## トポロジー火曜セミナー

過去の記録 ～12/11｜次回の予定｜今後の予定 12/12～

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

担当者 | 河澄 響矢, 北山 貴裕, 逆井卓也 |

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

**過去の記録**

### 2007年12月11日(火)

16:30-18:40 数理科学研究科棟(駒場) 056号室

Tea: 16:00 - 16:30 コモンルーム

A Singular Version of The Poincar\\'e-Hopf Theorem

Chen Ruan cohomology of cotangent orbifolds and Chas-Sullivan string topology

Tea: 16:00 - 16:30 コモンルーム

**Xavier G\'omez-Mont 氏**(CIMAT, Mexico) 16:30-17:30A Singular Version of The Poincar\\'e-Hopf Theorem

[ 講演概要 ]

The Poincar\\'e-Hopf Theorem asserts that the Euler Characteristic of a compact manifold is the sum of the indices of any vector field on it with isolated singularities.

A hypersurface in real or complex number space may be considered as the limit of the smooth hypersurfaces obtained from nearby regular values. The singularity contains “hidden” topology, which is unfolded by a smooth regeneration. At the singularity one has an algebraic invariant, the Jacobi Algebra, which is obtained by considering analytic functions modulo the partial derivatives. It contains topological information of the singularity.

One may consider vector fields tangent to a hypersurface with isolated singularities, and define topologically an index, which coincides with the sum of the Poincar\\'e-Hopf indices of a regeneration of it tangent to a nearby smooth hypersurface.

I will explain how to compute the index of a vector field X tangent to an isolated hypersurface singularity V using Homological Algebra, as the Euler Characteristic of the homology of the complex obtained by contracting differential forms on V with the vector field X. The formula contains several terms, but the higher order terms may be translated from the invariants of the singular point to invariants in the Jacobi Algebra, making this translation a local version of the Poincar\\'e-Hopf Theorem.

I will also explain how some of these ideas can be extended to complete intersections.

The Poincar\\'e-Hopf Theorem asserts that the Euler Characteristic of a compact manifold is the sum of the indices of any vector field on it with isolated singularities.

A hypersurface in real or complex number space may be considered as the limit of the smooth hypersurfaces obtained from nearby regular values. The singularity contains “hidden” topology, which is unfolded by a smooth regeneration. At the singularity one has an algebraic invariant, the Jacobi Algebra, which is obtained by considering analytic functions modulo the partial derivatives. It contains topological information of the singularity.

One may consider vector fields tangent to a hypersurface with isolated singularities, and define topologically an index, which coincides with the sum of the Poincar\\'e-Hopf indices of a regeneration of it tangent to a nearby smooth hypersurface.

I will explain how to compute the index of a vector field X tangent to an isolated hypersurface singularity V using Homological Algebra, as the Euler Characteristic of the homology of the complex obtained by contracting differential forms on V with the vector field X. The formula contains several terms, but the higher order terms may be translated from the invariants of the singular point to invariants in the Jacobi Algebra, making this translation a local version of the Poincar\\'e-Hopf Theorem.

I will also explain how some of these ideas can be extended to complete intersections.

**Miguel A. Xicotencatl 氏**(CINVESTAV, Mexico) 17:40-18:40Chen Ruan cohomology of cotangent orbifolds and Chas-Sullivan string topology

[ 講演概要 ]

(Joint with: A. Gonzalez, E. Lupercio, C. Segovia, and B. Uribe)

At the end of 90's, two theories of topology were invented roughly at the same time and attracted considerable interest in the mathematical community. One is the Chas-Sullivan's loop product on the homology of loop space and the second one is Chen-Ruan's stringy cohomology of orbifold. It was an observation of Chen that inertia orbifold (which carries Chen-Ruan cohomology) is the space of constant loops of an orbifold. Therefore, two theories should interact. In this work we show that for an interesting family of orbifolds, the virtual orbifold cohomology, turns out to be a subalgebra of the homology of the loop orbifold, and is isomorphic, as algebras, to the Chen-Ruan orbifold cohomology of its cotangent orbifold.

