## トポロジー火曜セミナー

過去の記録 ～06/22｜次回の予定｜今後の予定 06/23～

開催情報 | 火曜日 17:00～18:30 数理科学研究科棟(駒場) 056号室 |
---|---|

担当者 | 河澄 響矢, 北山 貴裕, 逆井卓也 |

セミナーURL | http://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index.html |

**過去の記録**

### 2017年11月28日(火)

17:00-18:30 数理科学研究科棟(駒場) 056号室

Tea: Common Room 16:30-17:00

Diffeomorphism Groups of One-Manifolds (ENGLISH)

Tea: Common Room 16:30-17:00

**Sang-hyun Kim 氏**(Seoul National University)Diffeomorphism Groups of One-Manifolds (ENGLISH)

[ 講演概要 ]

Let a>=2 be a real number and k = [a]. We denote by Diff^a(S^1) the group of C^k diffeomorphisms such that the k--th derivatives are Hölder--continuous of exponent (a - k). For each real number a>=2, we prove that there exists a finitely generated group G < Diff^a(S^1) such that G admits no injective homomorphisms into Diff^b(S^1) for any b>a. This is joint work with Thomas Koberda.

Let a>=2 be a real number and k = [a]. We denote by Diff^a(S^1) the group of C^k diffeomorphisms such that the k--th derivatives are Hölder--continuous of exponent (a - k). For each real number a>=2, we prove that there exists a finitely generated group G < Diff^a(S^1) such that G admits no injective homomorphisms into Diff^b(S^1) for any b>a. This is joint work with Thomas Koberda.

### 2017年11月21日(火)

17:00-18:30 数理科学研究科棟(駒場) 056号室

Tea: Common Room 16:30-17:00

The space of short ropes and the classifying space of the space of long knots (JAPANESE)

Tea: Common Room 16:30-17:00

**境 圭一 氏**(信州大学)The space of short ropes and the classifying space of the space of long knots (JAPANESE)

[ 講演概要 ]

We prove affirmatively the conjecture raised by J. Mostovoy; the space of short ropes is weakly homotopy equivalent to the classifying space of the topological monoid (or category) of long knots in R^3. We make use of techniques developed by S. Galatius and O. Randal-Williams to construct a manifold space model of the classifying space of the space of long knots, and we give an explicit map from the space of short ropes to the model in a geometric way. This is joint work with Syunji Moriya (Osaka Prefecture University).

We prove affirmatively the conjecture raised by J. Mostovoy; the space of short ropes is weakly homotopy equivalent to the classifying space of the topological monoid (or category) of long knots in R^3. We make use of techniques developed by S. Galatius and O. Randal-Williams to construct a manifold space model of the classifying space of the space of long knots, and we give an explicit map from the space of short ropes to the model in a geometric way. This is joint work with Syunji Moriya (Osaka Prefecture University).

### 2017年11月07日(火)

17:00-18:30 数理科学研究科棟(駒場) 056号室

Tea: Common Room 16:30-17:00

On an explicit example of topologically protected corner states (JAPANESE)

Tea: Common Room 16:30-17:00

**林 晋 氏**(産総研・東北大オープンイノベーションラボラトリ)On an explicit example of topologically protected corner states (JAPANESE)

[ 講演概要 ]

In condensed matter physics, topologically protected (codimension-one) edge states are known to appear on the surface of some insulators reflecting some topology of its bulk. Such phenomena can be understood from the point of view of an index theory associated to the Toeplitz extension and are called the bulk-edge correspondence. In this talk, we consider instead the quarter-plane Toeplitz extension and index theory associated with it. As a result, we show that topologically protected (codimension-two) corner states appear reflecting some topology of the gapped bulk and two edges. Such new topological phases can be obtained by taking a ``product’’ of two classically known topological phases (2d type A and 1d type AIII topological phases). By using this construction, we obtain an example of a continuous family of bounded self-adjoint Fredholm quarter-plane Toeplitz operators whose spectral flow is nontrivial, which gives an explicit example of topologically protected corner states.

In condensed matter physics, topologically protected (codimension-one) edge states are known to appear on the surface of some insulators reflecting some topology of its bulk. Such phenomena can be understood from the point of view of an index theory associated to the Toeplitz extension and are called the bulk-edge correspondence. In this talk, we consider instead the quarter-plane Toeplitz extension and index theory associated with it. As a result, we show that topologically protected (codimension-two) corner states appear reflecting some topology of the gapped bulk and two edges. Such new topological phases can be obtained by taking a ``product’’ of two classically known topological phases (2d type A and 1d type AIII topological phases). By using this construction, we obtain an example of a continuous family of bounded self-adjoint Fredholm quarter-plane Toeplitz operators whose spectral flow is nontrivial, which gives an explicit example of topologically protected corner states.

