## Tuesday Seminar on Topology

Seminar information archive ～05/28｜Next seminar｜Future seminars 05/29～

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**

### 2017/01/10

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

Stability of anti-canonically balanced metrics (JAPANESE)

**Shunsuke Saito**(The University of Tokyo)Stability of anti-canonically balanced metrics (JAPANESE)

[ Abstract ]

Donaldson introduced "anti-canonically balanced metrics" on Fano manifolds, which is a finite dimensional analogue of Kähler-Einstein metrics. It is proved that anti-canonically balanced metrics are critical points of the quantized Ding functional.

We first study the slope at infinity of the quantized Ding functional along Bergman geodesic rays. Then, we introduce a new algebro-geometric stability of Fano manifolds based on the slope formula, and show that the existence of anti-canonically balanced metrics implies our stability. The relationship between the stability and others is also discussed.

This talk is based on a joint work with R. Takahashi (Tohoku Univ).

Donaldson introduced "anti-canonically balanced metrics" on Fano manifolds, which is a finite dimensional analogue of Kähler-Einstein metrics. It is proved that anti-canonically balanced metrics are critical points of the quantized Ding functional.

We first study the slope at infinity of the quantized Ding functional along Bergman geodesic rays. Then, we introduce a new algebro-geometric stability of Fano manifolds based on the slope formula, and show that the existence of anti-canonically balanced metrics implies our stability. The relationship between the stability and others is also discussed.

This talk is based on a joint work with R. Takahashi (Tohoku Univ).

### 2017/01/10

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

Topological Invariants and Corner States for Hamiltonians on a Three Dimensional Lattice (JAPANESE)

**Shin Hayashi**(The University of Tokyo)Topological Invariants and Corner States for Hamiltonians on a Three Dimensional Lattice (JAPANESE)

[ Abstract ]

In condensed matter physics, a correspondence between two topological invariants defined for a gapped Hamiltonian is well-known. One is defined for such a Hamiltonian on a lattice (bulk invariant), and the other is defined for its restriction onto a subsemigroup (edge invariant). The edge invariant is related to the wave functions localized near the edge. This correspondence is known as the bulk-edge correspondence. In this talk, we consider a variant of this correspondence. We consider a periodic Hamiltonian on a three dimensional lattice (bulk) and its restrictions onto two subsemigroups (edges) and their intersection (corner). We will show that, if our Hamiltonian is "gapped" in some sense, we can define a topological invariant for the bulk and edges. We will also define another topological invariant related to the wave functions localized near the corner. We will explain that there is a correspondence between these two topological invariants by using the six-term exact sequence associated to the quarter-plane Toeplitz extension obtained by E. Park.

In condensed matter physics, a correspondence between two topological invariants defined for a gapped Hamiltonian is well-known. One is defined for such a Hamiltonian on a lattice (bulk invariant), and the other is defined for its restriction onto a subsemigroup (edge invariant). The edge invariant is related to the wave functions localized near the edge. This correspondence is known as the bulk-edge correspondence. In this talk, we consider a variant of this correspondence. We consider a periodic Hamiltonian on a three dimensional lattice (bulk) and its restrictions onto two subsemigroups (edges) and their intersection (corner). We will show that, if our Hamiltonian is "gapped" in some sense, we can define a topological invariant for the bulk and edges. We will also define another topological invariant related to the wave functions localized near the corner. We will explain that there is a correspondence between these two topological invariants by using the six-term exact sequence associated to the quarter-plane Toeplitz extension obtained by E. Park.

### 2016/12/20

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

Deligne-Mostow lattices and cone metrics on the sphere (ENGLISH)

**Irene Pasquinelli**(Durham University)Deligne-Mostow lattices and cone metrics on the sphere (ENGLISH)

[ Abstract ]

Finding lattices in PU(n,1) has been one of the major challenges of the last decades. One way of constructing lattices is to give a fundamental domain for its action on the complex hyperbolic space.

One approach, successful for some lattices, consists of seeing the complex hyperbolic space as the configuration space of cone metrics on the sphere and of studying the action of some maps exchanging the cone points with same cone angle.

