pGNbwpZOpG-o2h4xJwmpma3zL

### 2020/09/29

17:00-18:00 Online

Pre-registration required. See our seminar webpage.

Witten-Reshetikhin-Turaev function for a knot in Seifert manifolds (JAPANESE)

https://zoom.us/meeting/register/tJcqdO6pqz0pGNbwpZOpG-o2h4xJwmpma3zL

Pre-registration required. See our seminar webpage.

**Kohei Iwaki**(The University of Tokyo)Witten-Reshetikhin-Turaev function for a knot in Seifert manifolds (JAPANESE)

[ Abstract ]

In 1998, Lawrence-Zagier introduced a certain q-series and proved that its limit value at root of unity q=exp(2π i / K) coincides with the SU(2) Witten-Reshetikhin-Turaev (WRT) invariant of the Poincare homology sphere Σ(2,3,5) at the level K. Employing the idea of Gukov-Marino-Putrov based on resurgent analysis, we generalize the result of Lawrence-Zagier for the Seifert loops (Seifert manifolds with a single loop inside). That is, for each Seifert loop, we introduce an explicit q-series (WRT function) and show that its limit value at the root of unity coincides with the WRT invariant of the Seifert loop. We will also discuss a q-difference equation satisfied by the WRT function. This talk is based on a joint work with H. Fuji, H. Murakami and Y. Terashima which is available on arXiv:2007.15872.

[ Reference URL ]In 1998, Lawrence-Zagier introduced a certain q-series and proved that its limit value at root of unity q=exp(2π i / K) coincides with the SU(2) Witten-Reshetikhin-Turaev (WRT) invariant of the Poincare homology sphere Σ(2,3,5) at the level K. Employing the idea of Gukov-Marino-Putrov based on resurgent analysis, we generalize the result of Lawrence-Zagier for the Seifert loops (Seifert manifolds with a single loop inside). That is, for each Seifert loop, we introduce an explicit q-series (WRT function) and show that its limit value at the root of unity coincides with the WRT invariant of the Seifert loop. We will also discuss a q-difference equation satisfied by the WRT function. This talk is based on a joint work with H. Fuji, H. Murakami and Y. Terashima which is available on arXiv:2007.15872.

https://zoom.us/meeting/register/tJcqdO6pqz0pGNbwpZOpG-o2h4xJwmpma3zL

### 2020/07/28

17:00-18:00 Online

Pre-registration required. See our seminar webpage.

A double filtration for the mapping class group and the Goeritz group of the sphere (ENGLISH)

https://zoom.us/webinar/register/WN_oS594Z6BRyaKNCvlm3yCoQ

Pre-registration required. See our seminar webpage.

**Anderson Vera**(RIMS, Kyoto University)A double filtration for the mapping class group and the Goeritz group of the sphere (ENGLISH)

[ Abstract ]

I will talk about a double-indexed filtration of the mapping class group and of the Goeritz group of the sphere, the latter is the group of isotopy classes of self-homeomorphisms of the 3-sphere which preserves the standard Heegaard splitting of $S^3$. In particular I will explain how this double filtration allows to write the Torelli group as a product of some subgroups of the mapping class group. A similar study could be done for the group of automorphisms of a free group. (work in progress with K. Habiro)

[ Reference URL ]I will talk about a double-indexed filtration of the mapping class group and of the Goeritz group of the sphere, the latter is the group of isotopy classes of self-homeomorphisms of the 3-sphere which preserves the standard Heegaard splitting of $S^3$. In particular I will explain how this double filtration allows to write the Torelli group as a product of some subgroups of the mapping class group. A similar study could be done for the group of automorphisms of a free group. (work in progress with K. Habiro)

https://zoom.us/webinar/register/WN_oS594Z6BRyaKNCvlm3yCoQ

### 2020/07/21

17:00-18:00 Online

Pre-registration required. See our seminar webpage.

Twisted arrow categories of operads and Segal conditions (ENGLISH)

https://zoom.us/webinar/register/WN_oS594Z6BRyaKNCvlm3yCoQ

Pre-registration required. See our seminar webpage.

