## トポロジー火曜セミナー

過去の記録 ～02/15｜次回の予定｜今後の予定 02/16～

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 | 河野 俊丈, 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 |

備考 | Tea: 16:30 - 17:00 コモンルーム |

**過去の記録**

### 2019年02月12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

Representations of knot groups (ENGLISH)

Tea: Common Room 16:30-17:00

**Anastasiia Tsvietkova 氏**(沖縄科学技術大学院大学, Rutgers University)Representations of knot groups (ENGLISH)

[ 講演概要 ]

We describe a new method of producing equations for the representation variety of a knot group into (P)SL(2,C). Unlike known methods, this does not involve any polyhedral decomposition or triangulation of the link complement, and uses only a link diagram satisfying a few mild restrictions. This results in a simple algorithm that can often be performed by hand, and in many cases, for an infinite family of knots at once. This is a joint work with Kathleen Peterson (Florida State University).

We describe a new method of producing equations for the representation variety of a knot group into (P)SL(2,C). Unlike known methods, this does not involve any polyhedral decomposition or triangulation of the link complement, and uses only a link diagram satisfying a few mild restrictions. This results in a simple algorithm that can often be performed by hand, and in many cases, for an infinite family of knots at once. This is a joint work with Kathleen Peterson (Florida State University).

### 2019年01月15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

Generalized Dehn twists on surfaces and homology cylinders (JAPANESE)

Tea: Common Room 16:30-17:00

**久野 雄介 氏**(津田塾大学)Generalized Dehn twists on surfaces and homology cylinders (JAPANESE)

[ 講演概要 ]

This is a joint work with Gwénaël Massuyeau (University of Burgundy). Lickorish's trick describes Dehn twists along simple closed curves on an oriented surface in terms of surgery of 3-manifolds. We discuss one possible generalization of this description to the situation where the curve under consideration may have self-intersections. Our result generalizes previously known computations related to the Johnson homomorphisms for the mapping class groups and for homology cylinders. In particular, we obtain an alternative and direct proof of the surjectivity of the Johnson homomorphisms for homology cylinders, which was proved by Garoufalidis-Levine and Habegger.

This is a joint work with Gwénaël Massuyeau (University of Burgundy). Lickorish's trick describes Dehn twists along simple closed curves on an oriented surface in terms of surgery of 3-manifolds. We discuss one possible generalization of this description to the situation where the curve under consideration may have self-intersections. Our result generalizes previously known computations related to the Johnson homomorphisms for the mapping class groups and for homology cylinders. In particular, we obtain an alternative and direct proof of the surjectivity of the Johnson homomorphisms for homology cylinders, which was proved by Garoufalidis-Levine and Habegger.

### 2019年01月08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 property (T) for $\mathrm{Aut}(F_n)$ and $\mathrm{SL}_n(\mathbb{Z})$ (ENGLISH)

Tea: Common Room 16:30-17:00

**Marek Kaluba 氏**(Adam Mickiewicz Univeristy)On property (T) for $\mathrm{Aut}(F_n)$ and $\mathrm{SL}_n(\mathbb{Z})$ (ENGLISH)

[ 講演概要 ]

We prove that $\mathrm{Aut}(F_n)$ has Kazhdan's property (T) for every $n \ge 6$. Together with a previous result of Kaluba, Nowak, and Ozawa, this gives the same statement for $n \ge 5$. We also provide explicit lower bounds for the Kazhdan constants of $\mathrm{SAut}(F_n)$ (with $n \ge 6$) and of $\mathrm{SL}_n(\mathbb{Z})$ (with $n \ge 3$) with respect to natural generating sets. In the latter case, these bounds improve upon previously known lower bounds whenever $n >6$.

We prove that $\mathrm{Aut}(F_n)$ has Kazhdan's property (T) for every $n \ge 6$. Together with a previous result of Kaluba, Nowak, and Ozawa, this gives the same statement for $n \ge 5$. We also provide explicit lower bounds for the Kazhdan constants of $\mathrm{SAut}(F_n)$ (with $n \ge 6$) and of $\mathrm{SL}_n(\mathbb{Z})$ (with $n \ge 3$) with respect to natural generating sets. In the latter case, these bounds improve upon previously known lower bounds whenever $n >6$.