In this talk we will see how this construction of fundamental polyhedra can be extended to almost all Deligne-Mostow lattices with three folding symmetry.

Finding lattices in PU(n,1) has been one of the major challenges of the last decades. One way of constructing lattices is to give a fundamental domain for its action on the complex hyperbolic space.

One approach, successful for some lattices, consists of seeing the complex hyperbolic space as the configuration space of cone metrics on the sphere and of studying the action of some maps exchanging the cone points with same cone angle.

In this talk we will see how this construction of fundamental polyhedra can be extended to almost all Deligne-Mostow lattices with three folding symmetry.

### 2016/12/13

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

Plane fields on 3-manifolds and asymptotic linking of the tangential incompressible flows (JAPANESE)

**Yoshihiko Mitsumatsu**(Chuo University)Plane fields on 3-manifolds and asymptotic linking of the tangential incompressible flows (JAPANESE)

[ Abstract ]

This is a report on a project in (a very slow) progress which aims to prove the tightness of contact structures associated with algebraic Anosov flows without using Bennequin's nor Gromov's results.

After introducing an interpretation of asymptotic linking pairing in terms of differential forms, we attach a subspaces of exact 2-forms to each plane field. We analyze this space in the case where the plane field is an algebraic Anosov foliation and explain what can be done using results from foliated cohomology and frameworks for secondary characteristic classes. We also show some explicit computations.

To close the talk, a quantization phenomenon which happens when a foliation is deformed into a contact structure is explained and we state some perspectives on applying the results on foliations to the tightness.

This is a report on a project in (a very slow) progress which aims to prove the tightness of contact structures associated with algebraic Anosov flows without using Bennequin's nor Gromov's results.

After introducing an interpretation of asymptotic linking pairing in terms of differential forms, we attach a subspaces of exact 2-forms to each plane field. We analyze this space in the case where the plane field is an algebraic Anosov foliation and explain what can be done using results from foliated cohomology and frameworks for secondary characteristic classes. We also show some explicit computations.

To close the talk, a quantization phenomenon which happens when a foliation is deformed into a contact structure is explained and we state some perspectives on applying the results on foliations to the tightness.

### 2016/12/06

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

Union of 3-punctured spheres in a hyperbolic 3-manifold (JAPANESE)

**Ken'ichi Yoshida**(The University of Tokyo)Union of 3-punctured spheres in a hyperbolic 3-manifold (JAPANESE)

[ Abstract ]

An essential 3-punctured sphere in a hyperbolic 3-manifold is isotopic to a totally geodesic one. We will classify the topological types for components of union of the totally geodesic 3-punctured spheres in an orientable hyperbolic 3-manifold. There are special types each of which appears in precisely one manifold.

An essential 3-punctured sphere in a hyperbolic 3-manifold is isotopic to a totally geodesic one. We will classify the topological types for components of union of the totally geodesic 3-punctured spheres in an orientable hyperbolic 3-manifold. There are special types each of which appears in precisely one manifold.

### 2016/11/29

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

Generalized spectral theory and its application to coupled oscillators on networks (JAPANESE)

**Hayato Chiba**(Kyushu University)Generalized spectral theory and its application to coupled oscillators on networks (JAPANESE)

[ Abstract ]

For a system of large coupled oscillators on networks, we show that the transition from the de-synchronous state to the synchronization occurs as the coupling strength increases. For the proof, the generalized spectral theory of linear operators is employed.

For a system of large coupled oscillators on networks, we show that the transition from the de-synchronous state to the synchronization occurs as the coupling strength increases. For the proof, the generalized spectral theory of linear operators is employed.

### 2016/11/22

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

Sullivan's coproduct on the reduced loop homology (JAPANESE)

**Takahito Naito**(The University of Tokyo)Sullivan's coproduct on the reduced loop homology (JAPANESE)

[ Abstract ]

In string topology, Sullivan introduced a coproduct on the reduced loop homology and showed that the homology has an infinitesimal bialgebra structure with respect to the coproduct and Chas-Sullivan loop product. In this talk, I will give a homotopy theoretic description of Sullivan's coproduct. By using the description, we obtain some computational examples of the structure over the rational number field. Moreover, I will also discuss a based loop space version of the coproduct.