### 2017年10月31日(火)

17:00-18:30 数理科学研究科棟(駒場) 056号室

Tea: Common Room 16:30-17:00

Nonamenable groups of piecewise projective homeomorphisms (ENGLISH)

Tea: Common Room 16:30-17:00

**Yash Lodha 氏**(École Polytechnique Fédérale de Lausanne)Nonamenable groups of piecewise projective homeomorphisms (ENGLISH)

[ 講演概要 ]

Groups of piecewise projective homeomorphisms provide elegant examples of groups that are non amenable, yet do not contain non abelian free subgroups. In this talk I will present a survey of these groups and discuss their striking properties. I will discuss properties such as (non)amenability, finiteness properties, normal subgroup structure, actions by various degrees of regularity and Tarski numbers.

Groups of piecewise projective homeomorphisms provide elegant examples of groups that are non amenable, yet do not contain non abelian free subgroups. In this talk I will present a survey of these groups and discuss their striking properties. I will discuss properties such as (non)amenability, finiteness properties, normal subgroup structure, actions by various degrees of regularity and Tarski numbers.

### 2017年10月24日(火)

17:30-18:30 数理科学研究科棟(駒場) 056号室

Tea: Common Room 17:00-17:30, Lie群論・表現論セミナーと合同

ラグランジュ交叉のフレアホモロジーに対する部分多様体論からのアプローチ (JAPANESE)

Tea: Common Room 17:00-17:30, Lie群論・表現論セミナーと合同

**宮岡 礼子 氏**(東北大学)ラグランジュ交叉のフレアホモロジーに対する部分多様体論からのアプローチ (JAPANESE)

[ 講演概要 ]

球面の等径超曲面のガウス写像による像は，複素２次超曲面Q_n(C)の極小ラグランジュ部分多様体の豊富な例を与える．簡単な場合，これはQ_n(C)の実形となり，そのフレアホモロジーは既知である．ここでは相異なる主曲率の個数が3,4,6の場合に得られた結果を報告する．当研究は，入江博（茨城大），Hui Ma（清華大学），大仁田義裕（大阪市大）との共同研究である．

球面の等径超曲面のガウス写像による像は，複素２次超曲面Q_n(C)の極小ラグランジュ部分多様体の豊富な例を与える．簡単な場合，これはQ_n(C)の実形となり，そのフレアホモロジーは既知である．ここでは相異なる主曲率の個数が3,4,6の場合に得られた結果を報告する．当研究は，入江博（茨城大），Hui Ma（清華大学），大仁田義裕（大阪市大）との共同研究である．

### 2017年10月17日(火)

17:00-18:30 数理科学研究科棟(駒場) 056号室

Tea: Common Room 16:30-17:00

Generalizations of twisted Alexander invariants and quandle cocycle invariants (JAPANESE)

Tea: Common Room 16:30-17:00

**石井 敦 氏**(筑波大学)Generalizations of twisted Alexander invariants and quandle cocycle invariants (JAPANESE)

[ 講演概要 ]

We introduce augmented Alexander matrices, and construct link invariants. An augmented Alexander matrix is defined with an augmented Alexander pair, which gives an extension of a quandle. This framework gives the twisted Alexander invariant and the quandle cocycle invariant. This is a joint work with Kanako Oshiro.

We introduce augmented Alexander matrices, and construct link invariants. An augmented Alexander matrix is defined with an augmented Alexander pair, which gives an extension of a quandle. This framework gives the twisted Alexander invariant and the quandle cocycle invariant. This is a joint work with Kanako Oshiro.

### 2017年10月10日(火)

17:30-18:30 数理科学研究科棟(駒場) 056号室

Tea: Common Room 17:00-17:30

Poset-stratified spaces and some applications (JAPANESE)

Tea: Common Room 17:00-17:30

**與倉 昭治 氏**(鹿児島大学)Poset-stratified spaces and some applications (JAPANESE)

[ 講演概要 ]

A poset-stratified space is a continuous map from a topological space to a poset with the Alexandroff topology. In this talk I will discuss some thoughts about poset-stratified spaces from a naive general-topological viewpoint, some applications such as hyperplane arrangements and poset-stratified space structures of hom-sets, and related topics such as characteristic classes of vector bundles, dependence of maps (by Borsuk) and dependence of cohomology classes (by Thom).

A poset-stratified space is a continuous map from a topological space to a poset with the Alexandroff topology. In this talk I will discuss some thoughts about poset-stratified spaces from a naive general-topological viewpoint, some applications such as hyperplane arrangements and poset-stratified space structures of hom-sets, and related topics such as characteristic classes of vector bundles, dependence of maps (by Borsuk) and dependence of cohomology classes (by Thom).

### 2017年10月03日(火)

17:00-18:00 数理科学研究科棟(駒場) 056号室

Tea: Common Room 16:30-17:00

Transitional geometry (ENGLISH)

Tea: Common Room 16:30-17:00

**Athanase Papadopoulos 氏**(IRMA, Université de Strasbourg)Transitional geometry (ENGLISH)

[ 講演概要 ]

I will describe transitions, that is, paths between hyperbolic and spherical geometry, passing through the Euclidean. This is based on joint work with Norbert A’Campo and recent joint work with A’Campo and Yi Huang.

