## トポロジー火曜セミナー

過去の記録 ～02/07｜次回の予定｜今後の予定 02/08～

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11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

The smooth Gromov space and the realization problem (ENGLISH)

Tea: Common Room 16:30-17:00

**Ramón Barral Lijó 氏**(立命館大学)The smooth Gromov space and the realization problem (ENGLISH)

[ 講演概要 ]

The n-dimensional smooth Gromov space consists of the pointed isometry classes of complete Riemannian n-manifolds. In this talk we will present the definition and basic properties of this space as well as two different applications: The first addresses the following classical problem in foliation theory.

Realization problem: what kind of manifolds can be leaves of compact foliations?

Our joint work with Álvarez López has produced the following solution in the context of foliated spaces.

Theorem. Every Riemannian manifold of bounded geometry is a leaf in a compact foliated space X endowed with a metric tensor. Moreover, we can assume that X has trivial holonomy and is transversely Cantor.

The second application is the recent research by Abert and Biringer on the subject of unimodular random Riemannian manifolds.

The n-dimensional smooth Gromov space consists of the pointed isometry classes of complete Riemannian n-manifolds. In this talk we will present the definition and basic properties of this space as well as two different applications: The first addresses the following classical problem in foliation theory.

Realization problem: what kind of manifolds can be leaves of compact foliations?

Our joint work with Álvarez López has produced the following solution in the context of foliated spaces.

Theorem. Every Riemannian manifold of bounded geometry is a leaf in a compact foliated space X endowed with a metric tensor. Moreover, we can assume that X has trivial holonomy and is transversely Cantor.

The second application is the recent research by Abert and Biringer on the subject of unimodular random Riemannian manifolds.

### 2019年11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

Magnitude homology of geodesic space (JAPANESE)

Tea: Common Room 16:30-17:00

**五味 清紀 氏**(東京工業大学)Magnitude homology of geodesic space (JAPANESE)

[ 講演概要 ]

Magnitude is an invariant which counts `effective number of points' on a metric space. Its categorification is magnitude homology. This notion is first formulated for metric spaces associated to simple graphs by Hepworth and Willerton, and then for any metric spaces by Leinster and Shulman. The definition of the magnitude homology is easy, but its calculation is rather difficult. For example, the magnitude homology of the circle with geodesic metric was known partially. In my talk, I will explain my result that fully determines the magnitude homology of any geodesic metric space subject to a certain non-branching assumption. In this result, the magnitude homology is described in terms of geodesics. Complete and connected Riemannian manifolds are examples of the geodesic metric spaces satisfying the assumption.

Magnitude is an invariant which counts `effective number of points' on a metric space. Its categorification is magnitude homology. This notion is first formulated for metric spaces associated to simple graphs by Hepworth and Willerton, and then for any metric spaces by Leinster and Shulman. The definition of the magnitude homology is easy, but its calculation is rather difficult. For example, the magnitude homology of the circle with geodesic metric was known partially. In my talk, I will explain my result that fully determines the magnitude homology of any geodesic metric space subject to a certain non-branching assumption. In this result, the magnitude homology is described in terms of geodesics. Complete and connected Riemannian manifolds are examples of the geodesic metric spaces satisfying the assumption.

### 2019年10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

Strong stability of minimal submanifolds (ENGLISH)

Tea: Common Room 16:30-17:00

**Chung-Jun Tsai 氏**(National Taiwan University)Strong stability of minimal submanifolds (ENGLISH)

[ 講演概要 ]

It is well known that the distance function to a totally geodesic submanifold of a negatively curved ambient manifold is a convex function. One can identify a strong stability condition on minimal submanifolds that generalizes the above scenario. Besides a strong local uniqueness property, a strongly stable minimal submanifold is also Lipschitz stable under the mean curvature flow. We will also discuss some famous local (complete, non-compact) models. This is based on a joint work with Mu-Tao Wang.

It is well known that the distance function to a totally geodesic submanifold of a negatively curved ambient manifold is a convex function. One can identify a strong stability condition on minimal submanifolds that generalizes the above scenario. Besides a strong local uniqueness property, a strongly stable minimal submanifold is also Lipschitz stable under the mean curvature flow. We will also discuss some famous local (complete, non-compact) models. This is based on a joint work with Mu-Tao Wang.