In string topology, Sullivan introduced a coproduct on the reduced loop homology and showed that the homology has an infinitesimal bialgebra structure with respect to the coproduct and Chas-Sullivan loop product. In this talk, I will give a homotopy theoretic description of Sullivan's coproduct. By using the description, we obtain some computational examples of the structure over the rational number field. Moreover, I will also discuss a based loop space version of the coproduct.

### 2016/11/15

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

Cohomology of the moduli space of graphs and groups of homology cobordisms of surfaces (JAPANESE)

**Takuya Sakasai**(The University of Tokyo)Cohomology of the moduli space of graphs and groups of homology cobordisms of surfaces (JAPANESE)

[ Abstract ]

We construct an abelian quotient of the symplectic derivation Lie algebra of the free Lie algebra generated by the fundamental representation of the symplectic group. It gives an alternative proof of the fact first shown by Bartholdi that the top rational homology group of the moduli space of metric graphs of rank 7 is one dimensional. As an application, we construct a non-trivial abelian quotient of the homology cobordism group of a surface of positive genus. This talk is based on joint works with Shigeyuki Morita, Masaaki Suzuki and Gwénaël Massuyeau.

We construct an abelian quotient of the symplectic derivation Lie algebra of the free Lie algebra generated by the fundamental representation of the symplectic group. It gives an alternative proof of the fact first shown by Bartholdi that the top rational homology group of the moduli space of metric graphs of rank 7 is one dimensional. As an application, we construct a non-trivial abelian quotient of the homology cobordism group of a surface of positive genus. This talk is based on joint works with Shigeyuki Morita, Masaaki Suzuki and Gwénaël Massuyeau.

### 2016/11/08

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

Second mod 2 homology of Artin groups (JAPANESE)

**Toshiyuki Akita**(Hokkaido University)Second mod 2 homology of Artin groups (JAPANESE)

[ Abstract ]

After a brief survey on the K($\pi$,1) conjecture and homology of Artin groups, I will introduce our recent result: we determined second mod 2 homology of arbitrary Artin groups without assuming the K($\pi$,1)-conjecture. The key ingredients are Hopf's formula and a result of Howlett on Schur multipliers of Coxeter groups. This is a joint work with Ye Liu.

After a brief survey on the K($\pi$,1) conjecture and homology of Artin groups, I will introduce our recent result: we determined second mod 2 homology of arbitrary Artin groups without assuming the K($\pi$,1)-conjecture. The key ingredients are Hopf's formula and a result of Howlett on Schur multipliers of Coxeter groups. This is a joint work with Ye Liu.

### 2016/11/01

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

Higher-dimensional contact manifolds with infinitely many Stein fillings (JAPANESE)

**Takahiro Oba**(Tokyo Institute of Technology)Higher-dimensional contact manifolds with infinitely many Stein fillings (JAPANESE)

[ Abstract ]

A Stein fillings of a given contact manifold is a Stein domain whose boundary is contactomorphic to the given contact manifold.

Open books, Lefschetz fibrations, and mapping class groups of their fibers in particular help us to produce various contact manifolds and their Stein fillings. However, little is known about mapping class groups of higher-dimensional manifolds. This is one of the reasons that it was unknown whether there is a contact manifold of dimension > 3 with infinitely many Stein fillings. In this talk, I will choose a certain symplectic manifold as fibers of open books and Lefschetz fibrations and by using them, construct an infinite family of higher-dimensional contact manifolds with infinitely many Stein fillings.

A Stein fillings of a given contact manifold is a Stein domain whose boundary is contactomorphic to the given contact manifold.

Open books, Lefschetz fibrations, and mapping class groups of their fibers in particular help us to produce various contact manifolds and their Stein fillings. However, little is known about mapping class groups of higher-dimensional manifolds. This is one of the reasons that it was unknown whether there is a contact manifold of dimension > 3 with infinitely many Stein fillings. In this talk, I will choose a certain symplectic manifold as fibers of open books and Lefschetz fibrations and by using them, construct an infinite family of higher-dimensional contact manifolds with infinitely many Stein fillings.