I will describe transitions, that is, paths between hyperbolic and spherical geometry, passing through the Euclidean. This is based on joint work with Norbert A’Campo and recent joint work with A’Campo and Yi Huang.

### 2017年09月26日(火)

17:00-18:30 数理科学研究科棟(駒場) 056号室

Tea: Common Room 16:30-17:00, Lie群論・表現論セミナーと合同

Representations of Semisimple Lie Groups and Penrose Transform (JAPANESE)

Tea: Common Room 16:30-17:00, Lie群論・表現論セミナーと合同

**関口 英子 氏**(東京大学大学院数理科学研究科)Representations of Semisimple Lie Groups and Penrose Transform (JAPANESE)

[ 講演概要 ]

The classical Penrose transform is generalized to an intertwining operator on Dolbeault cohomologies of complex homogeneous spaces $X$ of (real) semisimple Lie groups.

I plan to discuss a detailed analysis when $X$ is an indefinite Grassmann manifold.

To be more precise, we determine the image of the Penrose transform, from the Dolbeault cohomology group on the indefinite Grassmann manifold consisting of maximally positive $k$-planes in ${\mathbb{C}}^{p,q}$ ($1 \le k \le \min(p,q)$) to the space of holomorphic functions over the bounded symmetric domain.

Furthermore, we prove that there is a duality between Dolbeault cohomology groups in two indefinite Grassmann manifolds,

namely, that of positive $k$-planes and that of negative $k$-planes.

The classical Penrose transform is generalized to an intertwining operator on Dolbeault cohomologies of complex homogeneous spaces $X$ of (real) semisimple Lie groups.

I plan to discuss a detailed analysis when $X$ is an indefinite Grassmann manifold.

To be more precise, we determine the image of the Penrose transform, from the Dolbeault cohomology group on the indefinite Grassmann manifold consisting of maximally positive $k$-planes in ${\mathbb{C}}^{p,q}$ ($1 \le k \le \min(p,q)$) to the space of holomorphic functions over the bounded symmetric domain.

Furthermore, we prove that there is a duality between Dolbeault cohomology groups in two indefinite Grassmann manifolds,

namely, that of positive $k$-planes and that of negative $k$-planes.

### 2017年07月11日(火)

17:00-18:30 数理科学研究科棟(駒場) 056号室

Tea: Common Room 16:30-17:00

Some remarkable quotients of virtual braid groups (ENGLISH)

Tea: Common Room 16:30-17:00

**Celeste Damiani 氏**(JSPS, 大阪市立大学)Some remarkable quotients of virtual braid groups (ENGLISH)

[ 講演概要 ]

Virtual braid groups are one of the most famous generalizations of braid groups. We introduce a family of quotients of virtual braid groups, called

Virtual braid groups are one of the most famous generalizations of braid groups. We introduce a family of quotients of virtual braid groups, called

*loop braid groups*. These groups have been an object of interest in different domains of mathematics and mathematical physics, and can be found in the literature also by names such as*groups of permutation-conjugacy automorphisms, braid- permutation groups, welded braid groups, weakly virtual braid groups, untwisted ring groups*, and others. We show that they share with braid groups the property of admitting many different definitions. After that we consider a further family of quotients called*unrestricted virtual braids*, describe their structure and explore their relations with fused links.### 2017年07月04日(火)

17:00-18:30 数理科学研究科棟(駒場) 056号室

Tea: Common Room 16:30-17:00

On link-homotopy for knotted surfaces in 4-space (ENGLISH)

Tea: Common Room 16:30-17:00

**Jean-Baptiste Meilhan 氏**(Université Grenoble Alpes)On link-homotopy for knotted surfaces in 4-space (ENGLISH)

[ 講演概要 ]

The purpose of this talk is to show how combinatorial objects (welded objects, which is a natural quotient of virtual knot theory) can be used to study knotted surfaces in 4-space.

We will first consider the case of 'ribbon' knotted surfaces, which are embedded surfaces bounding immersed 3-manifolds with only ribbon singularities. More precisely, we will consider ribbon knotted annuli ; these objects act naturally on the reduced free group, and we prove, using welded theory, that this action gives a classification up to link-homotopy, that is, up to continuous deformations leaving distinct component disjoint. This in turns implies a classification result for ribbon knotted tori.

Next, we will show how to extend this classification result beyond the ribbon case.

This is based on joint works with B. Audoux, P. Bellingeri and E. Wagner.

The purpose of this talk is to show how combinatorial objects (welded objects, which is a natural quotient of virtual knot theory) can be used to study knotted surfaces in 4-space.

We will first consider the case of 'ribbon' knotted surfaces, which are embedded surfaces bounding immersed 3-manifolds with only ribbon singularities. More precisely, we will consider ribbon knotted annuli ; these objects act naturally on the reduced free group, and we prove, using welded theory, that this action gives a classification up to link-homotopy, that is, up to continuous deformations leaving distinct component disjoint. This in turns implies a classification result for ribbon knotted tori.