### 2018年12月20日(木)

13:00-14:30 数理科学研究科棟(駒場) 056号室

開催日時にご注意下さい

Johnson-type homomorphisms and the LMO functor (ENGLISH)

開催日時にご注意下さい

**Anderson Vera 氏**(Université de Strasbourg)Johnson-type homomorphisms and the LMO functor (ENGLISH)

[ 講演概要 ]

One of the main objects associated to a surface S is the mapping class group MCG(S). This group plays an important role in the study of 3-manifolds. Reciprocally, the topological invariants of 3-manifolds can be used to obtain interesting representations of MCG(S).

One possible approach to the study of MCG(S) is to consider its action on the fundamental group P of the surface or on some subgroups of P. This way, we can obtain some kind of filtrations of MCG(S) and homomorphisms, called Johnson type homomorphisms, which take values in certain spaces of diagrams. These spaces of diagrams are quotients of the target space of the LMO functor. Hence it is natural to ask what is the relation between the Johnson type homomorphisms and the LMO functor. The answer is well known in the case of the Torelli group and the usual Johnson homomorphisms. In this talk we consider two other different filtrations of MCG(S) introduced by Levine and Habiro-Massuyeau. We show that the respective Johnson homomorphisms can also be deduced from the LMO functor.

One of the main objects associated to a surface S is the mapping class group MCG(S). This group plays an important role in the study of 3-manifolds. Reciprocally, the topological invariants of 3-manifolds can be used to obtain interesting representations of MCG(S).

One possible approach to the study of MCG(S) is to consider its action on the fundamental group P of the surface or on some subgroups of P. This way, we can obtain some kind of filtrations of MCG(S) and homomorphisms, called Johnson type homomorphisms, which take values in certain spaces of diagrams. These spaces of diagrams are quotients of the target space of the LMO functor. Hence it is natural to ask what is the relation between the Johnson type homomorphisms and the LMO functor. The answer is well known in the case of the Torelli group and the usual Johnson homomorphisms. In this talk we consider two other different filtrations of MCG(S) introduced by Levine and Habiro-Massuyeau. We show that the respective Johnson homomorphisms can also be deduced from the LMO functor.

### 2018年12月18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

離散GスペクトラムとK(n)局所安定ホモトピー圏のモデルについて (JAPANESE)

Tea: Common Room 17:00-17:30

**鳥居 猛 氏**(岡山大学)離散GスペクトラムとK(n)局所安定ホモトピー圏のモデルについて (JAPANESE)

[ 講演概要 ]

K(n)局所安定ホモトピー圏はスペクトラムの安定ホモトピー圏の基本構成単位と考えられる。この講演ではMorava E理論とその安定化群との関係が明確になるようなK(n)局所安定ホモトピー圏のモデルを構成する。そのために、Behrens-Davisにより研究された副有限群Gに対する離散対称Gスペクトラムについて考える。そして、K(n)局所安定ホモトピー圏が、離散対称G_nスペクトラムの圏におけるE_nの離散モデル上の加群のホモトピー圏の中に実現されることを示す。

K(n)局所安定ホモトピー圏はスペクトラムの安定ホモトピー圏の基本構成単位と考えられる。この講演ではMorava E理論とその安定化群との関係が明確になるようなK(n)局所安定ホモトピー圏のモデルを構成する。そのために、Behrens-Davisにより研究された副有限群Gに対する離散対称Gスペクトラムについて考える。そして、K(n)局所安定ホモトピー圏が、離散対称G_nスペクトラムの圏におけるE_nの離散モデル上の加群のホモトピー圏の中に実現されることを示す。

### 2018年12月11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

On non-singular solutions to the normalized Ricci flow on four-manifolds (JAPANESE)

Tea: Common Room 17:00-17:30

**石田 政司 氏**(大阪大学)On non-singular solutions to the normalized Ricci flow on four-manifolds (JAPANESE)

[ 講演概要 ]

A solution to the normalized Ricci flow is called non-singular if the solution exists for all time and the Riemannian curvature tensor is uniformly bounded. In 1999, Richard Hamilton introduced it as an important special class of solutions and proved that the underlying 3-manifold is geometrizable in the sense of Thurston. In this talk, we will discuss properties of 4-dimensional non-singular solutions from a gauge theoretical point of view. In particular, we would like to explain gauge theoretical invariants give rise to obstructions to the existence of 4-dimensional non-singular solutions.

A solution to the normalized Ricci flow is called non-singular if the solution exists for all time and the Riemannian curvature tensor is uniformly bounded. In 1999, Richard Hamilton introduced it as an important special class of solutions and proved that the underlying 3-manifold is geometrizable in the sense of Thurston. In this talk, we will discuss properties of 4-dimensional non-singular solutions from a gauge theoretical point of view. In particular, we would like to explain gauge theoretical invariants give rise to obstructions to the existence of 4-dimensional non-singular solutions.

