## Tuesday Seminar on Topology

Seminar information archive ～09/17｜Next seminar｜Future seminars 09/18～

Date, time & place | Tuesday 17:00 - 18:30 056Room #056 (Graduate School of Math. Sci. Bldg.) |
---|---|

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

Remarks | Tea: 16:30 - 17:00 Common Room |

**Seminar information archive**

### 2018/01/16

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

A new relationship between the dilatation of pseudo-Anosov braids and fixed point theory (JAPANESE)

**Yumehito Kawashima**(The University of Tokyo)A new relationship between the dilatation of pseudo-Anosov braids and fixed point theory (JAPANESE)

[ Abstract ]

A relation between the dilatation of pseudo-Anosov braids and fixed point theory was studied by Ivanov. In this talk we reveal a new relationship between the above two subjects by showing a formula for the dilatation of pseudo-Anosov braids by means of the representations of braid groups due to B. Jiang and H. Zheng.

A relation between the dilatation of pseudo-Anosov braids and fixed point theory was studied by Ivanov. In this talk we reveal a new relationship between the above two subjects by showing a formula for the dilatation of pseudo-Anosov braids by means of the representations of braid groups due to B. Jiang and H. Zheng.

### 2017/12/19

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

Deformation of holomorphic quadratic differentials and its applications (JAPANESE)

**Hideki Miyachi**(Osaka university)Deformation of holomorphic quadratic differentials and its applications (JAPANESE)

[ Abstract ]

Quadratic differentials are standard and important objects in Teichmuller theory. The deformation space (moduli space) of the quadratic differentials is applied to many fields of mathematics. In this talk, I will develop the deformation of quadratic differentials. Indeed, following pioneer works by A. Douady, J. Hubbard, H. Masur and W. Veech, we describe the infinitesimal deformations in the odd (co)homology groups on the double covering spaces defined from the square roots of the quadratic differentials. We formulate the decomposition theorem for the infinitesimal deformations with keeping in mind of the induced deformation of the moduli of underlying complex structures. As applications, we obtain the Levi form of the Teichmuller distance, and an alternate proof of the Krushkal formula on the pluricomplex Green function on the Teichmuller space.

Quadratic differentials are standard and important objects in Teichmuller theory. The deformation space (moduli space) of the quadratic differentials is applied to many fields of mathematics. In this talk, I will develop the deformation of quadratic differentials. Indeed, following pioneer works by A. Douady, J. Hubbard, H. Masur and W. Veech, we describe the infinitesimal deformations in the odd (co)homology groups on the double covering spaces defined from the square roots of the quadratic differentials. We formulate the decomposition theorem for the infinitesimal deformations with keeping in mind of the induced deformation of the moduli of underlying complex structures. As applications, we obtain the Levi form of the Teichmuller distance, and an alternate proof of the Krushkal formula on the pluricomplex Green function on the Teichmuller space.

### 2017/12/12

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

On the self-intersection of singular sets of maps and the signature defect (JAPANESE)

**Tatsuro Shimizu**(RIMS, Kyoto university)On the self-intersection of singular sets of maps and the signature defect (JAPANESE)

[ Abstract ]

Let $M$ be a closed oriented $n$-dimensional manifold. We give a geometric proof of that the $k$-times self-intersection of singular set of a Morin map from $M$ to $R^p$ coincides with the corank $k$ singular set of any generic map from $M$ to $R^{p+k-1}$ as homology classes with $Z/2$ coefficient ($n>p+k-2$). As an application we give a description of the signature defect of framed 3-manifold from the point of view of singular sets of maps.

Let $M$ be a closed oriented $n$-dimensional manifold. We give a geometric proof of that the $k$-times self-intersection of singular set of a Morin map from $M$ to $R^p$ coincides with the corank $k$ singular set of any generic map from $M$ to $R^{p+k-1}$ as homology classes with $Z/2$ coefficient ($n>p+k-2$). As an application we give a description of the signature defect of framed 3-manifold from the point of view of singular sets of maps.

### 2017/12/05

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

Derivations and cohomologies of Lipschitz algebras (JAPANESE)

**Kazuhiro Kawamura**(University of Tsukuba)Derivations and cohomologies of Lipschitz algebras (JAPANESE)

[ Abstract ]

For a compact metric space M, Lip(M) denotes the Banach algebra of all complex-valued Lipschitz functions on M. Motivated by a classical work of de Leeuw, we define a compact, not necessarily metrizable, Hausdorff space \hat{M} so that each point of \hat{M} induces a derivation on Lip(M). To some extent, \hat{M} may be regarded as "the space of directions." We study, by an elementary method, the space of derivations and continuous Hochschild cohomologies (in the sense of B.E. Johnson and A.Y. Helemskii) of Lip(M) with coefficients C(\hat{M}) and C(M). The results so obtained show that the behavior of Lip(M) is (naturally) rather different than that of the algebra of smooth/C^1 functions on M.

For a compact metric space M, Lip(M) denotes the Banach algebra of all complex-valued Lipschitz functions on M. Motivated by a classical work of de Leeuw, we define a compact, not necessarily metrizable, Hausdorff space \hat{M} so that each point of \hat{M} induces a derivation on Lip(M). To some extent, \hat{M} may be regarded as "the space of directions." We study, by an elementary method, the space of derivations and continuous Hochschild cohomologies (in the sense of B.E. Johnson and A.Y. Helemskii) of Lip(M) with coefficients C(\hat{M}) and C(M). The results so obtained show that the behavior of Lip(M) is (naturally) rather different than that of the algebra of smooth/C^1 functions on M.