Next, we will show how to extend this classification result beyond the ribbon case.

This is based on joint works with B. Audoux, P. Bellingeri and E. Wagner.

### 2017年06月27日(火)

17:00-18:30 数理科学研究科棟(駒場) 056号室

Tea: Common Room 16:30-17:00, RIKEN iTHEMS と共催

Braids and hyperbolic 3-manifolds from simple mixing devices (JAPANESE)

Tea: Common Room 16:30-17:00, RIKEN iTHEMS と共催

**金 英子 氏**(大阪大学)Braids and hyperbolic 3-manifolds from simple mixing devices (JAPANESE)

[ 講演概要 ]

Taffy pullers are devices for pulling candy. One can build braids from the motion of rods for taffy pullers. According to a beautiful article ``A mathematical history of taffy pullers" by Jean-Luc Thiffeault, all taffy pullers (except the first one) give rise to pseudo-Anosov braids. This means that the devices mix candies effectively. Following a study of Thiffeault, I will discuss which pseudo-Anosov braid is realized by taffy pullers. I will explain an interesting connection between braids coming from taffy pullers. I also discuss the hyperbolic mapping tori obtained from taffy pullers. Intriguingly, the two most common taffy pullers give rise to the complements of the the minimally twisted 4-chain link and 5-chain link which are important examples for the study of cusped hyperbolic 3-manifolds with small volumes.

Reference: A mathematical history of taffy pullers, Jean-Luc Thiffeault, https://arxiv.org/pdf/1608.00152.pdf

Taffy pullers are devices for pulling candy. One can build braids from the motion of rods for taffy pullers. According to a beautiful article ``A mathematical history of taffy pullers" by Jean-Luc Thiffeault, all taffy pullers (except the first one) give rise to pseudo-Anosov braids. This means that the devices mix candies effectively. Following a study of Thiffeault, I will discuss which pseudo-Anosov braid is realized by taffy pullers. I will explain an interesting connection between braids coming from taffy pullers. I also discuss the hyperbolic mapping tori obtained from taffy pullers. Intriguingly, the two most common taffy pullers give rise to the complements of the the minimally twisted 4-chain link and 5-chain link which are important examples for the study of cusped hyperbolic 3-manifolds with small volumes.

Reference: A mathematical history of taffy pullers, Jean-Luc Thiffeault, https://arxiv.org/pdf/1608.00152.pdf

### 2017年06月20日(火)

17:00-18:30 数理科学研究科棟(駒場) 056号室

Tea: Common Room 16:30-17:00

Introduction to the AJ conjecture (ENGLISH)

Tea: Common Room 16:30-17:00

**Anh Tran 氏**(The University of Texas at Dallas)Introduction to the AJ conjecture (ENGLISH)

[ 講演概要 ]

The AJ conjecture was proposed by Garoufalidis about 15 years ago. It predicts a strong connection between two important knot invariants derived from very different background, namely the colored Jones function (a quantum invariant) and the A-polynomial (a geometric invariant). The colored Jones function is a sequence of Laurent polynomials which is known to satisfy a linear q-difference equation. The AJ conjecture states that by writing the linear q-difference equation into an operator form and setting q=1, one gets the A-polynomial. In this talk, I will give an introduction to this conjecture.

The AJ conjecture was proposed by Garoufalidis about 15 years ago. It predicts a strong connection between two important knot invariants derived from very different background, namely the colored Jones function (a quantum invariant) and the A-polynomial (a geometric invariant). The colored Jones function is a sequence of Laurent polynomials which is known to satisfy a linear q-difference equation. The AJ conjecture states that by writing the linear q-difference equation into an operator form and setting q=1, one gets the A-polynomial. In this talk, I will give an introduction to this conjecture.

### 2017年06月13日(火)

17:00-18:30 数理科学研究科棟(駒場) 056号室

Tea: Common Room 16:30-17:00

Local criteria for non-embeddability of Levi-flat manifolds (JAPANESE)

Tea: Common Room 16:30-17:00

**小川 竜 氏**(東海大学)Local criteria for non-embeddability of Levi-flat manifolds (JAPANESE)

[ 講演概要 ]

In this talk, we consider the Levi-flat embedding problem. Barrett showed that a smooth Reeb foliation on S^3 cannot be realized as a Levi-flat hypersurface in any complex surfaces. To do this, he focused the relationship between the holonomy along the compact leaf and a system of its defining functions. We will show a new criterion for non-embeddability of Levi-flat manifolds. Our result is a higher dimensional analogue of Barrett's theorem. In particular, this enables us to weaken the compactness assumption. For this purpose, we pose a partial generalization of Ueda theory on the analytic neighborhood structure of complex hypersurfaces. This talk is based on a joint work with Takayuki Koike (Kyoto University).