**Sergei Burkin**(The University of Tokyo)Twisted arrow categories of operads and Segal conditions (ENGLISH)

[ Abstract ]

We generalize twisted arrow category construction from categories to operads, and show that several important categories, including the simplex category $\Delta$, Segal's category $\Gamma$ and Moerdijk--Weiss category $\Omega$ are twisted arrow categories of operads. Twisted arrow categories of operads are closely connected with Segal conditions, and the corresponding theory can be generalized from multi-object associative algebras (i.e. categories) to multi-object P-algebras for reasonably nice operads P.

[ Reference URL ]We generalize twisted arrow category construction from categories to operads, and show that several important categories, including the simplex category $\Delta$, Segal's category $\Gamma$ and Moerdijk--Weiss category $\Omega$ are twisted arrow categories of operads. Twisted arrow categories of operads are closely connected with Segal conditions, and the corresponding theory can be generalized from multi-object associative algebras (i.e. categories) to multi-object P-algebras for reasonably nice operads P.

https://zoom.us/webinar/register/WN_oS594Z6BRyaKNCvlm3yCoQ

### 2020/07/21

18:00-19:00 Online

Pre-registration required. See our seminar webpage.

Monopole Floer homology for codimension-3 Riemannian foliation (ENGLISH)

https://zoom.us/webinar/register/WN_oS594Z6BRyaKNCvlm3yCoQ

Pre-registration required. See our seminar webpage.

**Dexie Lin**(The University of Tokyo)Monopole Floer homology for codimension-3 Riemannian foliation (ENGLISH)

[ Abstract ]

In this paper, we give a systematic study of Seiberg-Witten theory on closed oriented manifold with codimension-3 oriented Riemannian foliation. Under a certain topological condition, we construct the basic monopole Floer homologies for a transverse spinc structure with a bundle-like metric, generic perturbation and a complete local system. We will show that these homologies are independent of the bundle-like metric and generic perturbation. The major difference between the basic monopole Floer homologies and the ones on manifolds is the necessity to use the complete local system to construct the monopole Floer homologies.

[ Reference URL ]In this paper, we give a systematic study of Seiberg-Witten theory on closed oriented manifold with codimension-3 oriented Riemannian foliation. Under a certain topological condition, we construct the basic monopole Floer homologies for a transverse spinc structure with a bundle-like metric, generic perturbation and a complete local system. We will show that these homologies are independent of the bundle-like metric and generic perturbation. The major difference between the basic monopole Floer homologies and the ones on manifolds is the necessity to use the complete local system to construct the monopole Floer homologies.

https://zoom.us/webinar/register/WN_oS594Z6BRyaKNCvlm3yCoQ

### 2020/07/14

17:30-18:30 Online

Joint with Lie Groups and Representation Theory Seminar. Pre-registration required. See our seminar webpage.

Kobayashi's properness criterion and totally geodesic submanifolds in locally symmetric spaces (JAPANESE)

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

Joint with Lie Groups and Representation Theory Seminar. Pre-registration required. See our seminar webpage.

**Takayuki Okuda**(Hiroshima University)Kobayashi's properness criterion and totally geodesic submanifolds in locally symmetric spaces (JAPANESE)

[ Abstract ]

Let G be a Lie group and X a homogeneous G-space. A discrete subgroup of G acting on X properly is called a discontinuous group for X. We are interested in constructions and classifications of discontinuous groups for a given X.

It is well-known that if the isotropies of G on X are compact, any closed subgroup acts on X properly. However, the cases where the isotropies are non-compact, the same claim does not hold in general.

Let us consider the case where G is a linear reductive. In this situation, T. Kobayashi [Math. Ann. (1989)], [J. Lie Theory (1996)]

gave a criterion for the properness of the action on a homogeneous G-space X of closed subgroups in G.

In this talk, we consider homogeneous G-spaces of reductive types realized as families of totally geodesic submanifolds in non-compact Riemannian symmetric spaces. As a main result, we give a translation of Kobayashi's criterion within the framework of Riemannian geometry. In particular, for a torsion-free discrete subgroup of G, the criterion can be stated in terms of totally geodesic submanifolds in the Riemannian locally symmetric space corresponding to the subgroup in G.

[ Reference URL ]Let G be a Lie group and X a homogeneous G-space. A discrete subgroup of G acting on X properly is called a discontinuous group for X. We are interested in constructions and classifications of discontinuous groups for a given X.

It is well-known that if the isotropies of G on X are compact, any closed subgroup acts on X properly. However, the cases where the isotropies are non-compact, the same claim does not hold in general.