(Joint with: A. Gonzalez, E. Lupercio, C. Segovia, and B. Uribe)

At the end of 90's, two theories of topology were invented roughly at the same time and attracted considerable interest in the mathematical community. One is the Chas-Sullivan's loop product on the homology of loop space and the second one is Chen-Ruan's stringy cohomology of orbifold. It was an observation of Chen that inertia orbifold (which carries Chen-Ruan cohomology) is the space of constant loops of an orbifold. Therefore, two theories should interact. In this work we show that for an interesting family of orbifolds, the virtual orbifold cohomology, turns out to be a subalgebra of the homology of the loop orbifold, and is isomorphic, as algebras, to the Chen-Ruan orbifold cohomology of its cotangent orbifold.

### 2007年12月04日(火)

16:30-18:00 数理科学研究科棟(駒場) 056号室

Tea: 16:00 - 16:30 コモンルーム

Morse theory for abelian hyperkahler quotients

Tea: 16:00 - 16:30 コモンルーム

**今野 宏 氏**(東京大学大学院数理科学研究科)Morse theory for abelian hyperkahler quotients

[ 講演概要 ]

In 1980's Kirwan computed Betti numbers of symplectic quotients by using Morse theory. In this talk, we develop this method to hyperkahler quotients by abelian Lie groups. In this method, many computations are much more simplified in the case of hyperkahler quotients than the case of symplectic quotients. As a result we compute not only the Betti numbers, but also the cohomology rings of abelian hyperkahler quotients.

In 1980's Kirwan computed Betti numbers of symplectic quotients by using Morse theory. In this talk, we develop this method to hyperkahler quotients by abelian Lie groups. In this method, many computations are much more simplified in the case of hyperkahler quotients than the case of symplectic quotients. As a result we compute not only the Betti numbers, but also the cohomology rings of abelian hyperkahler quotients.

### 2007年11月27日(火)

16:30-18:00 数理科学研究科棟(駒場) 056号室

Tea: 16:00 - 16:30 コモンルーム

A quandle cocycle invariant for handlebody-links

Tea: 16:00 - 16:30 コモンルーム

**石井 敦 氏**(京都大学数理解析研究所)A quandle cocycle invariant for handlebody-links

[ 講演概要 ]

[joint work with Masahide Iwakiri (Osaka City University)]

A handlebody-link is a disjoint union of circles and a

finite trivalent graph embedded in a closed 3-manifold.

We consider it up to isotopies and IH-moves.

Then it represents an ambient isotopy class of

handlebodies embedded in the closed 3-manifold.

In this talk, I explain how a quandle cocycle invariant

is defined for handlebody-links.

[joint work with Masahide Iwakiri (Osaka City University)]

A handlebody-link is a disjoint union of circles and a

finite trivalent graph embedded in a closed 3-manifold.

We consider it up to isotopies and IH-moves.

Then it represents an ambient isotopy class of

handlebodies embedded in the closed 3-manifold.

In this talk, I explain how a quandle cocycle invariant

is defined for handlebody-links.

### 2007年11月20日(火)

16:30-18:00 数理科学研究科棟(駒場) 056号室

Tea: 16:00 - 16:30 コモンルーム

A certain slice of the character variety of a knot group

and the knot contact homology

Tea: 16:00 - 16:30 コモンルーム

**長郷 文和 氏**(東京工業大学大学院理工学研究科)A certain slice of the character variety of a knot group

and the knot contact homology

[ 講演概要 ]

For a knot $K$ in 3-sphere, we can consider representations of

the knot group $G_K$ into $SL(2,\\mathbb{C})$.

Their characters construct an algebraic set.

This is so-called the $SL(2,\\mathbb{C})$-character variety of

$G_K$ and denoted by $X(G_K)$.