In this talk, we consider the Levi-flat embedding problem. Barrett showed that a smooth Reeb foliation on S^3 cannot be realized as a Levi-flat hypersurface in any complex surfaces. To do this, he focused the relationship between the holonomy along the compact leaf and a system of its defining functions. We will show a new criterion for non-embeddability of Levi-flat manifolds. Our result is a higher dimensional analogue of Barrett's theorem. In particular, this enables us to weaken the compactness assumption. For this purpose, we pose a partial generalization of Ueda theory on the analytic neighborhood structure of complex hypersurfaces. This talk is based on a joint work with Takayuki Koike (Kyoto University).

### 2017年06月06日(火)

17:00-18:30 数理科学研究科棟(駒場) 056号室

Tea: Common Room 16:30-17:00

A formula for the action of Dehn twists on the HOMFLY-PT type skein algebra and its application (JAPANESE)

Tea: Common Room 16:30-17:00

**辻 俊輔 氏**(東京大学大学院数理科学研究科)A formula for the action of Dehn twists on the HOMFLY-PT type skein algebra and its application (JAPANESE)

[ 講演概要 ]

We give an explicit formula for the action of the Dehn twist along a simple closed curve of a surface on the completed HOMFLY-PT type skein modules of the surface in terms of the action of the completed HOMFLY-PT type skein algebra of the surface. As an application, using this formula, we construct an invariant for an integral homology 3-sphere which is an element of Q[ρ] [[h]].

We give an explicit formula for the action of the Dehn twist along a simple closed curve of a surface on the completed HOMFLY-PT type skein modules of the surface in terms of the action of the completed HOMFLY-PT type skein algebra of the surface. As an application, using this formula, we construct an invariant for an integral homology 3-sphere which is an element of Q[ρ] [[h]].

### 2017年05月30日(火)

17:00-18:30 数理科学研究科棟(駒場) 056号室

Tea: Common Room 16:30-17:00

On a conjecture of Dunfield, Friedl and Jackson for hyperbolic knots (JAPANESE)

Tea: Common Room 16:30-17:00

**森藤 孝之 氏**(慶應義塾大学)On a conjecture of Dunfield, Friedl and Jackson for hyperbolic knots (JAPANESE)

[ 講演概要 ]

The hyperbolic torsion polynomial is defined to be the twisted Alexander polynomial associated to the holonomy representation of a hyperbolic knot. Dunfield, Friedl and Jackson conjecture that the hyperbolic torsion polynomial determines the genus and fiberedness of a hyperbolic knot. In this talk we will survey recent results on the conjecture and explain its generalization to hyperbolic links.

The hyperbolic torsion polynomial is defined to be the twisted Alexander polynomial associated to the holonomy representation of a hyperbolic knot. Dunfield, Friedl and Jackson conjecture that the hyperbolic torsion polynomial determines the genus and fiberedness of a hyperbolic knot. In this talk we will survey recent results on the conjecture and explain its generalization to hyperbolic links.

### 2017年05月23日(火)

17:00-18:30 数理科学研究科棟(駒場) 大講義室号室

Tea: 大講義室前ホワイエ 16:40-17:00

Johnson homomorphisms, stable and unstable (ENGLISH)

Tea: 大講義室前ホワイエ 16:40-17:00

**Richard Hain 氏**(Duke University)Johnson homomorphisms, stable and unstable (ENGLISH)

[ 講演概要 ]

In this talk I will recall how motivic structures (Hodge and/or Galois) on the relative completions of mapping class groups yield non-trivial information about Johnson homomorphisms. I will explain how these motivic structures can be pasted, and why I believe that the genus 1 case is of fundamental importance in studying the higher genus (stable) case.

In this talk I will recall how motivic structures (Hodge and/or Galois) on the relative completions of mapping class groups yield non-trivial information about Johnson homomorphisms. I will explain how these motivic structures can be pasted, and why I believe that the genus 1 case is of fundamental importance in studying the higher genus (stable) case.

### 2017年05月16日(火)

17:00-18:30 数理科学研究科棟(駒場) 056号室

Tea: Common Room 16:30-17:00

Twisted Alexander invariants and Hyperbolic volume of knots (JAPANESE)

Tea: Common Room 16:30-17:00

**合田 洋 氏**(東京農工大学)Twisted Alexander invariants and Hyperbolic volume of knots (JAPANESE)

[ 講演概要 ]

In [1], Müller investigated the asymptotics of the Ray-Singer analytic torsion of hyperbolic 3-manifolds, and then Menal-Ferrer and Porti [2] have obtained a formula on the volume of a hyperbolic 3-manifold using the Higher-dimensional Reidemeister torsion.

On the other hand, Yoshikazu Yamaguchi has shown that a relationship between the twisted Alexander polynomial and the Reidemeister torsion associated with the adjoint representation of the holonomy representation of a hyperbolic 3-manifold in his thesis [3].

In this talk, we observe that Yamaguchi's idea is applicable to the Higher-dimensional Reidemeister torsion, then we give a volume formula of a hyperbolic knot using the twisted Alexander polynomial.