### 2018年12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

Slopes and concordance of links (ENGLISH)

Tea: Common Room 16:30-17:00

**Vincent Florens 氏**(Université de Pau et des Pays de l'Adour)Slopes and concordance of links (ENGLISH)

[ 講演概要 ]

We define the slope of a link associated to admissible characters on the link group. Away from a certain singular locus, the slope is a rational function which can be regarded as a multivariate generalization of the Kojima-Yamasaki η-function. It is the ratio of two Conway potentials, provided that the latter makes sense; otherwise, it is a new invariant. We present several examples and discuss the invariance by concordance. Joint with A. Degtyarev and A. Lecuona.

We define the slope of a link associated to admissible characters on the link group. Away from a certain singular locus, the slope is a rational function which can be regarded as a multivariate generalization of the Kojima-Yamasaki η-function. It is the ratio of two Conway potentials, provided that the latter makes sense; otherwise, it is a new invariant. We present several examples and discuss the invariance by concordance. Joint with A. Degtyarev and A. Lecuona.

### 2018年11月27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

Fixed points for group actions on non-positively curved spaces (JAPANESE)

Tea: Common Room 16:30-17:00

**加藤 本子 氏**(東京大学大学院数理科学研究科)Fixed points for group actions on non-positively curved spaces (JAPANESE)

[ 講演概要 ]

In this talk, we introduce a fixed point property of groups which is a broad generalization of Serre's property FA, and give a criterion for groups to have such a property. We also apply the criterion to show that various generalizations of Thompson's group T have fixed points whenever they act on finite dimensional non-positively curved metric spaces, including CAT(0) spaces. Since Thompson's group T is known to have fixed point free actions on infinite dimensional CAT(0) spaces, it follows that there is a group which acts on infinite dimensional CAT(0) spaces without global fixed points, but not on finite dimensional ones.

In this talk, we introduce a fixed point property of groups which is a broad generalization of Serre's property FA, and give a criterion for groups to have such a property. We also apply the criterion to show that various generalizations of Thompson's group T have fixed points whenever they act on finite dimensional non-positively curved metric spaces, including CAT(0) spaces. Since Thompson's group T is known to have fixed point free actions on infinite dimensional CAT(0) spaces, it follows that there is a group which acts on infinite dimensional CAT(0) spaces without global fixed points, but not on finite dimensional ones.

### 2018年11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

Torelli group, Johnson kernel and invariants of homology 3-spheres (JAPANESE)

Tea: Common Room 16:30-17:00

**逆井 卓也 氏**(東京大学大学院数理科学研究科)Torelli group, Johnson kernel and invariants of homology 3-spheres (JAPANESE)

[ 講演概要 ]

There are two filtrations of the Torelli group: One is the lower central series and the other is the Johnson filtration. They are closely related to Johnson homomorphisms as well as finite type invariants of homology 3-spheres. We compare the associated graded Lie algebras of the filtrations and report our explicit computational results. Then we discuss some applications of our computations. In particular, we give an explicit description of the rational abelianization of the Johnson kernel. This is a joint work with Shigeyuki Morita and Masaaki Suzuki.

There are two filtrations of the Torelli group: One is the lower central series and the other is the Johnson filtration. They are closely related to Johnson homomorphisms as well as finite type invariants of homology 3-spheres. We compare the associated graded Lie algebras of the filtrations and report our explicit computational results. Then we discuss some applications of our computations. In particular, we give an explicit description of the rational abelianization of the Johnson kernel. This is a joint work with Shigeyuki Morita and Masaaki Suzuki.

### 2018年11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

On continuity of drifts of the mapping class group (JAPANESE)

Tea: Common Room 16:30-17:00

**正井 秀俊 氏**(東京工業大学)On continuity of drifts of the mapping class group (JAPANESE)

[ 講演概要 ]

When a group is acting on a space isometrically, we may consider the "translation distance" of random walks, which is called the drift of the random walk. In this talk we consider mapping class group acting on the Teichmüller space. We first recall several characterizations of the drift. The drift is determined by the transition probability of the random walk. The goal of this talk is to show that the drift varies continuously with the transition probability measure.

When a group is acting on a space isometrically, we may consider the "translation distance" of random walks, which is called the drift of the random walk. In this talk we consider mapping class group acting on the Teichmüller space. We first recall several characterizations of the drift. The drift is determined by the transition probability of the random walk. The goal of this talk is to show that the drift varies continuously with the transition probability measure.