### 2016/10/18

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

Conformal field theory for extended W-algebras (JAPANESE)

**Yoshitake Hashimoto**(Tokyo City University)Conformal field theory for extended W-algebras (JAPANESE)

[ Abstract ]

This talk is based on a joint work with A. Tsuchiya (Kavli IPMU) and T. Matsumoto (Nagoya Univ). In 2006 Feigin-Gainutdinov-Semikhatov-Tipunin introduced vertex operator algebras M called extended W-algebras. Tsuchiya-Wood developed representation theory of M by the method of

"infinitesimal deformation of parameter" and Jack symmetric polynomials.

In this talk I will discuss the following subjects:

1. "factorization" in conformal field theory,

2. tensor structure of the category of M-modules and "module-bimodule correspondence".

This talk is based on a joint work with A. Tsuchiya (Kavli IPMU) and T. Matsumoto (Nagoya Univ). In 2006 Feigin-Gainutdinov-Semikhatov-Tipunin introduced vertex operator algebras M called extended W-algebras. Tsuchiya-Wood developed representation theory of M by the method of

"infinitesimal deformation of parameter" and Jack symmetric polynomials.

In this talk I will discuss the following subjects:

1. "factorization" in conformal field theory,

2. tensor structure of the category of M-modules and "module-bimodule correspondence".

### 2016/10/11

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

The Kashiwara-Vergne problem and the Goldman-Turaev Lie bialgebra in genus zero (JAPANESE)

**Nariya Kawazumi**(The University of Tokyo)The Kashiwara-Vergne problem and the Goldman-Turaev Lie bialgebra in genus zero (JAPANESE)

[ Abstract ]

In view of results of Goldman and Turaev, the free vector space over the free loops on an oriented surface has a natural Lie bialgebra structure. The Goldman bracket has a formal description by using a special (or symplectic) expansion of the fundamental group of the surface. It is natural to ask for a formal description of the Turaev cobracket. We will show how to obtain a formal description of the Goldman-Turaev Lie bialgebra for genus 0 using a solution of the Kashiwara-Vergne problem. A similar description was recently obtained by Massuyeau using the Kontsevich integral. Moreover we propose a generalization of the Kashiwara-Vergne problem in the context of the Goldman-Turaev Lie bialgebra. This talk is based on a joint work with A. Alekseev, Y. Kuno and F. Naef.

In view of results of Goldman and Turaev, the free vector space over the free loops on an oriented surface has a natural Lie bialgebra structure. The Goldman bracket has a formal description by using a special (or symplectic) expansion of the fundamental group of the surface. It is natural to ask for a formal description of the Turaev cobracket. We will show how to obtain a formal description of the Goldman-Turaev Lie bialgebra for genus 0 using a solution of the Kashiwara-Vergne problem. A similar description was recently obtained by Massuyeau using the Kontsevich integral. Moreover we propose a generalization of the Kashiwara-Vergne problem in the context of the Goldman-Turaev Lie bialgebra. This talk is based on a joint work with A. Alekseev, Y. Kuno and F. Naef.

### 2016/09/27

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

CAT(0) properties for orthoscheme complexes (JAPANESE)

**Shouta Tounai**(The University of Tokyo)CAT(0) properties for orthoscheme complexes (JAPANESE)

[ Abstract ]

Gromov showed that a cubical complex is locally CAT(0) if and only if the link of every vertex is a flag complex. Brady and MacCammond introduced an orthoscheme complex as a generalization of cubical complexes. It is, however, difficult to tell whether an orthoscheme complex is (locally) CAT(0) or not. In this talk, I will discuss a translation of Gromov's characterization for orthoscheme complexes. As a generalization of Gromov's characterization, I will show that the orthoscheme complex of locally distributive semilattice is CAT(0) if and only if it is a flag semilattice.

Gromov showed that a cubical complex is locally CAT(0) if and only if the link of every vertex is a flag complex. Brady and MacCammond introduced an orthoscheme complex as a generalization of cubical complexes. It is, however, difficult to tell whether an orthoscheme complex is (locally) CAT(0) or not. In this talk, I will discuss a translation of Gromov's characterization for orthoscheme complexes. As a generalization of Gromov's characterization, I will show that the orthoscheme complex of locally distributive semilattice is CAT(0) if and only if it is a flag semilattice.