References

[1] Müller, W., The asymptotics of the Ray-Singer analytic torsion of hyperbolic 3-manifolds, Metric and differential geometry, 317--352, Progr. Math., 297, Birkhäuser/Springer, Basel, 2012.

[2] Menal-Ferrer, P. and Porti, J., Higher-dimensional Reidemeister torsion invariants for cusped hyperbolic 3-manifolds. J. Topol., 7 (2014), no. 1, 69--119.

[3] Yamaguchi, Y., On the non-acyclic Reidemeister torsion for knots, Dissertation at the University of Tokyo, 2007.

In [1], Müller investigated the asymptotics of the Ray-Singer analytic torsion of hyperbolic 3-manifolds, and then Menal-Ferrer and Porti [2] have obtained a formula on the volume of a hyperbolic 3-manifold using the Higher-dimensional Reidemeister torsion.

On the other hand, Yoshikazu Yamaguchi has shown that a relationship between the twisted Alexander polynomial and the Reidemeister torsion associated with the adjoint representation of the holonomy representation of a hyperbolic 3-manifold in his thesis [3].

In this talk, we observe that Yamaguchi's idea is applicable to the Higher-dimensional Reidemeister torsion, then we give a volume formula of a hyperbolic knot using the twisted Alexander polynomial.

References

[1] Müller, W., The asymptotics of the Ray-Singer analytic torsion of hyperbolic 3-manifolds, Metric and differential geometry, 317--352, Progr. Math., 297, Birkhäuser/Springer, Basel, 2012.

[2] Menal-Ferrer, P. and Porti, J., Higher-dimensional Reidemeister torsion invariants for cusped hyperbolic 3-manifolds. J. Topol., 7 (2014), no. 1, 69--119.

[3] Yamaguchi, Y., On the non-acyclic Reidemeister torsion for knots, Dissertation at the University of Tokyo, 2007.

### 2017年05月09日(火)

17:00-18:30 数理科学研究科棟(駒場) 056号室

Tea: Common Room 16:30-17:00

Local and global coincidence homology classes (JAPANESE)

Tea: Common Room 16:30-17:00

**諏訪 立雄 氏**(北海道大学)Local and global coincidence homology classes (JAPANESE)

[ 講演概要 ]

We consider two differentiable maps between two oriented manifolds. In the case the manifolds are compact with the same dimension and the coincidence points are isolated, there is the Lefschetz coincidence point formula, which generalizes his fixed point formula. In this talk we discuss the case where the dimensions of the manifolds may possible be different so that the coincidence points are not isolated in general. In fact it seems that Lefschetz himself already considered this case (cf. [4]).

We introduce the local and global coincidence homology classes and state a general coincidence point theorem.

We then give some explicit expressions for the local class. We also take up the case of several maps as considered in [1] and perform similar tasks. These all together lead to a generalization of the aforementioned Lefschetz formula. The key ingredients are the Alexander duality in combinatorial topology, intersection theory with maps and the Thom class in Čech-de Rham cohomology. The contents of the talk are in [2] and [3].

References

[1] C. Biasi, A.K.M. Libardi and T.F.M. Monis,

[2] C. Bisi, F. Bracci, T. Izawa and T. Suwa,

[3] J.-P. Brasselet and T. Suwa,

[4] N.E. Steenrod,

We consider two differentiable maps between two oriented manifolds. In the case the manifolds are compact with the same dimension and the coincidence points are isolated, there is the Lefschetz coincidence point formula, which generalizes his fixed point formula. In this talk we discuss the case where the dimensions of the manifolds may possible be different so that the coincidence points are not isolated in general. In fact it seems that Lefschetz himself already considered this case (cf. [4]).

We introduce the local and global coincidence homology classes and state a general coincidence point theorem.

We then give some explicit expressions for the local class. We also take up the case of several maps as considered in [1] and perform similar tasks. These all together lead to a generalization of the aforementioned Lefschetz formula. The key ingredients are the Alexander duality in combinatorial topology, intersection theory with maps and the Thom class in Čech-de Rham cohomology. The contents of the talk are in [2] and [3].

References

[1] C. Biasi, A.K.M. Libardi and T.F.M. Monis,

*The Lefschetz coincidence class of p maps*, Forum Math. 27 (2015), 1717-1728.[2] C. Bisi, F. Bracci, T. Izawa and T. Suwa,

*Localized intersection of currents and the Lefschetz coincidence point theorem*, Annali di Mat. Pura ed Applicata 195 (2016), 601-621.[3] J.-P. Brasselet and T. Suwa,

*Local and global coincidence homology classes*, arXiv:1612.02105.[4] N.E. Steenrod,

*The work and influence of Professor Lefschetz in algebraic topology*, Algebraic Geometry and Topology: A Symposium in Honor of Solomon Lefschetz, Princeton Univ. Press 1957, 24-43.### 2017年04月25日(火)

17:00-18:30 数理科学研究科棟(駒場) 056号室

Tea: Common Room 16:30-17:00

Formality of the Goldman-Turaev Lie bialgebra and the Kashiwara-Vergne problem in positive genus (JAPANESE)