### 2019年10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 surgeries in 3-manifolds (ENGLISH)

Tea: Common Room 16:30-17:00

**Gwénaël Massuyeau 氏**(Université de Bourgogne)Generalized Dehn twists on surfaces and surgeries in 3-manifolds (ENGLISH)

[ 講演概要 ]

(Joint work with Yusuke Kuno.) Given an oriented surface S and a simple closed curve C in S, the "Dehn twist" along C is the homeomorphism of S defined by "twisting" S around C by a full twist. If the curve C is not simple, this transformation of S does not make sense anymore, but one can consider two possible generalizations: one possibility is to use the homotopy intersection form of S to "simulate" the action of a Dehn twist on the (Malcev completion of) the fundamental group of S; another possibility is to view C as a curve on the top boundary of the cylinder S×[0,1], to push it arbitrarily into the interior so as to obtain, by surgery along the resulting knot, a new 3-manifold. In this talk, we will relate two those possible generalizations of a Dehn twist and we will give explicit formulas using a "symplectic expansion" of the fundamental group of S.

(Joint work with Yusuke Kuno.) Given an oriented surface S and a simple closed curve C in S, the "Dehn twist" along C is the homeomorphism of S defined by "twisting" S around C by a full twist. If the curve C is not simple, this transformation of S does not make sense anymore, but one can consider two possible generalizations: one possibility is to use the homotopy intersection form of S to "simulate" the action of a Dehn twist on the (Malcev completion of) the fundamental group of S; another possibility is to view C as a curve on the top boundary of the cylinder S×[0,1], to push it arbitrarily into the interior so as to obtain, by surgery along the resulting knot, a new 3-manifold. In this talk, we will relate two those possible generalizations of a Dehn twist and we will give explicit formulas using a "symplectic expansion" of the fundamental group of S.

### 2019年10月08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

いかにして双曲的力学系を群作用に拡張するか？ (JAPANESE)

Tea: Common Room 17:00-17:30

**塚本 真輝 氏**(九州大学)いかにして双曲的力学系を群作用に拡張するか？ (JAPANESE)

[ 講演概要 ]

双曲性は通常の力学系（1パラメータ群作用の研究）において最も基本的な重要性を持つ概念です．それは，十分な豊かさ（拡大性や正エントロピー）を持ちながらも，同時に制御可能（安定性や適切な意味での良い測度の一意性）な力学系の例を与えます．ではこれを群作用に拡張できるでしょうか？

ナイーブには困難です．例えば $Z^2$ の作用を考えましょう（つまり可換な 2 パラメータ作用）・簡単にわかるのは，有限次元のコンパクト多様体に $Z^2$ が可微分に作用するとき，その $Z^2$ 作用としてのエントロピーはゼロになります．つまり，通常の有限次元の状況には，豊かな $Z^2$ 作用は存在しません．言い換えると，十分に豊かな群作用を得るためには無限次元の世界に行かざるを得ません．しかし，無限次元の世界でどのような構造を見出せばよいのでしょうか？

この講演では，このような方向性にアプローチする際に，平均次元と呼ばれる量が大きな役割を果たす可能性を説明します．特に，次のような原理についてお話します：

$Z^k$（可換な $k$ パラメータ群）が空間 $X$ に何らかの「双曲性」を持って作用するとき，$Z^k$ のランク $k-1$ の部分群 $G$ の部分作用に対する平均次元が制御できる．

この講演はTom Meyerovitch，篠田万穂との共同研究に基づきます．

双曲性は通常の力学系（1パラメータ群作用の研究）において最も基本的な重要性を持つ概念です．それは，十分な豊かさ（拡大性や正エントロピー）を持ちながらも，同時に制御可能（安定性や適切な意味での良い測度の一意性）な力学系の例を与えます．ではこれを群作用に拡張できるでしょうか？