### 2017/11/28

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

Diffeomorphism Groups of One-Manifolds (ENGLISH)

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

[ Abstract ]

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.

### 2017/11/21

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

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

**Keiichi Sakai**(Shinshu University)The space of short ropes and the classifying space of the space of long knots (JAPANESE)

[ Abstract ]

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).

### 2017/11/07

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

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

**Shin Hayashi**(AIST-TohokuU MathAM-OIL)On an explicit example of topologically protected corner states (JAPANESE)

[ Abstract ]

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.

### 2017/10/31

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

Nonamenable groups of piecewise projective homeomorphisms (ENGLISH)

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

[ Abstract ]

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.

### 2017/10/24

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

Approach from the submanifold theory to the Floer homology of Lagrangian intersections (JAPANESE)

**Reiko Miyaoka**(Tohoku University)Approach from the submanifold theory to the Floer homology of Lagrangian intersections (JAPANESE)

[ Abstract ]

The Gauss map images of isoparametric hypersurfaces in the spheres supply a rich family of minimal Lagrangian submanifolds of the complex hyperquadric Q_n(C). In simple cases, these are real forms of Q_n(C), and their Floer homology is known. In this talk, we consider the case when the number of distinct principal curvatures is 3,4,6, and report our results. This is a joint work with Hiroshi Iriyeh (Ibaraki U.), Hui Ma (Tsinghua U.) and Yoshihiro Ohnita (Osaka City U.).

The Gauss map images of isoparametric hypersurfaces in the spheres supply a rich family of minimal Lagrangian submanifolds of the complex hyperquadric Q_n(C). In simple cases, these are real forms of Q_n(C), and their Floer homology is known. In this talk, we consider the case when the number of distinct principal curvatures is 3,4,6, and report our results. This is a joint work with Hiroshi Iriyeh (Ibaraki U.), Hui Ma (Tsinghua U.) and Yoshihiro Ohnita (Osaka City U.).

### 2017/10/17

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

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

**Atsushi Ishii**(University of Tsukuba)Generalizations of twisted Alexander invariants and quandle cocycle invariants (JAPANESE)

[ Abstract ]

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.

### 2017/10/10

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

Poset-stratified spaces and some applications (JAPANESE)

**Shoji Yokura**(Kagoshima University)Poset-stratified spaces and some applications (JAPANESE)

[ Abstract ]

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).

### 2017/10/03

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

Transitional geometry (ENGLISH)

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

[ Abstract ]

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.

### 2017/09/26

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

Representations of Semisimple Lie Groups and Penrose Transform (JAPANESE)

**Hideko Sekiguchi**(The University of Tokyo)Representations of Semisimple Lie Groups and Penrose Transform (JAPANESE)

[ Abstract ]

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.

### 2017/07/11

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

Some remarkable quotients of virtual braid groups (ENGLISH)

**Celeste Damiani**(JSPS, Osaka City University)Some remarkable quotients of virtual braid groups (ENGLISH)

[ Abstract ]

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.### 2017/07/04

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

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

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

[ Abstract ]

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.

### 2017/06/27

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

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

**Eiko Kin**(Osaka University)Braids and hyperbolic 3-manifolds from simple mixing devices (JAPANESE)

[ Abstract ]

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

### 2017/06/20

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

Introduction to the AJ conjecture (ENGLISH)

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

[ Abstract ]

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.

### 2017/06/13

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

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

**Noboru Ogawa**(Tokai University)Local criteria for non-embeddability of Levi-flat manifolds (JAPANESE)

[ Abstract ]

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).

### 2017/06/06

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

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

**Shunsuke Tsuji**(The University of Tokyo)A formula for the action of Dehn twists on the HOMFLY-PT type skein algebra and its application (JAPANESE)

[ Abstract ]

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]].

### 2017/05/30

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

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

**Takayuki Morifuji**(Keio University)On a conjecture of Dunfield, Friedl and Jackson for hyperbolic knots (JAPANESE)

[ Abstract ]

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.

### 2017/05/23

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

Johnson homomorphisms, stable and unstable (ENGLISH)

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

[ Abstract ]

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.

### 2017/05/16

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

Twisted Alexander invariants and Hyperbolic volume of knots (JAPANESE)

**Hiroshi Goda**(Tokyo University of Agriculture and Technology)Twisted Alexander invariants and Hyperbolic volume of knots (JAPANESE)

[ Abstract ]

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.

### 2017/05/09

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

Local and global coincidence homology classes (JAPANESE)

**Tatsuo Suwa**(Hokkaido University)Local and global coincidence homology classes (JAPANESE)

[ Abstract ]

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.### 2017/04/25

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

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

**Yusuke Kuno**(Tsuda University)Formality of the Goldman-Turaev Lie bialgebra and the Kashiwara-Vergne problem in positive genus (JAPANESE)

[ Abstract ]

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.

### 2017/04/18

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

Milnor invariants via unipotent Magnus embeddings (JAPANESE)

**Takefumi Nosaka**(Tokyo institute of Technology)Milnor invariants via unipotent Magnus embeddings (JAPANESE)

[ Abstract ]

We reconfigured the Milnor invariant, in terms of central group extensions and unipotent Magnus embeddings, and develop a diagrammatic computation of the invariant. In this talk, we explain the reconfiguration and the computation with mentioning some examples. I also introduce some properties of the unipotent embeddings. This is a joint work with Hisatoshi Kodani.

We reconfigured the Milnor invariant, in terms of central group extensions and unipotent Magnus embeddings, and develop a diagrammatic computation of the invariant. In this talk, we explain the reconfiguration and the computation with mentioning some examples. I also introduce some properties of the unipotent embeddings. This is a joint work with Hisatoshi Kodani.