Tea: Common Room 16:30-17:00

**久野 雄介 氏**(津田塾大学)Formality of the Goldman-Turaev Lie bialgebra and the Kashiwara-Vergne problem in positive genus (JAPANESE)

[ 講演概要 ]

This talk is based on a joint work with A. Alekseev, N. Kawazumi and F. Naef. Given a compact oriented surface with non-empty boundary and a framing of the surface, one can define the Lie bracket (Goldman bracket) and the Lie cobracket (Turaev bracket) on the vector space spanned by free homotopy classes of loops on the surface. These maps are of degree minus two with respect to a certain filtration. Then one can ask the formality of this Lie bialgebra: is the Goldman-Turaev Lie bialgebra isomorphic to its associated graded?

For surfaces of genus zero, we showed that this question is closely related to the Kashiwara-Vergne (KV) problem in Lie theory (arXiv:1703.05813). A similar result was obtained by G. Massuyeau by using the Kontsevich integral.

Our new topological interpretation of the classical KV problem leads us to introduce a generalization of the KV problem in connection with the formality of the Goldman-Turaev Lie bialgebra for surfaces of positive genus. We will discuss the existence and uniqueness of solutions to the generalized KV problem.

This talk is based on a joint work with A. Alekseev, N. Kawazumi and F. Naef. Given a compact oriented surface with non-empty boundary and a framing of the surface, one can define the Lie bracket (Goldman bracket) and the Lie cobracket (Turaev bracket) on the vector space spanned by free homotopy classes of loops on the surface. These maps are of degree minus two with respect to a certain filtration. Then one can ask the formality of this Lie bialgebra: is the Goldman-Turaev Lie bialgebra isomorphic to its associated graded?

For surfaces of genus zero, we showed that this question is closely related to the Kashiwara-Vergne (KV) problem in Lie theory (arXiv:1703.05813). A similar result was obtained by G. Massuyeau by using the Kontsevich integral.

Our new topological interpretation of the classical KV problem leads us to introduce a generalization of the KV problem in connection with the formality of the Goldman-Turaev Lie bialgebra for surfaces of positive genus. We will discuss the existence and uniqueness of solutions to the generalized KV problem.

### 2017年04月18日(火)

17:00-18:30 数理科学研究科棟(駒場) 056号室

Tea: Common Room 16:30-17:00

冪単マグナス展開によるミルナー不変量 (JAPANESE)

Tea: Common Room 16:30-17:00

**野坂 武史 氏**(東京工業大学)冪単マグナス展開によるミルナー不変量 (JAPANESE)

[ 講演概要 ]

われわれは、ミルナー不変量を、群の中心拡大と冪単マグナス展開をもちいて再構成した。それにより当不変量の図式計算方法を確立した。本講演ではその再構成と計算法を説明し、いくつか例示をする。また冪零的マグナス展開の性質も紹介したい。本研究は九大の小谷久寿氏との共同研究である。

われわれは、ミルナー不変量を、群の中心拡大と冪単マグナス展開をもちいて再構成した。それにより当不変量の図式計算方法を確立した。本講演ではその再構成と計算法を説明し、いくつか例示をする。また冪零的マグナス展開の性質も紹介したい。本研究は九大の小谷久寿氏との共同研究である。

### 2017年04月11日(火)

17:00-18:30 数理科学研究科棟(駒場) 056号室

Tea: Common Room 16:30-17:00

Homotopy Lie algebroids (ENGLISH)

Tea: Common Room 16:30-17:00

**Alexander Voronov 氏**(University of Minnesota)Homotopy Lie algebroids (ENGLISH)

[ 講演概要 ]

A well-known result of A. Vaintrob [Vai97] characterizes Lie algebroids and their morphisms in terms of homological vector fields on supermanifolds. We give an interpretation of Lie bialgebroids and their morphisms in terms of odd symplectic dg-manifolds, building on the approach of D. Roytenberg [Roy99]. This extends naturally to the homotopy Lie case and leads to the notion of L

A well-known result of A. Vaintrob [Vai97] characterizes Lie algebroids and their morphisms in terms of homological vector fields on supermanifolds. We give an interpretation of Lie bialgebroids and their morphisms in terms of odd symplectic dg-manifolds, building on the approach of D. Roytenberg [Roy99]. This extends naturally to the homotopy Lie case and leads to the notion of L

_{∞}-bialgebroids and L_{∞}-morphisms between them.### 2017年03月10日(金)

17:00-18:30 数理科学研究科棟(駒場) 056号室

Tea: Common Room 16:30-17:00, Lie群論・表現論セミナーと合同

Satake compactifications and metric Schottky problems (ENGLISH)

Tea: Common Room 16:30-17:00, Lie群論・表現論セミナーと合同

**Lizhen Ji 氏**(University of Michigan)Satake compactifications and metric Schottky problems (ENGLISH)

[ 講演概要 ]

The quotient of the Poincare upper half plane by the modular group SL(2, Z) is a basic locally symmetric space and also the moduli space of compact Riemann surfaces of genus 1, and it admits two important classes of generalization:

(1) Moduli spaces M_g of compact Riemann surfaces of genus g>1,

(2) Arithmetic locally symmetric spaces Γ \ G/K such as the Siegel modular variety A_g, which is also the moduli of principally polarized abelian varieties of dimension g.