Let us consider the case where G is a linear reductive. In this situation, T. Kobayashi [Math. Ann. (1989)], [J. Lie Theory (1996)]

gave a criterion for the properness of the action on a homogeneous G-space X of closed subgroups in G.

In this talk, we consider homogeneous G-spaces of reductive types realized as families of totally geodesic submanifolds in non-compact Riemannian symmetric spaces. As a main result, we give a translation of Kobayashi's criterion within the framework of Riemannian geometry. In particular, for a torsion-free discrete subgroup of G, the criterion can be stated in terms of totally geodesic submanifolds in the Riemannian locally symmetric space corresponding to the subgroup in G.

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

### 2020/07/07

17:00-18:00 Online

Pre-registration required. See our seminar webpage.

Abelian quotients of the Y-filtration on the homology cylinders via the LMO functor (JAPANESE)

https://zoom.us/webinar/register/WN_oS594Z6BRyaKNCvlm3yCoQ

Pre-registration required. See our seminar webpage.

**Yuta Nozaki**(Hiroshima University)Abelian quotients of the Y-filtration on the homology cylinders via the LMO functor (JAPANESE)

[ Abstract ]

We construct a series of homomorphisms on the Y-filtration on the homology cylinders via the mod $\mathbb{Z}$ reduction of the LMO functor. The restriction of our homomorphism to the lower central series of the Torelli group does not factor through Morita's refinement of the Johnson homomorphism. We use it to show that the abelianization of the Johnson kernel of a closed surface has torsion elements. This is the joint work with Masatoshi Sato and Masaaki Suzuki.

[ Reference URL ]We construct a series of homomorphisms on the Y-filtration on the homology cylinders via the mod $\mathbb{Z}$ reduction of the LMO functor. The restriction of our homomorphism to the lower central series of the Torelli group does not factor through Morita's refinement of the Johnson homomorphism. We use it to show that the abelianization of the Johnson kernel of a closed surface has torsion elements. This is the joint work with Masatoshi Sato and Masaaki Suzuki.

https://zoom.us/webinar/register/WN_oS594Z6BRyaKNCvlm3yCoQ

### 2020/06/30

17:00-18:00 Online

Pre-registration required. See our seminar webpage.

Homology of right-angled Artin kernels (ENGLISH)

https://zoom.us/webinar/register/WN_oS594Z6BRyaKNCvlm3yCoQ

Pre-registration required. See our seminar webpage.

**Daniel Matei**(IMAR Bucharest)Homology of right-angled Artin kernels (ENGLISH)

[ Abstract ]

The right-angled Artin groups A(G) are the finitely presented groups associated to a finite simplicial graph G=(V,E), which are generated by the vertices V satisfying commutator relations vw=wv for every edge vw in E. An Artin kernel

N

[ Reference URL ]The right-angled Artin groups A(G) are the finitely presented groups associated to a finite simplicial graph G=(V,E), which are generated by the vertices V satisfying commutator relations vw=wv for every edge vw in E. An Artin kernel

N

_{h}(G) is defined by an epimorphism h of A(G) onto the integers. In this talk, we discuss the module structure over the Laurent polynomial ring of the homology groups of N_{h}(G).https://zoom.us/webinar/register/WN_oS594Z6BRyaKNCvlm3yCoQ

### 2020/06/23

17:00-18:00 Online

Pre-registration required. See our seminar webpage.

Gauge theory and the diffeomorphism and homeomorphism groups of 4-manifolds (JAPANESE)

https://zoom.us/webinar/register/WN_oS594Z6BRyaKNCvlm3yCoQ

Pre-registration required. See our seminar webpage.

**Hokuto Konno**(The University of Tokyo)Gauge theory and the diffeomorphism and homeomorphism groups of 4-manifolds (JAPANESE)

[ Abstract ]

I will explain my recent collaboration with several groups that develops gauge theory for families

to extract difference between the diffeomorphism groups and the homeomorphism groups of 4-manifolds.

After Donaldson’s celebrated diagonalization theorem, gauge theory has given strong constraints on the topology of smooth 4-manifolds. Combining such constraints with Freedman’s theory, one may find many non-smoothable topological 4-manifolds.