ナイーブには困難です．例えば $Z^2$ の作用を考えましょう（つまり可換な 2 パラメータ作用）・簡単にわかるのは，有限次元のコンパクト多様体に $Z^2$ が可微分に作用するとき，その $Z^2$ 作用としてのエントロピーはゼロになります．つまり，通常の有限次元の状況には，豊かな $Z^2$ 作用は存在しません．言い換えると，十分に豊かな群作用を得るためには無限次元の世界に行かざるを得ません．しかし，無限次元の世界でどのような構造を見出せばよいのでしょうか？

この講演では，このような方向性にアプローチする際に，平均次元と呼ばれる量が大きな役割を果たす可能性を説明します．特に，次のような原理についてお話します：

$Z^k$（可換な $k$ パラメータ群）が空間 $X$ に何らかの「双曲性」を持って作用するとき，$Z^k$ のランク $k-1$ の部分群 $G$ の部分作用に対する平均次元が制御できる．

この講演はTom Meyerovitch，篠田万穂との共同研究に基づきます．

### 2019年10月01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

Quantized SL(2) representations of knot groups (JAPANESE)

Tea: Common Room 16:30-17:00

**村上 順 氏**(早稲田大学)Quantized SL(2) representations of knot groups (JAPANESE)

[ 講演概要 ]

Let K be a knot and G be a group. The representation space of K for the group G means the space of homomorphisms from the knot group to G and is defined by using the group ring C[G], where C[G] is the ring of functions on G and has a commutative Hopf algebra structure. This construction can be generalized to any commutative Hopf algebras.

In this talk, we extend this construction to any braided Hopf algebras with braided commutativity. A typical example is BSL(2), which is the braided SL(2) introduced by S. Majid. Applying the above construction to BSL(2), we get the space of BSL(2) representations, which provides a quantization of SL(2) representations of a knot. This is joint work with Roloand van der Veen.

Let K be a knot and G be a group. The representation space of K for the group G means the space of homomorphisms from the knot group to G and is defined by using the group ring C[G], where C[G] is the ring of functions on G and has a commutative Hopf algebra structure. This construction can be generalized to any commutative Hopf algebras.

In this talk, we extend this construction to any braided Hopf algebras with braided commutativity. A typical example is BSL(2), which is the braided SL(2) introduced by S. Majid. Applying the above construction to BSL(2), we get the space of BSL(2) representations, which provides a quantization of SL(2) representations of a knot. This is joint work with Roloand van der Veen.

### 2019年07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

Seifert vs. slice genera of knots in twist families and a characterization of braid axes (JAPANESE)

Tea: Common Room 16:30-17:00

**茂手木 公彦 氏**(日本大学)Seifert vs. slice genera of knots in twist families and a characterization of braid axes (JAPANESE)

[ 講演概要 ]

Twisting a knot $K$ in $S^3$ along a disjoint unknot $c$ produces a twist family of knots $\{K_n\}$ indexed by the integers. Comparing the behaviors of the Seifert genus $g(K_n)$ and the slice genus $g_4(K_n)$ under twistings, we prove that if $g(K_n) - g_4(K_n) < C$ for some constant $C$ for infinitely many integers $n > 0$ or $g(K_n) / g_4(K_n)$ limits to $1$, then the winding number of $K$ about $c$ equals either zero or the wrapping number. As a key application, if $\{K_n\}$ or the mirror twist family $\{\overline{K_n}\}$ contains infinitely many tight fibered knots, then the latter must occur. This leads to the characterization that $c$ is a braid axis of $K$ if and only if both $\{K_n\}$ and $\{\overline{K_n}\}$ each contain infinitely many tight fibered knots. We also give a necessary and sufficient condition for $\{K_n\}$ to contain infinitely many L-space knots, and apply the characterization to prove that satellite L-space knots have braided patterns, which answers a question of both Baker-Moore and Hom in the positive. This result also implies an absence of essential Conway spheres for satellite L-space knots, which gives a partial answer to a conjecture of Lidman-Moore.

This is joint work with Kenneth Baker (University of Miami).