### 2016/07/19

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

The geometry of the curve graphs and beyond (JAPANESE)

**Yohsuke Watanabe**(University of Hawaii)The geometry of the curve graphs and beyond (JAPANESE)

[ Abstract ]

The curve graphs are locally infinite. However, by using Masur-Minsky's tight geodesics, one could view them as locally finite graphs. Bell-Fujiwara used a special property of tight geodesics and showed that the asymptotic dimension of the curve graphs is finite. In this talk, I will introduce a new class of geodesics which also has the property. If time permits, I will explain how such geodesics can be adapted in Out(F_n) setting.

The curve graphs are locally infinite. However, by using Masur-Minsky's tight geodesics, one could view them as locally finite graphs. Bell-Fujiwara used a special property of tight geodesics and showed that the asymptotic dimension of the curve graphs is finite. In this talk, I will introduce a new class of geodesics which also has the property. If time permits, I will explain how such geodesics can be adapted in Out(F_n) setting.

### 2016/07/12

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

Non-arithmetic lattices (ENGLISH)

**John Parker**(Durham University)Non-arithmetic lattices (ENGLISH)

[ Abstract ]

In this talk I will discuss arithmetic and non-arithmetic lattices and I will give a history of the problem of finding non-arithmetic lattices. I will also briefly describe the construction of new non-arithmetic lattices in SU(2,1) found in my joint workwith Martin Deraux and Julien Paupert.

In this talk I will discuss arithmetic and non-arithmetic lattices and I will give a history of the problem of finding non-arithmetic lattices. I will also briefly describe the construction of new non-arithmetic lattices in SU(2,1) found in my joint workwith Martin Deraux and Julien Paupert.

### 2016/06/28

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

Strong algebraization of fixed point properties (JAPANESE)

**Masato Mimura**(Tohoku University)Strong algebraization of fixed point properties (JAPANESE)

[ Abstract ]

Concerning proofs of fixed point properties, we succeed in removing all forms of "bounded generation" assumptions from previous celebrated strategy of "algebraization" by Y. Shalom ([Publ. IHES, 1999] and [ICM proceedings, 2006]). Our condition is stated as the existence of winning strategies for a certain "Game."

Concerning proofs of fixed point properties, we succeed in removing all forms of "bounded generation" assumptions from previous celebrated strategy of "algebraization" by Y. Shalom ([Publ. IHES, 1999] and [ICM proceedings, 2006]). Our condition is stated as the existence of winning strategies for a certain "Game."

### 2016/06/21

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

Spaces of chord diagrams of spherical curves (JAPANESE)

**Noboru Ito**(The University of Tokyo)Spaces of chord diagrams of spherical curves (JAPANESE)

[ Abstract ]

In this talk, the speaker introduces a framework to obtain (possibly infinitely many) new topological invariants of spherical curves under local homotopy moves (several types of Reidemeister moves). They are defined by chord diagrams, each of which is a configurations of even paired points on a circle. We see that these invariants have useful properties.

In this talk, the speaker introduces a framework to obtain (possibly infinitely many) new topological invariants of spherical curves under local homotopy moves (several types of Reidemeister moves). They are defined by chord diagrams, each of which is a configurations of even paired points on a circle. We see that these invariants have useful properties.

### 2016/06/14

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

Non-Kähler complex structures on R^4 (JAPANESE)

**Naohiko Kasuya**(Aoyama Gakuin University)Non-Kähler complex structures on R^4 (JAPANESE)

[ Abstract ]

We consider the following problem. "Is there any non-Kähler complex structure on R^{2n}?" If n=1, the answer is clearly negative. On the other hand, Calabi and Eckmann constructed non-Kähler complex structures on R^{2n} for n ≥ 3. In this talk, I will construct uncountably many non-Kähler complex structures on R^4, and give the affirmative answer to the case where n=2. For the construction, it is important to understand the genus-one achiral Lefschetz fibration S^4 → S^2 found by Yukio Matsumoto and Kenji Fukaya. This is a joint work with Antonio Jose Di Scala and Daniele Zuddas.