Recently, a family version of this argument was started by T. Kato, N. Nakamura and myself, and soon later it was developed also by D. Baraglia and his collaborating work with myself. More precisely, considering gauge theory for smooth fiber bundles of 4-manifolds, they obtained some constraints on the topology of smooth 4-manifold bundles. Using such constraints, they detected non-smoothable topological fiber bundles of smooth 4-manifolds. The existence of such bundles implies that there is homotopical difference between the diffeomorphism and homeomorphism groups of the 4-manifolds given as the fibers.

If time permits, I will also mention my collaboration with Baraglia which shows that a K3 surface gives a counterexample to the Nielsen realization problem in dimension 4. This example reveals also that there is difference between the Nielsen realization problems asked in the smooth category and the topological category.

[ Reference URL ]I will explain my recent collaboration with several groups that develops gauge theory for families

to extract difference between the diffeomorphism groups and the homeomorphism groups of 4-manifolds.

After Donaldson’s celebrated diagonalization theorem, gauge theory has given strong constraints on the topology of smooth 4-manifolds. Combining such constraints with Freedman’s theory, one may find many non-smoothable topological 4-manifolds.

Recently, a family version of this argument was started by T. Kato, N. Nakamura and myself, and soon later it was developed also by D. Baraglia and his collaborating work with myself. More precisely, considering gauge theory for smooth fiber bundles of 4-manifolds, they obtained some constraints on the topology of smooth 4-manifold bundles. Using such constraints, they detected non-smoothable topological fiber bundles of smooth 4-manifolds. The existence of such bundles implies that there is homotopical difference between the diffeomorphism and homeomorphism groups of the 4-manifolds given as the fibers.

If time permits, I will also mention my collaboration with Baraglia which shows that a K3 surface gives a counterexample to the Nielsen realization problem in dimension 4. This example reveals also that there is difference between the Nielsen realization problems asked in the smooth category and the topological category.

https://zoom.us/webinar/register/WN_oS594Z6BRyaKNCvlm3yCoQ

### 2020/01/28

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

Existence problems for fibered links (JAPANESE)

**Nozomu Sekino**(The University of Tokyo)Existence problems for fibered links (JAPANESE)

[ Abstract ]

It is known that every connected orientable closed 3-manifold has a fibered knot. However, finding (and classifying) fibered links whose fiber surfaces are fixed homeomorphism type in a given 3-manifold is difficult in general. We give a criterion of a simple closed curve on a genus 2g Heegaard surface being a genus g fibered knot in terms of its Heegaard diagram. As an application, we can prove the non-existence of genus one fibered knots in some Seifert manifolds.

There is one generalization of fibered links, homologically fibered links. This requests that the complement of the "fiber surface" is a homologically product of a surface and an interval. We give a necessary and sufficient condition for a connected sums of lens spaces of having a homologically fibered link whose fiber surfaces are some fixed types as some algebraic equations.

It is known that every connected orientable closed 3-manifold has a fibered knot. However, finding (and classifying) fibered links whose fiber surfaces are fixed homeomorphism type in a given 3-manifold is difficult in general. We give a criterion of a simple closed curve on a genus 2g Heegaard surface being a genus g fibered knot in terms of its Heegaard diagram. As an application, we can prove the non-existence of genus one fibered knots in some Seifert manifolds.

There is one generalization of fibered links, homologically fibered links. This requests that the complement of the "fiber surface" is a homologically product of a surface and an interval. We give a necessary and sufficient condition for a connected sums of lens spaces of having a homologically fibered link whose fiber surfaces are some fixed types as some algebraic equations.

### 2020/01/28

18:00-19:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Fibred cusp b-pseudodifferential operators and its applications (JAPANESE)

**Jun Watanabe**(The University of Tokyo)Fibred cusp b-pseudodifferential operators and its applications (JAPANESE)

[ Abstract ]

Melrose's b-calculus and its variants are important tools to study index problems on manifolds with singularities. In this talk, we introduce a new variant "fibred cusp b-calculus", which is a generalization of fibred cusp calculus of Mazzeo-Melrose and b-calculus of Melrose. We discuss the basic property of this calculus and give a relative index formula. As its application, we prove the index theorem for a Z/k manifold with boundary, which is a generalization of the mod k index theorem of Freed-Melrose.