There have been a lot of fruitful work to explore the similarity between these two classes of spaces, and there is also a direct interaction between them through the Jacobian (or period) map J: M_g --> A_g. In this talk, I will discuss some results along these lines related to the Stake compactifications and the Schottky problems on understanding the image J(M_g) in A_g from the metric perspective.

The quotient of the Poincare upper half plane by the modular group SL(2, Z) is a basic locally symmetric space and also the moduli space of compact Riemann surfaces of genus 1, and it admits two important classes of generalization:

(1) Moduli spaces M_g of compact Riemann surfaces of genus g>1,

(2) Arithmetic locally symmetric spaces Γ \ G/K such as the Siegel modular variety A_g, which is also the moduli of principally polarized abelian varieties of dimension g.

There have been a lot of fruitful work to explore the similarity between these two classes of spaces, and there is also a direct interaction between them through the Jacobian (or period) map J: M_g --> A_g. In this talk, I will discuss some results along these lines related to the Stake compactifications and the Schottky problems on understanding the image J(M_g) in A_g from the metric perspective.

### 2017年03月08日(水)

17:00-18:30 数理科学研究科棟(駒場) 056号室

Tea: Common Room 16:30-17:00

Action of the Long-Moody Construction on Polynomial Functors (ENGLISH)

Tea: Common Room 16:30-17:00

**Arthur Soulié 氏**(Université de Strasbourg)Action of the Long-Moody Construction on Polynomial Functors (ENGLISH)

[ 講演概要 ]

In 2016, Randal-Williams and Wahl proved homological stability with certain twisted coefficients for different families of groups, in particular the one of braid groups. In fact, they obtain the stability for coefficients given by functors satisfying polynomial conditions. We only know few examples of such functors. Among them, we have the functor given by the unreduced Burau representations. In 1994, Long and Moody gave a construction on representations of braid groups which associates a representation of Bn with a representation of Bn+1. This construction complexifies in a sense the initial representation: for instance, starting from a dimension one representation, one obtains the unreduced Burau representation. In this talk, I will present this construction from a functorial point of view. I will explain that the construction of Long and Moody defines an endofunctor, called the Long-Moody functor, between a suitable category of functors. Then, after defining strong polynomial functors in this context, I will prove that the Long-Moody functor increases by one the degree of strong polynomiality of a strong polynomial functor. Thus, the Long-Moody construction will provide new examples of twisted coefficients entering in the framework of Randal-Williams and Wahl.

In 2016, Randal-Williams and Wahl proved homological stability with certain twisted coefficients for different families of groups, in particular the one of braid groups. In fact, they obtain the stability for coefficients given by functors satisfying polynomial conditions. We only know few examples of such functors. Among them, we have the functor given by the unreduced Burau representations. In 1994, Long and Moody gave a construction on representations of braid groups which associates a representation of Bn with a representation of Bn+1. This construction complexifies in a sense the initial representation: for instance, starting from a dimension one representation, one obtains the unreduced Burau representation. In this talk, I will present this construction from a functorial point of view. I will explain that the construction of Long and Moody defines an endofunctor, called the Long-Moody functor, between a suitable category of functors. Then, after defining strong polynomial functors in this context, I will prove that the Long-Moody functor increases by one the degree of strong polynomiality of a strong polynomial functor. Thus, the Long-Moody construction will provide new examples of twisted coefficients entering in the framework of Randal-Williams and Wahl.

### 2017年02月20日(月)

17:00-18:30 数理科学研究科棟(駒場) 056号室

Tea: Common Room 16:30-17:00

The Verlinde formula for Higgs bundles (ENGLISH)

Tea: Common Room 16:30-17:00

**Jørgen Ellegaard Andersen 氏**(Aarhus University)The Verlinde formula for Higgs bundles (ENGLISH)

[ 講演概要 ]

In this talk we will present a Verlinde formula for the quantization of the Higgs bundle moduli spaces and stacks for any simple and simply-connected group. We further present a Verlinde formula for the quantization of parabolic Higgs bundle moduli spaces and stacks. We will explain how all these dimensions fit into a one parameter family of 2D TQFT's, encoded in a one parameter family of Frobenius algebras, which we will construct.

In this talk we will present a Verlinde formula for the quantization of the Higgs bundle moduli spaces and stacks for any simple and simply-connected group. We further present a Verlinde formula for the quantization of parabolic Higgs bundle moduli spaces and stacks. We will explain how all these dimensions fit into a one parameter family of 2D TQFT's, encoded in a one parameter family of Frobenius algebras, which we will construct.