We consider the following problem. "Is there any non-Kähler complex structure on R^{2n}?" If n=1, the answer is clearly negative. On the other hand, Calabi and Eckmann constructed non-Kähler complex structures on R^{2n} for n ≥ 3. In this talk, I will construct uncountably many non-Kähler complex structures on R^4, and give the affirmative answer to the case where n=2. For the construction, it is important to understand the genus-one achiral Lefschetz fibration S^4 → S^2 found by Yukio Matsumoto and Kenji Fukaya. This is a joint work with Antonio Jose Di Scala and Daniele Zuddas.

### 2016/06/07

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

Topology of holomorphic Lefschetz pencils on the four-torus (JAPANESE)

**Kenta Hayano**(Keio University)Topology of holomorphic Lefschetz pencils on the four-torus (JAPANESE)

[ Abstract ]

In this talk, we will show that two holomorphic Lefschetz pencils on the four-torus are (smoothly) isomorphic if they have the same genus and divisibility. The proof relies on the theory of moduli spaces of polarized abelian surfaces. We will also give vanishing cycles of some holomorphic pencils of the four-torus explicitly. This is joint work with Noriyuki Hamada (The University of Tokyo).

In this talk, we will show that two holomorphic Lefschetz pencils on the four-torus are (smoothly) isomorphic if they have the same genus and divisibility. The proof relies on the theory of moduli spaces of polarized abelian surfaces. We will also give vanishing cycles of some holomorphic pencils of the four-torus explicitly. This is joint work with Noriyuki Hamada (The University of Tokyo).

### 2016/05/31

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

A linking invariant for algebraic curves (ENGLISH)

**Benoît Guerville-Ballé**(Tokyo Gakugei University)A linking invariant for algebraic curves (ENGLISH)

[ Abstract ]

We construct a topological invariant of algebraic plane curves, which is in some sense an adaptation of the linking number of knot theory. As an application, we show that this invariant distinguishes a new Zariski pair of curves (ie a pair of curves having same combinatorics, yet different topology).

We construct a topological invariant of algebraic plane curves, which is in some sense an adaptation of the linking number of knot theory. As an application, we show that this invariant distinguishes a new Zariski pair of curves (ie a pair of curves having same combinatorics, yet different topology).

### 2016/05/24

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

Independence of Roseman moves for surface-knot diagrams (JAPANESE)

**Kokoro Tanaka**(Tokyo Gakugei University)Independence of Roseman moves for surface-knot diagrams (JAPANESE)

[ Abstract ]

Roseman moves are seven types of local modifications for surface-knot diagrams in 3-space which generate ambient isotopies of surface-knots in 4-space. In this talk, I will discuss independence among the seven Roseman moves. In particular, I will focus on Roseman moves involving triple points and on those involving branch points. The former is joint work with Kanako Oshiro (Sophia University) and Kengo Kawamura (Osaka City University), and the latter is joint work with Masamichi Takase (Seikei University).

Roseman moves are seven types of local modifications for surface-knot diagrams in 3-space which generate ambient isotopies of surface-knots in 4-space. In this talk, I will discuss independence among the seven Roseman moves. In particular, I will focus on Roseman moves involving triple points and on those involving branch points. The former is joint work with Kanako Oshiro (Sophia University) and Kengo Kawamura (Osaka City University), and the latter is joint work with Masamichi Takase (Seikei University).

### 2016/05/17

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

Some dynamics of random walks on the mapping class groups (JAPANESE)

**Hidetoshi Masai**(The University of Tokyo)Some dynamics of random walks on the mapping class groups (JAPANESE)

[ Abstract ]

The dynamics of random walks on the mapping class groups on closed surfaces of genus >1 will be discussed. We define the topological entropy of random walks. Then we prove that the drift with respect to Thurston or Teichmüller metrics and the Lyapunov exponent all coincide with the topological entropy. This is a "random version" of pseudo-Anosov dynamics observed by Thurston and I will begin this talk by recalling the work of Thurston.

The dynamics of random walks on the mapping class groups on closed surfaces of genus >1 will be discussed. We define the topological entropy of random walks. Then we prove that the drift with respect to Thurston or Teichmüller metrics and the Lyapunov exponent all coincide with the topological entropy. This is a "random version" of pseudo-Anosov dynamics observed by Thurston and I will begin this talk by recalling the work of Thurston.