Melrose's b-calculus and its variants are important tools to study index problems on manifolds with singularities. In this talk, we introduce a new variant "fibred cusp b-calculus", which is a generalization of fibred cusp calculus of Mazzeo-Melrose and b-calculus of Melrose. We discuss the basic property of this calculus and give a relative index formula. As its application, we prove the index theorem for a Z/k manifold with boundary, which is a generalization of the mod k index theorem of Freed-Melrose.

### 2020/01/14

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

SO(3)-invariant G

**Ryohei Chihara**(The University of Tokyo)SO(3)-invariant G

_{2}-geometry (JAPANESE)
[ Abstract ]

Berger's classification of holonomy groups of Riemannian manifolds includes exceptional cases of the Lie groups G

Berger's classification of holonomy groups of Riemannian manifolds includes exceptional cases of the Lie groups G

_{2}and Spin(7). Many authors have studied G_{2}- and Spin(7)-manifolds with torus symmetry. In this talk, we generalize the celebrated examples due to Bryant and Salamon and study G_{2}-manifolds with SO(3)-symmetry. Such torsion-free G_{2}-structures are described as a dynamical system of SU(3)-structures on an SO(3)-fibration over a 3-manifold. As a main result, we reduce this system into a constrained Hamiltonian dynamical system on the cotangent bundle over the space of all Riemannian metrics on the 3-manifold. The Hamiltonian function is very similar to that of the Hamiltonian formulation of general relativity.### 2020/01/14

18:00-19:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Algebraic entropy of sign-stable mutation loops (JAPANESE)

**Tsukasa Ishibashi**(The University of Tokyo)Algebraic entropy of sign-stable mutation loops (JAPANESE)

[ Abstract ]

Since its discovery, the cluster algebra has been developed with friutful connections with other branches of mathematics, unifying several combinatorial operations as well as their positivity notions. A mutation loop induces several dynamical systems via cluster transformations, and they form a group which can be seen as a combinatorial generalization of the mapping class groups of marked surfaces.

We introduce a new property of mutation loops called the sign stability, with a focus on an asymptotic behavior of the iteration of the tropicalized cluster X-transformation. A sign-stable mutation loop has a numerical invariant which we call the "cluster stretch factor", in analogy with the stretch factor of a pseudo-Anosov mapping class on a marked surface. We compute the algebraic entropies of the cluster A- and X-transformations induced by a sign-stable mutation loop, and conclude that these two coincide with the logarithm of the cluster stretch factor. This talk is based on a joint work with Shunsuke Kano.

Since its discovery, the cluster algebra has been developed with friutful connections with other branches of mathematics, unifying several combinatorial operations as well as their positivity notions. A mutation loop induces several dynamical systems via cluster transformations, and they form a group which can be seen as a combinatorial generalization of the mapping class groups of marked surfaces.

We introduce a new property of mutation loops called the sign stability, with a focus on an asymptotic behavior of the iteration of the tropicalized cluster X-transformation. A sign-stable mutation loop has a numerical invariant which we call the "cluster stretch factor", in analogy with the stretch factor of a pseudo-Anosov mapping class on a marked surface. We compute the algebraic entropies of the cluster A- and X-transformations induced by a sign-stable mutation loop, and conclude that these two coincide with the logarithm of the cluster stretch factor. This talk is based on a joint work with Shunsuke Kano.

### 2020/01/07

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

Magnitude homology of crushable spaces (JAPANESE)

**Yasuhiko Asao**(The University of Tokyo)Magnitude homology of crushable spaces (JAPANESE)

[ Abstract ]

The magnitude homology and the blurred magnitude homology are novel notions of homology theory for general metric spaces coined by Leinster et al. They are expected to be dealt with in the context of Topological Data Analysis since its original idea is based on a kind of "persistence of points clouds". However, little property of them has been revealed. In this talk, we see that the blurred magnitude homology is trivial when a metric space is contractible by a distance decreasing homotopy. We use techniques from singular homology theory.

The magnitude homology and the blurred magnitude homology are novel notions of homology theory for general metric spaces coined by Leinster et al. They are expected to be dealt with in the context of Topological Data Analysis since its original idea is based on a kind of "persistence of points clouds". However, little property of them has been revealed. In this talk, we see that the blurred magnitude homology is trivial when a metric space is contractible by a distance decreasing homotopy. We use techniques from singular homology theory.