Twisting a knot $K$ in $S^3$ along a disjoint unknot $c$ produces a twist family of knots $\{K_n\}$ indexed by the integers. Comparing the behaviors of the Seifert genus $g(K_n)$ and the slice genus $g_4(K_n)$ under twistings, we prove that if $g(K_n) - g_4(K_n) < C$ for some constant $C$ for infinitely many integers $n > 0$ or $g(K_n) / g_4(K_n)$ limits to $1$, then the winding number of $K$ about $c$ equals either zero or the wrapping number. As a key application, if $\{K_n\}$ or the mirror twist family $\{\overline{K_n}\}$ contains infinitely many tight fibered knots, then the latter must occur. This leads to the characterization that $c$ is a braid axis of $K$ if and only if both $\{K_n\}$ and $\{\overline{K_n}\}$ each contain infinitely many tight fibered knots. We also give a necessary and sufficient condition for $\{K_n\}$ to contain infinitely many L-space knots, and apply the characterization to prove that satellite L-space knots have braided patterns, which answers a question of both Baker-Moore and Hom in the positive. This result also implies an absence of essential Conway spheres for satellite L-space knots, which gives a partial answer to a conjecture of Lidman-Moore.

This is joint work with Kenneth Baker (University of Miami).

### 2019年07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

Mod 2 cohomology of moduli stacks of real vector bundles (ENGLISH)

Tea: Common Room 16:30-17:00

**Florent Schaffhauser 氏**(Université de Strasbourg)Mod 2 cohomology of moduli stacks of real vector bundles (ENGLISH)

[ 講演概要 ]

The rational cohomology ring of the moduli stack of holomorphic vector bundles of fixed rank and degree over a compact Riemann surface was studied by Atiyah and Bott using tools of differential geometry and algebraic topology: they found generators of that ring and computed its Poincaré series. In joint work with Chiu-Chu Melissa Liu, we study in a similar way the mod 2 cohomology ring of the moduli stack of real vector bundles of fixed topological type over a compact Riemann surface with real structure. The goal of the talk is to explain the principle of that computation, emphasizing the analogies and differences between the real and complex cases, and discuss applications of the method. In particular, we provide explicit generators of mod 2 cohomology rings of moduli stacks of vector bundles over a real algebraic curve.

The rational cohomology ring of the moduli stack of holomorphic vector bundles of fixed rank and degree over a compact Riemann surface was studied by Atiyah and Bott using tools of differential geometry and algebraic topology: they found generators of that ring and computed its Poincaré series. In joint work with Chiu-Chu Melissa Liu, we study in a similar way the mod 2 cohomology ring of the moduli stack of real vector bundles of fixed topological type over a compact Riemann surface with real structure. The goal of the talk is to explain the principle of that computation, emphasizing the analogies and differences between the real and complex cases, and discuss applications of the method. In particular, we provide explicit generators of mod 2 cohomology rings of moduli stacks of vector bundles over a real algebraic curve.

### 2019年07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

Brane coproducts and their applications (JAPANESE)

Tea: Common Room 16:30-17:00

**若月 駿 氏**(東京大学大学院数理科学研究科)Brane coproducts and their applications (JAPANESE)

[ 講演概要 ]

The loop coproduct is a coproduct on the homology of the free loop space of a Poincaré duality space (or more generally a Gorenstein space). In this talk, I will introduce two kinds of brane coproducts which are generalizations of the loop coproduct to the homology of a sphere space (i.e. the mapping space from a sphere). Their constructions are based on the finiteness of the dimensions of mapping spaces in some sense. As an application, I will show the vanishing of some cup products on sphere spaces by comparing these two brane coproducts. This gives a generalization of a result of Menichi for the case of free loop spaces.

The loop coproduct is a coproduct on the homology of the free loop space of a Poincaré duality space (or more generally a Gorenstein space). In this talk, I will introduce two kinds of brane coproducts which are generalizations of the loop coproduct to the homology of a sphere space (i.e. the mapping space from a sphere). Their constructions are based on the finiteness of the dimensions of mapping spaces in some sense. As an application, I will show the vanishing of some cup products on sphere spaces by comparing these two brane coproducts. This gives a generalization of a result of Menichi for the case of free loop spaces.