### 2018年11月08日(木)

10:30-12:00 数理科学研究科棟(駒場) 056号室

開催日，時刻にご注意下さい

Deformations of diagonal representations of knot groups into $\mathrm{SL}(n,\mathbb{C})$ (ENGLISH)

開催日，時刻にご注意下さい

**Michael Heusener 氏**(Université Clermont Auvergne)Deformations of diagonal representations of knot groups into $\mathrm{SL}(n,\mathbb{C})$ (ENGLISH)

[ 講演概要 ]

This is joint work with Leila Ben Abdelghani, Monastir (Tunisia).

Given a manifold $M$, the variety of representations of $\pi_1(M)$ into $\mathrm{SL}(2,\mathbb{C})$ and the variety of characters of such representations both contain information of the topology of $M$. Since the foundational work of W.P. Thurston and Culler & Shalen, the varieties of $\mathrm{SL}(2,\mathbb{C})$-characters have been extensively studied. This is specially interesting for $3$-dimensional manifolds, where the fundamental group and the geometrical properties of the manifold are strongly related.

However, much less is known of the character varieties for other groups, notably for $\mathrm{SL}(n,\mathbb{C})$ with $n\geq 3$. The $\mathrm{SL}(n,\mathbb{C})$-character varieties for free groups have been studied by S. Lawton and P. Will, and the $\mathrm{SL}(3,\mathbb{C})$-character variety of torus knot groups has been determined by V. Munoz and J. Porti.

In this talk I will present some results concerning the deformations of diagonal representations of knot groups in basic notations and some recent results concerning the representation and character varieties of $3$-manifold groups and in particular knot groups. In particular, we are interested in the local structure of the $\mathrm{SL}(n,\mathbb{C})$-representation variety at the diagonal representation.

This is joint work with Leila Ben Abdelghani, Monastir (Tunisia).

Given a manifold $M$, the variety of representations of $\pi_1(M)$ into $\mathrm{SL}(2,\mathbb{C})$ and the variety of characters of such representations both contain information of the topology of $M$. Since the foundational work of W.P. Thurston and Culler & Shalen, the varieties of $\mathrm{SL}(2,\mathbb{C})$-characters have been extensively studied. This is specially interesting for $3$-dimensional manifolds, where the fundamental group and the geometrical properties of the manifold are strongly related.

However, much less is known of the character varieties for other groups, notably for $\mathrm{SL}(n,\mathbb{C})$ with $n\geq 3$. The $\mathrm{SL}(n,\mathbb{C})$-character varieties for free groups have been studied by S. Lawton and P. Will, and the $\mathrm{SL}(3,\mathbb{C})$-character variety of torus knot groups has been determined by V. Munoz and J. Porti.

In this talk I will present some results concerning the deformations of diagonal representations of knot groups in basic notations and some recent results concerning the representation and character varieties of $3$-manifold groups and in particular knot groups. In particular, we are interested in the local structure of the $\mathrm{SL}(n,\mathbb{C})$-representation variety at the diagonal representation.

### 2018年11月06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

Coarsely convex spaces and a coarse Cartan-Hadamard theorem (JAPANESE)

Tea: Common Room 17:00-17:30

**尾國 新一 氏**(愛媛大学)Coarsely convex spaces and a coarse Cartan-Hadamard theorem (JAPANESE)

[ 講演概要 ]

A coarse version of negatively-curved spaces have been very well studied as Gromov hyperbolic spaces. Recently we introduced a coarse version of non-positively curved spaces, named them coarsely convex spaces and showed a coarse version of the Cartan-Hadamard theorem for such spaces in a joint-work with Tomohiro Fukaya (arXiv:1705.05588). Based on the work, I introduce coarsely convex spaces and explain a coarse Cartan-Hadamard theorem, ideas for proof and its applications to differential topology.

A coarse version of negatively-curved spaces have been very well studied as Gromov hyperbolic spaces. Recently we introduced a coarse version of non-positively curved spaces, named them coarsely convex spaces and showed a coarse version of the Cartan-Hadamard theorem for such spaces in a joint-work with Tomohiro Fukaya (arXiv:1705.05588). Based on the work, I introduce coarsely convex spaces and explain a coarse Cartan-Hadamard theorem, ideas for proof and its applications to differential topology.