In this talk, we introduce a slice (a subset) $S_0(K)$ of $X(G_K)$.

In fact, this slice is closely related to the A-polynomial

and the abelian knot contact homology.

For example, the A-polynomial $A_K(m,l)$ of a knot $K$ is

a two-variable polynomial knot invariant defined by using

the character variety $X(G_K)$.

Then we can show that for any {\\it small knot} $K$, the number of

irreducible components of $S_0(K)$ gives an upper bound of

the maximal degree of the A-polynomial $A_K(m,l)$ in terms of

the variable $l$.

Moreover, for any 2-bridge knot $K$, we can show that

the coordinate ring of $S_0(K)$ is exactly the degree 0

abelian knot contact homology $HC_0^{ab}(K)$.

We will mainly explain these facts.

For a knot $K$ in 3-sphere, we can consider representations of

the knot group $G_K$ into $SL(2,\\mathbb{C})$.

Their characters construct an algebraic set.

This is so-called the $SL(2,\\mathbb{C})$-character variety of

$G_K$ and denoted by $X(G_K)$.

In this talk, we introduce a slice (a subset) $S_0(K)$ of $X(G_K)$.

In fact, this slice is closely related to the A-polynomial

and the abelian knot contact homology.

For example, the A-polynomial $A_K(m,l)$ of a knot $K$ is

a two-variable polynomial knot invariant defined by using

the character variety $X(G_K)$.

Then we can show that for any {\\it small knot} $K$, the number of

irreducible components of $S_0(K)$ gives an upper bound of

the maximal degree of the A-polynomial $A_K(m,l)$ in terms of

the variable $l$.

Moreover, for any 2-bridge knot $K$, we can show that

the coordinate ring of $S_0(K)$ is exactly the degree 0

abelian knot contact homology $HC_0^{ab}(K)$.

We will mainly explain these facts.

### 2007年11月06日(火)

16:30-18:00 数理科学研究科棟(駒場) 056号室

Tea: 16:00 - 16:30 コモンルーム

Thustion's inequality and open book foliations

Tea: 16:00 - 16:30 コモンルーム

**児玉 大樹 氏**(東京大学大学院数理科学研究科)Thustion's inequality and open book foliations

[ 講演概要 ]

We will study codimension 1 foliations on 3-manifolds.

Thurston's inequality, which implies convexity of the foliation in

some sense, folds for Reebless foliations [Th]. We will discuss

whether the inequality holds or not for open book foliations.

[Th] W. Thurston: Norm on the homology of 3-manifolds, Memoirs of the

AMS, 339 (1986), 99--130.

We will study codimension 1 foliations on 3-manifolds.

Thurston's inequality, which implies convexity of the foliation in

some sense, folds for Reebless foliations [Th]. We will discuss

whether the inequality holds or not for open book foliations.

[Th] W. Thurston: Norm on the homology of 3-manifolds, Memoirs of the

AMS, 339 (1986), 99--130.

### 2007年10月30日(火)

17:00-18:00 数理科学研究科棟(駒場) 056号室

Tea: 16:40 - 17:00 コモンルーム

$L_{\\infty}$ action on Lagrangian filtered $A_{\\infty}$ algebras.

Tea: 16:40 - 17:00 コモンルーム

**太田 啓史 氏**(名大多元数理)$L_{\\infty}$ action on Lagrangian filtered $A_{\\infty}$ algebras.

[ 講演概要 ]

I will discuss $L_{\\infty}$ actions on Lagrangian filtered

$A_{\\infty}$ algebras by cycles of the ambient symplectic

manifold. In the course of the construction,

I like to remark that the stable map compactification is not

sufficient in some case when we consider ones from genus zero

bordered Riemann surface. Also, if I have time, I like to discuss

some relation to (absolute) Gromov-Witten invariant and other

applications.