### 2019年06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

Geometry of symplectic log Calabi-Yau surfaces (ENGLISH)

Tea: Common Room 16:30-17:00

**Tian-Jun Li 氏**(University of Minnesota)Geometry of symplectic log Calabi-Yau surfaces (ENGLISH)

[ 講演概要 ]

This is a survey on the geometry of symplectic log Calabi-Yau surfaces, which are the symplectic analogues of Looijenga pairs. We address the classification up to symplectic deformation, the relations between symplectic circular sequences and anti-canonical sequences, contact trichotomy, and symplectic fillings. This is a joint work with Cheuk Yu Mak.

This is a survey on the geometry of symplectic log Calabi-Yau surfaces, which are the symplectic analogues of Looijenga pairs. We address the classification up to symplectic deformation, the relations between symplectic circular sequences and anti-canonical sequences, contact trichotomy, and symplectic fillings. This is a joint work with Cheuk Yu Mak.

### 2019年06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

Filtered instanton homology and the homology cobordism group (JAPANESE)

Tea: Common Room 16:30-17:00

**谷口 正樹 氏**(東京大学大学院数理科学研究科)Filtered instanton homology and the homology cobordism group (JAPANESE)

[ 講演概要 ]

We give a new family of real-valued invariants {r_s} of oriented homology 3-spheres. The invariants are defined by using some filtered version of instanton Floer homology. The invariants are closely related to the existence of solutions to ASD equations on Y×R for a given homology sphere Y. We show some properties of {r_s} containing a connected sum formula and a negative definite inequality. As applications of such properties of {r_s}, we obtain several new results on the homology cobordism group and the knot concordance group. As one of such results, we show that if the 1-surgery of a knot has the Froyshov invariant negative, then all positive 1/n-surgeries of the knot are linearly independent in the homology cobordism group. This theorem gives a generalization of the theorem shown by Furuta and Fintushel-Stern in ’90. Moreover, we estimate the values of {r_s} for a hyperbolic manifold Y with an error of at most 10^{-50}. It seems the values are irrational. If the values are irrational, we can conclude that the homology cobordism group is not generated by Seifert homology spheres. This is joint work with Yuta Nozaki and Kouki Sato.

We give a new family of real-valued invariants {r_s} of oriented homology 3-spheres. The invariants are defined by using some filtered version of instanton Floer homology. The invariants are closely related to the existence of solutions to ASD equations on Y×R for a given homology sphere Y. We show some properties of {r_s} containing a connected sum formula and a negative definite inequality. As applications of such properties of {r_s}, we obtain several new results on the homology cobordism group and the knot concordance group. As one of such results, we show that if the 1-surgery of a knot has the Froyshov invariant negative, then all positive 1/n-surgeries of the knot are linearly independent in the homology cobordism group. This theorem gives a generalization of the theorem shown by Furuta and Fintushel-Stern in ’90. Moreover, we estimate the values of {r_s} for a hyperbolic manifold Y with an error of at most 10^{-50}. It seems the values are irrational. If the values are irrational, we can conclude that the homology cobordism group is not generated by Seifert homology spheres. This is joint work with Yuta Nozaki and Kouki Sato.

### 2019年06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

Gluck twist on branched twist spins (JAPANESE)

Tea: Common Room 16:30-17:00

**福田 瑞季 氏**(東京学芸大学)Gluck twist on branched twist spins (JAPANESE)

[ 講演概要 ]

Branched twist spin とは４次元球面上の円作用の特異点集合として定義される埋め込まれた２次元球面であり，スパン結び目やツイストスパン結び目などの２次元結び目の一般化となっている．Gluck は４次元多様体内の２次元結び目に沿った向きを保つ手術は微分同相類を除いて２種類のみであることを示しており，自明でない手術を Gluck twist と呼ぶ．一般に Gluck twist が全空間の微分同相を保つかどうかは知られていないが，Pao によって branched twist spin に沿った Gluck twist は 再び４次元球面と微分同相になることが知られている．本講演では，Pao の結果の別証明として円作用を用いて４次元球面の分解を与え，各ピースが Gluck twist を通してどのように変化するかを説明する．また，２次元結び目に注目したとき，Gluck twist によって branched twist spin は再び branched twist spin になることを証明する．