### 2018年10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

The quasiconformal equivalence of Riemann surfaces and a universality of Schottky spaces (JAPANESE)

Tea: Common Room 16:30-17:00

**志賀 啓成 氏**(東京工業大学)The quasiconformal equivalence of Riemann surfaces and a universality of Schottky spaces (JAPANESE)

[ 講演概要 ]

In the theory of Teichmüller space of Riemann surfaces, we consider the set of Riemann surfaces which are quasiconformally equivalent. For topologically finite Riemann surfaces, it is quite easy to examine if they are quasiconformally equivalent or not. On the other hand, for Riemann surfaces of topologically infinite type, the situation is rather complicated.

In this talk, after constructing an example which shows the complexity of the problem, we give some geometric conditions for Riemann surfaces to be quasiconformally equivalent. Our argument enables us to see a universality of Schottky spaces.

In the theory of Teichmüller space of Riemann surfaces, we consider the set of Riemann surfaces which are quasiconformally equivalent. For topologically finite Riemann surfaces, it is quite easy to examine if they are quasiconformally equivalent or not. On the other hand, for Riemann surfaces of topologically infinite type, the situation is rather complicated.

In this talk, after constructing an example which shows the complexity of the problem, we give some geometric conditions for Riemann surfaces to be quasiconformally equivalent. Our argument enables us to see a universality of Schottky spaces.

### 2018年10月23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

Co-Minkowski space and hyperbolic surfaces (ENGLISH)

Tea: Common Room 16:30-17:00

**François Fillastre 氏**(Université de Cergy-Pontoise)Co-Minkowski space and hyperbolic surfaces (ENGLISH)

[ 講演概要 ]

There are many ways to parametrize two copies of Teichmueller space by constant curvature -1 Riemannian or Lorentzian 3d manifolds (for example the Bers double uniformization theorem). We present the co-Minkowski space (or half-pipe space), which is a constant curvature -1 degenerated 3d space, and which is related to the tangent space of Teichmueller space. As an illustration, we give a new proof of a theorem of Thurston saying that, once the space of measured geodesic laminations on a compact hyperbolic surface is identified with the tangent space of Teichmueller space via infinitesimal earthquake, then the length of laminations is an asymmetric norm. Joint work with Thierry Barbot (Avignon).

There are many ways to parametrize two copies of Teichmueller space by constant curvature -1 Riemannian or Lorentzian 3d manifolds (for example the Bers double uniformization theorem). We present the co-Minkowski space (or half-pipe space), which is a constant curvature -1 degenerated 3d space, and which is related to the tangent space of Teichmueller space. As an illustration, we give a new proof of a theorem of Thurston saying that, once the space of measured geodesic laminations on a compact hyperbolic surface is identified with the tangent space of Teichmueller space via infinitesimal earthquake, then the length of laminations is an asymmetric norm. Joint work with Thierry Barbot (Avignon).

### 2018年10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

Resonance varieties and matrix tree theorems (ENGLISH)

Tea: Common Room 16:30-17:00

**Daniel Matei 氏**(IMAR Bucharest)Resonance varieties and matrix tree theorems (ENGLISH)

[ 講演概要 ]

We discuss the resonance varieties, encoding vanishing of cohomology cup products, of various classes of finitely presented groups of geometric and combinatorial origin. We describe the ideals defining those varieties in terms spanning trees in a similar vein with the classical matrix tree theorem in graph theory. We present applications of this description to 3-manifold groups and Artin groups.

We discuss the resonance varieties, encoding vanishing of cohomology cup products, of various classes of finitely presented groups of geometric and combinatorial origin. We describe the ideals defining those varieties in terms spanning trees in a similar vein with the classical matrix tree theorem in graph theory. We present applications of this description to 3-manifold groups and Artin groups.

### 2018年10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

Foulon surgery, new contact flows, and dynamical complexity (ENGLISH)

Tea: Common Room 16:30-17:00

**Boris Hasselblatt 氏**(Tufts University)Foulon surgery, new contact flows, and dynamical complexity (ENGLISH)

[ 講演概要 ]

A refinement of Dehn surgery produces new contact flows that are unusual and interesting in several ways. The geodesic flow of a hyperbolic surface becomes a nonalgebraic contact Anosov flow with larger orbit growth, and the purely periodic fiber flow becomes parabolic or hyperbolic. Moreover, Reeb flows for other contact forms for the same contact structure have the same complexity. Finally, an idea by Vinhage promises a quantification of the complexity increase.

A refinement of Dehn surgery produces new contact flows that are unusual and interesting in several ways. The geodesic flow of a hyperbolic surface becomes a nonalgebraic contact Anosov flow with larger orbit growth, and the purely periodic fiber flow becomes parabolic or hyperbolic. Moreover, Reeb flows for other contact forms for the same contact structure have the same complexity. Finally, an idea by Vinhage promises a quantification of the complexity increase.