### 2016/05/10

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

On Milnor's link-homotopy invariants for handlebody-links (JAPANESE)

**Yuka Kotorii**(The University of Tokyo)On Milnor's link-homotopy invariants for handlebody-links (JAPANESE)

[ Abstract ]

A handlebody-link is a disjoint union of handlebodies embedded in $S^3$ and HL-homotopy is an equivalence relation on handlebody-links generated by self-crossing changes. A. Mizusawa and R. Nikkuni classified the set of HL-homotopy classes of 2-component handlebody-links completely using the linking numbers for handlebody-links. In this talk, by using Milnor's link-homotopy invariants, we construct an invariant for handlebody-links and give a bijection between the set of HL-homotopy classes of n-component handlebody-links with some assumption and a quotient of the action of the general linear group on a tensor product of modules. This is joint work with Atsuhiko Mizusawa at Waseda University.

A handlebody-link is a disjoint union of handlebodies embedded in $S^3$ and HL-homotopy is an equivalence relation on handlebody-links generated by self-crossing changes. A. Mizusawa and R. Nikkuni classified the set of HL-homotopy classes of 2-component handlebody-links completely using the linking numbers for handlebody-links. In this talk, by using Milnor's link-homotopy invariants, we construct an invariant for handlebody-links and give a bijection between the set of HL-homotopy classes of n-component handlebody-links with some assumption and a quotient of the action of the general linear group on a tensor product of modules. This is joint work with Atsuhiko Mizusawa at Waseda University.

### 2016/04/26

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

Arithmetic topology on branched covers of 3-manifolds (JAPANESE)

**Jun Ueki**(The University of Tokyo)Arithmetic topology on branched covers of 3-manifolds (JAPANESE)

[ Abstract ]

The analogy between 3-dimensional topology and number theory was first pointed out by Mazur in the 1960s, and it has been studied systematically by Kapranov, Reznikov, Morishita, and others. In their analogies, for example, knots and 3-manifolds correspond to primes and number rings respectively. The study of these analogies is called arithmetic topology now.

In my talk, based on their dictionary of analogies, we study analogues of idelic class field theory, Iwasawa theory, and Galois deformation theory in the context of 3-dimensional topology, and establish various foundational analogies in arithmetic topology.

The analogy between 3-dimensional topology and number theory was first pointed out by Mazur in the 1960s, and it has been studied systematically by Kapranov, Reznikov, Morishita, and others. In their analogies, for example, knots and 3-manifolds correspond to primes and number rings respectively. The study of these analogies is called arithmetic topology now.

In my talk, based on their dictionary of analogies, we study analogues of idelic class field theory, Iwasawa theory, and Galois deformation theory in the context of 3-dimensional topology, and establish various foundational analogies in arithmetic topology.

### 2016/04/19

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

Topological rigidity of finite cyclic group actions on compact surfaces (ENGLISH)

**Błażej Szepietowski**(Gdansk University)Topological rigidity of finite cyclic group actions on compact surfaces (ENGLISH)

[ Abstract ]

Two actions of a group on a surface are called topologically equivalent if they are conjugate by a homeomorphism of the surface. I will describe a method of enumeration (and classification) of topological equivalence classes of actions of a finite group on a compact surface, based on the combinatorial theory of noneuclidean crystallographic groups (NEC groups in short) and a relationship between the outer automorphism group of an NEC group and certain mapping class group. By this method we study topological equivalence of actions of a finite cyclic group on a compact surface, in the situation where the order of the group is large relative to the genus of the surface.

Two actions of a group on a surface are called topologically equivalent if they are conjugate by a homeomorphism of the surface. I will describe a method of enumeration (and classification) of topological equivalence classes of actions of a finite group on a compact surface, based on the combinatorial theory of noneuclidean crystallographic groups (NEC groups in short) and a relationship between the outer automorphism group of an NEC group and certain mapping class group. By this method we study topological equivalence of actions of a finite cyclic group on a compact surface, in the situation where the order of the group is large relative to the genus of the surface.