Branched twist spin とは４次元球面上の円作用の特異点集合として定義される埋め込まれた２次元球面であり，スパン結び目やツイストスパン結び目などの２次元結び目の一般化となっている．Gluck は４次元多様体内の２次元結び目に沿った向きを保つ手術は微分同相類を除いて２種類のみであることを示しており，自明でない手術を Gluck twist と呼ぶ．一般に Gluck twist が全空間の微分同相を保つかどうかは知られていないが，Pao によって branched twist spin に沿った Gluck twist は 再び４次元球面と微分同相になることが知られている．本講演では，Pao の結果の別証明として円作用を用いて４次元球面の分解を与え，各ピースが Gluck twist を通してどのように変化するかを説明する．また，２次元結び目に注目したとき，Gluck twist によって branched twist spin は再び branched twist spin になることを証明する．

### 2019年05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

Exotic four-manifolds via positive factorizations (ENGLISH)

Tea: Common Room 16:30-17:00

**R. Inanc Baykur 氏**(University of Massachusetts)Exotic four-manifolds via positive factorizations (ENGLISH)

[ 講演概要 ]

We will discuss several new ideas and techniques for producing positive Dehn twist factorizations of surface mapping classes, which yield novel constructions of various interesting four-manifolds, such as symplectic Calabi-Yau surfaces and exotic rational surfaces, via Lefschetz pencils.

We will discuss several new ideas and techniques for producing positive Dehn twist factorizations of surface mapping classes, which yield novel constructions of various interesting four-manifolds, such as symplectic Calabi-Yau surfaces and exotic rational surfaces, via Lefschetz pencils.

### 2019年05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

On the dealternating number and the alternation number (ENGLISH)

Tea: Common Room 16:30-17:00

**Maria de los Angeles Guevara 氏**(大阪市立大学)On the dealternating number and the alternation number (ENGLISH)

[ 講演概要 ]

Links can be divided into alternating and non-alternating depending on if they possess an alternating diagram or not. After the proof of the Tait flype conjecture on alternating links, it became an important question to ask how a non-alternating link is “close to” alternating links. The dealternating and alternation numbers, which are invariants introduced by C. Adams et al. and A. Kawauchi, respectively, can deal with this question. By definitions, for any link, its alternation number is less than or equal to its dealternating number. It is known that in general the equality does not hold. However, in general, it is not easy to show a gap between these invariants. In this seminar, we will show some results regarding these invariants. In particular, for each pair of positive integers, we will construct infinitely many knots, which have dealternating and alternation numbers determined for these integers. Therefore, an arbitrary gap between the values of these invariants will be obtained.

Links can be divided into alternating and non-alternating depending on if they possess an alternating diagram or not. After the proof of the Tait flype conjecture on alternating links, it became an important question to ask how a non-alternating link is “close to” alternating links. The dealternating and alternation numbers, which are invariants introduced by C. Adams et al. and A. Kawauchi, respectively, can deal with this question. By definitions, for any link, its alternation number is less than or equal to its dealternating number. It is known that in general the equality does not hold. However, in general, it is not easy to show a gap between these invariants. In this seminar, we will show some results regarding these invariants. In particular, for each pair of positive integers, we will construct infinitely many knots, which have dealternating and alternation numbers determined for these integers. Therefore, an arbitrary gap between the values of these invariants will be obtained.

### 2019年05月14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

Diagrammatic Algebra (ENGLISH)

Tea: Common Room 16:30-17:00

**J. Scott Carter 氏**(University of South Alabama, 大阪市立大学)Diagrammatic Algebra (ENGLISH)

[ 講演概要 ]

Three main ideas will be explored. First, a higher dimensional category (a category that has arrows, double arrows, triple arrows, and quadruple arrows) that is based upon the axioms of a Frobenius algebra will be outlined. Then these structures will be promoted into one higher dimension so that braiding can be introduced. Second, relationships between braiding and multiplication will be studied from a homological perspective. Third, the next order relations will be used to formulate a system of abstract tensor equations that are analogous to the Yang-Baxter relation. In this way, a broad outline of the notion of diagrammatic algebra will be presented.