### 2018年10月02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

An Alexander polynomial for MOY graphs (JAPANESE)

Tea: Common Room 16:30-17:00

**鮑 園園 氏**(東京大学大学院数理科学研究科)An Alexander polynomial for MOY graphs (JAPANESE)

[ 講演概要 ]

An MOY graph is a trivalent graph equipped with a balanced coloring. In this talk, we define a version of Alexander polynomial for an MOY graph. This polynomial is the Euler characteristic of the Heegaard Floer homology of an MOY graph. We give a characterization of the polynomial, which we call MOY-type relations, and show that it is equivalent to Viro’s gl(1 | 1)-Alexander polynomial of a graph. (A part of the talk is a joint work of Zhongtao Wu)

An MOY graph is a trivalent graph equipped with a balanced coloring. In this talk, we define a version of Alexander polynomial for an MOY graph. This polynomial is the Euler characteristic of the Heegaard Floer homology of an MOY graph. We give a characterization of the polynomial, which we call MOY-type relations, and show that it is equivalent to Viro’s gl(1 | 1)-Alexander polynomial of a graph. (A part of the talk is a joint work of Zhongtao Wu)

### 2018年07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

Positive flow-spines and contact 3-manifolds (JAPANESE)

Tea: Common Room 16:30-17:00

**石川 昌治 氏**(慶應義塾大学)Positive flow-spines and contact 3-manifolds (JAPANESE)

[ 講演概要 ]

A contact structure is a smooth distribution of hyperplanes on an odd-dimensional manifold that is non-integrable everywhere. In the case of dimension 3, there is a nice relationship between open book decompositions of 3-manifolds and contact structures up to contactomorphisms, called Giroux correspondence. A flow-spine is a spine of a 3-manifold admitting a flow such that it is transverse to the spine and the flow in the complement of the spine is diffeomorphic to a constant flow in an open ball. In this talk, we introduce some results in progress that give a correspondence between contact structures and positive flow-spines by regarding Reeb vector fields as flows of spines. This is a joint work with Y. Koda (Hiroshima) and H. Naoe (Tohoku).

A contact structure is a smooth distribution of hyperplanes on an odd-dimensional manifold that is non-integrable everywhere. In the case of dimension 3, there is a nice relationship between open book decompositions of 3-manifolds and contact structures up to contactomorphisms, called Giroux correspondence. A flow-spine is a spine of a 3-manifold admitting a flow such that it is transverse to the spine and the flow in the complement of the spine is diffeomorphic to a constant flow in an open ball. In this talk, we introduce some results in progress that give a correspondence between contact structures and positive flow-spines by regarding Reeb vector fields as flows of spines. This is a joint work with Y. Koda (Hiroshima) and H. Naoe (Tohoku).

### 2018年07月10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

Loose Legendrians and arboreal singularities (ENGLISH)

Tea: Common Room 16:30-17:00

**Emmy Murphy 氏**(Northwestern University)Loose Legendrians and arboreal singularities (ENGLISH)

[ 講演概要 ]

Given a Stein manifold X, under what conditions can we ensure that X is symplectomorphic to C^n? For n>2 the condition of X being diffeomorphic to C^n does not suffice, and many counterexamples have been constructed which are detected by symplectic cohomology and the Fukaya category. One might conjecture that the diffeomorphism type together with a vanishing Fukaya category characterizes C^n. While this question is currently well of of reach, we present some new partial results. The main tools we'll discuss are arboreal singularities, constructable sheaf theory, and loose Legendrians -- and how they fit together to approach this question.

Given a Stein manifold X, under what conditions can we ensure that X is symplectomorphic to C^n? For n>2 the condition of X being diffeomorphic to C^n does not suffice, and many counterexamples have been constructed which are detected by symplectic cohomology and the Fukaya category. One might conjecture that the diffeomorphism type together with a vanishing Fukaya category characterizes C^n. While this question is currently well of of reach, we present some new partial results. The main tools we'll discuss are arboreal singularities, constructable sheaf theory, and loose Legendrians -- and how they fit together to approach this question.