(This talk is based on my joint work with K.Fukaya, Y-G Oh and K. Ono.)

I will discuss $L_{\\infty}$ actions on Lagrangian filtered

$A_{\\infty}$ algebras by cycles of the ambient symplectic

manifold. In the course of the construction,

I like to remark that the stable map compactification is not

sufficient in some case when we consider ones from genus zero

bordered Riemann surface. Also, if I have time, I like to discuss

some relation to (absolute) Gromov-Witten invariant and other

applications.

(This talk is based on my joint work with K.Fukaya, Y-G Oh and K. Ono.)

### 2007年10月23日(火)

16:30-18:00 数理科学研究科棟(駒場) 002号室

Tea: 16:00 - 16:30 コモンルーム

Spaces of subspheres and their applications

Tea: 16:00 - 16:30 コモンルーム

**Jun O'Hara 氏**(首都大学東京)Spaces of subspheres and their applications

[ 講演概要 ]

The set of q-dimensional subspheres in S^n is a Grassmann manifold which has natural pseudo-Riemannian structure, and in some cases, symplectic structure as well. Both of them are conformally invariant.

I will explain some examples of their applications to geometric aspects of knots and links.

The set of q-dimensional subspheres in S^n is a Grassmann manifold which has natural pseudo-Riemannian structure, and in some cases, symplectic structure as well. Both of them are conformally invariant.

I will explain some examples of their applications to geometric aspects of knots and links.

### 2007年10月16日(火)

17:00-18:00 数理科学研究科棟(駒場) 056号室

Tea: 16:40 - 17:00 コモンルーム

Toric Sasaki-Einstein manifolds

Tea: 16:40 - 17:00 コモンルーム

**二木 昭人 氏**(東京工業大学大学院理工学研究科)Toric Sasaki-Einstein manifolds

[ 講演概要 ]

A compact toric Sasaki manifold admits a Sasaki-Einstein metric if and only if it is obtained by the Delzant construction from a toric diagram of a constant height. As an application we see that the canonical line bundle of a toric Fano manifold admits a complete Ricci-flat K\\"ahler metric.

A compact toric Sasaki manifold admits a Sasaki-Einstein metric if and only if it is obtained by the Delzant construction from a toric diagram of a constant height. As an application we see that the canonical line bundle of a toric Fano manifold admits a complete Ricci-flat K\\"ahler metric.

### 2007年10月09日(火)

16:30-18:00 数理科学研究科棟(駒場) 056号室

Tea: 16:00 - 16:30 コモンルーム

Classification of codimension-one locally free actions of the affine group of the real line.

Tea: 16:00 - 16:30 コモンルーム

**浅岡 正幸 氏**(京都大学大学院理学研究科)Classification of codimension-one locally free actions of the affine group of the real line.

[ 講演概要 ]

By GA, we denote the group of affine and orientation-preserving transformations

of the real line. In this talk, I will report on classification of locally free action of

GA on closed three manifolds, which I obtained recently. In 1979, E.Ghys proved

that if such an action preserves a volume, then it is smoothly conjugate to a homogeneous action. However, it was unknown whether non-homogeneous action exists. As a consequence of the classification, we will see that the unit tangent bundle of a closed surface of higher genus admits a finite-parameter family of

non-homogeneous actions.

By GA, we denote the group of affine and orientation-preserving transformations

of the real line. In this talk, I will report on classification of locally free action of

GA on closed three manifolds, which I obtained recently. In 1979, E.Ghys proved

that if such an action preserves a volume, then it is smoothly conjugate to a homogeneous action. However, it was unknown whether non-homogeneous action exists. As a consequence of the classification, we will see that the unit tangent bundle of a closed surface of higher genus admits a finite-parameter family of

non-homogeneous actions.