Three main ideas will be explored. First, a higher dimensional category (a category that has arrows, double arrows, triple arrows, and quadruple arrows) that is based upon the axioms of a Frobenius algebra will be outlined. Then these structures will be promoted into one higher dimension so that braiding can be introduced. Second, relationships between braiding and multiplication will be studied from a homological perspective. Third, the next order relations will be used to formulate a system of abstract tensor equations that are analogous to the Yang-Baxter relation. In this way, a broad outline of the notion of diagrammatic algebra will be presented.

### 2019年04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

Higher Hochschild homology as a functor (ENGLISH)

Tea: Common Room 16:30-17:00

**Christine Vespa 氏**(Université de Strasbourg)Higher Hochschild homology as a functor (ENGLISH)

[ 講演概要 ]

Higher Hochschild homology generalizes classical Hochschild homology for rings. Recently, Turchin and Willwacher computed higher Hochschild homology of a finite wedge of circles with coefficients in the Loday functor associated to the ring of dual numbers over the rationals. In particular, they obtained linear representations of the groups Out(F_n) which do not factorize through GL(n,Z).

In this talk, I will begin by recalling what is Hochschild homology and higher Hochschild homology. Then I will explain how viewing higher Hochschild homology of a finite wedge of circles as a functor on the category of free groups provides a conceptual framework which allows powerful tools such as exponential functors and polynomial functors to be used. In particular, this allows the generalization of the results of Turchin and Willwacher; this gives rise to new linear representations of Out(F_n) which do not factorize through GL(n,Z).

(This is joint work with Geoffrey Powell.)

Higher Hochschild homology generalizes classical Hochschild homology for rings. Recently, Turchin and Willwacher computed higher Hochschild homology of a finite wedge of circles with coefficients in the Loday functor associated to the ring of dual numbers over the rationals. In particular, they obtained linear representations of the groups Out(F_n) which do not factorize through GL(n,Z).

In this talk, I will begin by recalling what is Hochschild homology and higher Hochschild homology. Then I will explain how viewing higher Hochschild homology of a finite wedge of circles as a functor on the category of free groups provides a conceptual framework which allows powerful tools such as exponential functors and polynomial functors to be used. In particular, this allows the generalization of the results of Turchin and Willwacher; this gives rise to new linear representations of Out(F_n) which do not factorize through GL(n,Z).

(This is joint work with Geoffrey Powell.)

### 2019年04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

Thurston’s bounded image theorem (ENGLISH)

Tea: Common Room 16:30-17:00

**大鹿 健一 氏**(学習院大学)Thurston’s bounded image theorem (ENGLISH)

[ 講演概要 ]

The bounded image theorem by Thurston constitutes an important step in the proof of his unifomisation theorem for Haken manifolds. Thurston’s original argument was never published and has been unknown up to now. It has turned out a weaker form of this theorem is enough for the proof, and books by Kappovich and by Otal use this weaker version. In this talk, I will show how to prove Thurston’s original version making use of more recent technology. This is joint work with Cyril Lecuire.

The bounded image theorem by Thurston constitutes an important step in the proof of his unifomisation theorem for Haken manifolds. Thurston’s original argument was never published and has been unknown up to now. It has turned out a weaker form of this theorem is enough for the proof, and books by Kappovich and by Otal use this weaker version. In this talk, I will show how to prove Thurston’s original version making use of more recent technology. This is joint work with Cyril Lecuire.

### 2019年04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

Coulomb branches of 3d SUSY gauge theories (JAPANESE)

Tea: Common Room 16:30-17:00

**中島 啓 氏**(Kavli IPMU, 東京大学大学院数理科学研究科)Coulomb branches of 3d SUSY gauge theories (JAPANESE)

[ 講演概要 ]

I will give an introduction to a mathematical definition of Coulomb branches of 3-dimensional SUSY gauge theories, given by my joint work with Braverman and Finkelberg. I will emphasize on the role of hypothetical 3d TQFT associated with gauge theories.

I will give an introduction to a mathematical definition of Coulomb branches of 3-dimensional SUSY gauge theories, given by my joint work with Braverman and Finkelberg. I will emphasize on the role of hypothetical 3d TQFT associated with gauge theories.