### 2018年07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

Symmetries on algebras and Hochschild homology in view of categories of operators (JAPANESE)

Tea: Common Room 16:30-17:00

**吉田 純 氏**(東京大学大学院数理科学研究科)Symmetries on algebras and Hochschild homology in view of categories of operators (JAPANESE)

[ 講演概要 ]

The categorical construction of Hochschild homology by Connes reveals that the symmetric structure on the tensor product of abelian groups is essential. It means that the categorical meaning of ad-hoc generalizations of Hochschild homology in less symmetric monoidal abelian categories remains unclear. In this talk, I will propose formulation of this problem in terms of group operads introduced by Zhang. Moreover, for each group operad G, G-symmetric versions of categories of operators will be discussed. The notion plays a key role in defining Hochschild homology for homotopy algebras; such as topological Hochschild homology.

The categorical construction of Hochschild homology by Connes reveals that the symmetric structure on the tensor product of abelian groups is essential. It means that the categorical meaning of ad-hoc generalizations of Hochschild homology in less symmetric monoidal abelian categories remains unclear. In this talk, I will propose formulation of this problem in terms of group operads introduced by Zhang. Moreover, for each group operad G, G-symmetric versions of categories of operators will be discussed. The notion plays a key role in defining Hochschild homology for homotopy algebras; such as topological Hochschild homology.

### 2018年06月19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

Characteristic classes via 4-dimensional gauge theory (JAPANESE)

Tea: Common Room 16:30-17:00

**今野 北斗 氏**(東京大学大学院数理科学研究科)Characteristic classes via 4-dimensional gauge theory (JAPANESE)

[ 講演概要 ]

Using gauge theory, more precisely SO(3)-Yang-Mills theory and Seiberg-Witten theory, I will construct characteristic classes of 4-manifold bundles. These characteristic classes are extensions of the SO(3)-Donaldson invariant and the Seiberg-Witten invariant to families of 4-manifolds, and can detect non-triviality of smooth 4-manifold bundles. The basic idea of the construction of these characteristic classes is to consider an infinite dimensional analogue of classical characteristic classes of manifold bundles, typified by the Mumford-Morita-Miller classes for surface bundles.

Using gauge theory, more precisely SO(3)-Yang-Mills theory and Seiberg-Witten theory, I will construct characteristic classes of 4-manifold bundles. These characteristic classes are extensions of the SO(3)-Donaldson invariant and the Seiberg-Witten invariant to families of 4-manifolds, and can detect non-triviality of smooth 4-manifold bundles. The basic idea of the construction of these characteristic classes is to consider an infinite dimensional analogue of classical characteristic classes of manifold bundles, typified by the Mumford-Morita-Miller classes for surface bundles.

### 2018年06月19日(火)

14:30-16:00 数理科学研究科棟(駒場) 056号室

RIKEN iTHEMS と共催

相対かつ同変ラグランジアンフレアーホモロジーとアティヤ-フレアー予想 (JAPANESE)

RIKEN iTHEMS と共催

**深谷 賢治 氏**(サイモンズセンター, SUNY)相対かつ同変ラグランジアンフレアーホモロジーとアティヤ-フレアー予想 (JAPANESE)

[ 講演概要 ]

アティヤ-フレアー予想は，ゲージ理論におけるフレアーホモロジーとラグランジアンフレアーホモロジーの間の関係に関するものである．その一つの困難は，ラグランジアンフレアーホモロジーを考えるシンプレクティック多様体が特異点を持つことである．双対かつ同変ラグランジアンフレアーホモロジーを考えることで，この困難が解消し，少なくともアティヤ-フレアー予想を数学的に厳密な予想として定式化できることを説明する．

アティヤ-フレアー予想は，ゲージ理論におけるフレアーホモロジーとラグランジアンフレアーホモロジーの間の関係に関するものである．その一つの困難は，ラグランジアンフレアーホモロジーを考えるシンプレクティック多様体が特異点を持つことである．双対かつ同変ラグランジアンフレアーホモロジーを考えることで，この困難が解消し，少なくともアティヤ-フレアー予想を数学的に厳密な予想として定式化できることを説明する．

### 2018年06月12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

Turbulization of 2-dimensional foliations on 4-manifolds (JAPANESE)

Tea: Common Room 16:30-17:00

**三松 佳彦 氏**(中央大学)Turbulization of 2-dimensional foliations on 4-manifolds (JAPANESE)

[ 講演概要 ]

This is a report on a joint work with Elmar VOGT(Freie Universität Berlin). For codimension 1 foliations, the process of turbulization, i.e., inserting a Reeb component along a closed transversal, is well-known, while for higher codimensional foliation, similar processes were not understood until around 2006.

In this talk, first we formulate the turbulization along a closed transversal. Then in our dimension setting, namely 2-dimensional foliations on 4-manifolds ((4,2)-foliations), a cohomological criterion is given for a given transversal to a foliation, which tells the turbulization is possible or not, relying on the Thurston's h-principle. Also we give cocrete geometric constructions of turbulizations.