### 2020/01/07

18:00-19:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Intersection number estimate of rational Lagrangian immersions in cotangent bundles via microlocal sheaf theory (JAPANESE)

**Tomohiro Asano**(The University of Tokyo)Intersection number estimate of rational Lagrangian immersions in cotangent bundles via microlocal sheaf theory (JAPANESE)

[ Abstract ]

Guillermou associated sheaves to exact Lagrangian submanifolds in cotangent bundles and proved topological properties of the Lagrangian submanifolds. In this talk, I will give an estimate on the displacement energy of rational Lagrangian immersions in cotangent bundles with intersection number estimates via microlocal sheaf theory. This result overlaps with results by Chekanov, Liu, and Akaho via Floer theory. This is joint work with Yuichi Ike.

Guillermou associated sheaves to exact Lagrangian submanifolds in cotangent bundles and proved topological properties of the Lagrangian submanifolds. In this talk, I will give an estimate on the displacement energy of rational Lagrangian immersions in cotangent bundles with intersection number estimates via microlocal sheaf theory. This result overlaps with results by Chekanov, Liu, and Akaho via Floer theory. This is joint work with Yuichi Ike.

### 2019/12/17

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

Symplectic homology of fiberwise convex sets and homology of loop spaces (JAPANESE)

**Kei Irie**(The University of Tokyo)Symplectic homology of fiberwise convex sets and homology of loop spaces (JAPANESE)

[ Abstract ]

For any (compact) subset in the symplectic vector space, one can define its symplectic capacity by using symplectic homology, which is a version of Floer homology.

In general, it is very difficult to compute or estimate this capacity directly from its definition, since the definition of Floer homology involves counting solutions of nonlinear PDEs (so called Floer equations). In this talk, we consider the symplectic vector space as the cotangent bundle of the Euclidean space, and show a formula which computes symplectic homology and capacity of fiberwise convex sets from homology of loop spaces. We also explain two applications of this formula.

For any (compact) subset in the symplectic vector space, one can define its symplectic capacity by using symplectic homology, which is a version of Floer homology.

In general, it is very difficult to compute or estimate this capacity directly from its definition, since the definition of Floer homology involves counting solutions of nonlinear PDEs (so called Floer equations). In this talk, we consider the symplectic vector space as the cotangent bundle of the Euclidean space, and show a formula which computes symplectic homology and capacity of fiberwise convex sets from homology of loop spaces. We also explain two applications of this formula.

### 2019/12/10

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

q-Deformation of a continued fraction and its applications (JAPANESE)

**Takeyoshi Kogiso**(Josai University)q-Deformation of a continued fraction and its applications (JAPANESE)

[ Abstract ]

A kind of q-Deformation of continued fractions was introduced by Morier-Genoud and Ovsienko. The most important application of this q-deformation of regular (or negative) continued fraction expansion of rational number r/s is to calculate the Jones polynomial of the rational link of r/s. Moreover we can apply the q-deformation of this continued fraction to quadratic irrational number theory and combinatorics.

On the other hand, there exist another recipes for determing the Jones polynomials by using Lee-Schiffler's snake graph and by using Kogiso-Wakui's Conway-Coxeter frieze method. Therefore, another approach of the result due to Morier-Genoud and Ovsienko can be considered from the viewpoint of a Conway-Coxeter frieze and a snake graph. Furthermore, we can consider a cluster-variable transformation of continued fractions as a further generalization by using ancestoral triangles used in the Kogiso-Wakui.

A kind of q-Deformation of continued fractions was introduced by Morier-Genoud and Ovsienko. The most important application of this q-deformation of regular (or negative) continued fraction expansion of rational number r/s is to calculate the Jones polynomial of the rational link of r/s. Moreover we can apply the q-deformation of this continued fraction to quadratic irrational number theory and combinatorics.

On the other hand, there exist another recipes for determing the Jones polynomials by using Lee-Schiffler's snake graph and by using Kogiso-Wakui's Conway-Coxeter frieze method. Therefore, another approach of the result due to Morier-Genoud and Ovsienko can be considered from the viewpoint of a Conway-Coxeter frieze and a snake graph. Furthermore, we can consider a cluster-variable transformation of continued fractions as a further generalization by using ancestoral triangles used in the Kogiso-Wakui.