### 2019年04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

A topological interpretation of symplectic fillings of a normal surface singularity (ENGLISH)

Tea: Common Room 16:30-17:00

**Jongil Park 氏**(Seoul National University)A topological interpretation of symplectic fillings of a normal surface singularity (ENGLISH)

[ 講演概要 ]

One of active research areas in symplectic 4-manifolds is to classify symplectic fillings of certain 3-manifolds equipped with a contact structure.

Among them, people have long studied symplectic fillings of the link of a normal surface singularity. Note that the link of a normal surface singularity carries a canonical contact structure which is also known as the Milnor fillable contact structure.

In this talk, I’d like to investigate a topological surgery description for minimal symplectic fillings of the link of quotient surface singularities and weighted homogeneous surface singularities with a canonical contact structure. Explicitly, I’ll show that every minimal symplectic filling of the link of quotient surface singularities and weighted homogeneous surface singularities satisfying certain conditions can be obtained by a sequence of rational blowdowns from the minimal resolution of the corresponding surface singularity. This is joint work with Hakho Choi.

One of active research areas in symplectic 4-manifolds is to classify symplectic fillings of certain 3-manifolds equipped with a contact structure.

Among them, people have long studied symplectic fillings of the link of a normal surface singularity. Note that the link of a normal surface singularity carries a canonical contact structure which is also known as the Milnor fillable contact structure.

In this talk, I’d like to investigate a topological surgery description for minimal symplectic fillings of the link of quotient surface singularities and weighted homogeneous surface singularities with a canonical contact structure. Explicitly, I’ll show that every minimal symplectic filling of the link of quotient surface singularities and weighted homogeneous surface singularities satisfying certain conditions can be obtained by a sequence of rational blowdowns from the minimal resolution of the corresponding surface singularity. This is joint work with Hakho Choi.

### 2019年03月27日(水)

17:00-18:30 数理科学研究科棟(駒場) 123号室

開催日, 会場にご注意下さい. Tea: Common Room 16:30-17:00

On a moduli space interpretation of the Turaev cobracket (ENGLISH)

開催日, 会場にご注意下さい. Tea: Common Room 16:30-17:00

**Florian Naef 氏**(Université de Genève)On a moduli space interpretation of the Turaev cobracket (ENGLISH)

[ 講演概要 ]

Given an oriented surface, Goldman defines a Lie bracket on the vector space spanned by free homotopy classes of loops in terms of intersections. This Lie bracket is the universal version of the Atiyah-Bott Poisson structure on the moduli space of flat connections. Using self-intersections Turaev defines a Lie cobracket on loops. We give a possible interpretation of this structure on moduli spaces of flat connections in the form of a natural BV operator on the moduli space of flat connection with values in a super Lie algebra equipped with an odd pairing. This is joint work with A. Alekseev, J. Pulmann and P. Ševera.

Given an oriented surface, Goldman defines a Lie bracket on the vector space spanned by free homotopy classes of loops in terms of intersections. This Lie bracket is the universal version of the Atiyah-Bott Poisson structure on the moduli space of flat connections. Using self-intersections Turaev defines a Lie cobracket on loops. We give a possible interpretation of this structure on moduli spaces of flat connections in the form of a natural BV operator on the moduli space of flat connection with values in a super Lie algebra equipped with an odd pairing. This is joint work with A. Alekseev, J. Pulmann and P. Ševera.

### 2019年03月26日(火)

17:00-18:30 数理科学研究科棟(駒場) 056号室

Cube capacities (ENGLISH)

**Michael Hutchings 氏**(University of California, Berkeley)Cube capacities (ENGLISH)

[ 講演概要 ]

We define a new series of symplectic capacities using equivariant symplectic homology. These capacities are conjecturally equal to the Ekeland-Hofer capacities, but can be computed in many more examples. In particular, we use these capacities to find many examples of symplectic embeddings of cubes where the cube is as large as possible. This is joint work with Jean Gutt.

We define a new series of symplectic capacities using equivariant symplectic homology. These capacities are conjecturally equal to the Ekeland-Hofer capacities, but can be computed in many more examples. In particular, we use these capacities to find many examples of symplectic embeddings of cubes where the cube is as large as possible. This is joint work with Jean Gutt.

### 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.