The cohomological criterion for turbulization is deduced from a more general criterion for a given embedded surface to be a compact leaf or a closed transversal of some foliation, which is stated in terms of the euler classes of tangent and normal bndle of the foliation to look for. The anormalous cohomological solutions for certain cases suggested the geometric realization of turbulization, while the cohomological criterion is obtained by the h-principle.

Some other modifications are also formulated for (4,2)-foliations and their possibility are assured by the anormalous solutions mentioned above. For some of them, good geometric realizations are not yet known. So far the difficulty lies on the problem of the connected components of the space of representations of the surface groups to Diff S^1.

If the time permits, some special features on the h-principle for 2-dimensional foliations are also explained.

This is a report on a joint work with Elmar VOGT(Freie Universität Berlin). For codimension 1 foliations, the process of turbulization, i.e., inserting a Reeb component along a closed transversal, is well-known, while for higher codimensional foliation, similar processes were not understood until around 2006.

In this talk, first we formulate the turbulization along a closed transversal. Then in our dimension setting, namely 2-dimensional foliations on 4-manifolds ((4,2)-foliations), a cohomological criterion is given for a given transversal to a foliation, which tells the turbulization is possible or not, relying on the Thurston's h-principle. Also we give cocrete geometric constructions of turbulizations.

The cohomological criterion for turbulization is deduced from a more general criterion for a given embedded surface to be a compact leaf or a closed transversal of some foliation, which is stated in terms of the euler classes of tangent and normal bndle of the foliation to look for. The anormalous cohomological solutions for certain cases suggested the geometric realization of turbulization, while the cohomological criterion is obtained by the h-principle.

Some other modifications are also formulated for (4,2)-foliations and their possibility are assured by the anormalous solutions mentioned above. For some of them, good geometric realizations are not yet known. So far the difficulty lies on the problem of the connected components of the space of representations of the surface groups to Diff S^1.

If the time permits, some special features on the h-principle for 2-dimensional foliations are also explained.

### 2018年06月05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

Topological full groups and generalizations of the Higman-Thompson groups (JAPANESE)

Tea: Common Room 16:30-17:00

**松井 宏樹 氏**(千葉大学)Topological full groups and generalizations of the Higman-Thompson groups (JAPANESE)

[ 講演概要 ]

For a topological dynamical system on the Cantor set, one can introduce its topological full group, which is a countable subgroup of the homeomorphism group of the Cantor set. The Higman-Thompson group V_n is regarded as the topological full group of the one-sided full shift over n symbols. Replacing the one-sided full shift with other dynamical systems, we obtain variants of the Higman-Thompson group. It is then natural to ask whether those generalized Higman-Thompson groups possess similar (or different) features. I would like to discuss isomorphism classes of these groups, finiteness properties, abelianizations, connections to C*-algebras and their K-theory, and so on.

For a topological dynamical system on the Cantor set, one can introduce its topological full group, which is a countable subgroup of the homeomorphism group of the Cantor set. The Higman-Thompson group V_n is regarded as the topological full group of the one-sided full shift over n symbols. Replacing the one-sided full shift with other dynamical systems, we obtain variants of the Higman-Thompson group. It is then natural to ask whether those generalized Higman-Thompson groups possess similar (or different) features. I would like to discuss isomorphism classes of these groups, finiteness properties, abelianizations, connections to C*-algebras and their K-theory, and so on.

### 2018年05月29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 partial order on nu+ equivalence classes (JAPANESE)

Tea: Common Room 16:30-17:00

**佐藤 光樹 氏**(東京大学大学院数理科学研究科)A partial order on nu+ equivalence classes (JAPANESE)

[ 講演概要 ]

The nu+ equivalence is an equivalence relation on the knot concordance group. Hom proves that many concordance invariants derived from Heegaard Floer homology are invariant under nu+ equivalence. In this work, we introduce a partial order on nu+ equivalence classes, and study its algebraic and geometrical properties. As an application, we prove that any genus one knot is nu+ equivalent to one of the unknot, the trefoil and its mirror.

The nu+ equivalence is an equivalence relation on the knot concordance group. Hom proves that many concordance invariants derived from Heegaard Floer homology are invariant under nu+ equivalence. In this work, we introduce a partial order on nu+ equivalence classes, and study its algebraic and geometrical properties. As an application, we prove that any genus one knot is nu+ equivalent to one of the unknot, the trefoil and its mirror.