### 2007年07月17日(火)

16:30-18:00 数理科学研究科棟(駒場) 056号室

Tea: 16:00 - 16:30 コモンルーム

Orbifold Cohomology of Wreath Product Orbifolds and

Cohomological HyperKahler Resolution Conjecture

Tea: 16:00 - 16:30 コモンルーム

**松村 朝雄 氏**(東京大学大学院数理科学研究科)Orbifold Cohomology of Wreath Product Orbifolds and

Cohomological HyperKahler Resolution Conjecture

[ 講演概要 ]

Chen-Ruan orbifold cohomology ring was introduced in 2000 as

the degree zero genus zero orbifold Gromov-Witten invariants with

three marked points. We will review its construction in the case of

global quotient orbifolds, following Fantechi-Gottsche and

Jarvis-Kaufmann-Kimura. We will describe the orbifold cohomology of

wreath product orbifolds and explain its application to Ruan's

cohomological hyperKahler resolution conjecture.

Chen-Ruan orbifold cohomology ring was introduced in 2000 as

the degree zero genus zero orbifold Gromov-Witten invariants with

three marked points. We will review its construction in the case of

global quotient orbifolds, following Fantechi-Gottsche and

Jarvis-Kaufmann-Kimura. We will describe the orbifold cohomology of

wreath product orbifolds and explain its application to Ruan's

cohomological hyperKahler resolution conjecture.

### 2007年07月10日(火)

16:30-18:00 数理科学研究科棟(駒場) 056号室

Tea: 16:00 - 16:30 コモンルーム

Combable functions, quasimorphisms, and the central limit theorem

(joint with Koji Fujiwara)

Tea: 16:00 - 16:30 コモンルーム

**Danny C. Calegari 氏**(California Institute of Technology)Combable functions, quasimorphisms, and the central limit theorem

(joint with Koji Fujiwara)

[ 講演概要 ]

Quasimorphisms on groups are dual to stable commutator length,

and detect extremal phenomena in topology and dynamics. In typical groups

(even in a free group) stable commutator length is very difficult to

calculate, because the space of quasimorphisms is too large to study

directly without adding more structure.

In this talk, we show that a large class of quasimorphisms - the so-called

"counting quasimorphisms" on word-hyperbolic groups - can be effectively

described using simple machines called finite state automata. From this,

and from the ergodic theory of finite directed graphs, one can deduce a

number of properties about the statistical distribution of the values of a

counting quasimorphism on elements of the group.

Quasimorphisms on groups are dual to stable commutator length,

and detect extremal phenomena in topology and dynamics. In typical groups

(even in a free group) stable commutator length is very difficult to

calculate, because the space of quasimorphisms is too large to study

directly without adding more structure.

In this talk, we show that a large class of quasimorphisms - the so-called

"counting quasimorphisms" on word-hyperbolic groups - can be effectively

described using simple machines called finite state automata. From this,

and from the ergodic theory of finite directed graphs, one can deduce a

number of properties about the statistical distribution of the values of a

counting quasimorphism on elements of the group.

### 2007年07月03日(火)

16:30-18:00 数理科学研究科棟(駒場) 056号室

Tea: 16:00 - 16:30 コモンルーム

Two invariants of pseudo--Anosov mapping classes: hyperbolic volume vs dilatation

(joint work with Mitsuhiko Takasawa)

Tea: 16:00 - 16:30 コモンルーム

**金 英子 氏**(東京工業大学情報理工学研究科)Two invariants of pseudo--Anosov mapping classes: hyperbolic volume vs dilatation

(joint work with Mitsuhiko Takasawa)

[ 講演概要 ]

We concern two invariants of pseudo-Anosov mapping classes.

One is the dilatation of pseudo--Anosov maps and the other is the volume

of mapping tori. To study how two invariants are related, fixing a surface

we represent a mapping class by using the standard generator set and compute

these two for all pseudo--Anosov mapping classes with up to some word length.

In the talk, we observe two properties:

(1) The ratio of the topological entropy (i.e. logarithm of the dilatation) to

the volume is bounded from below by some positive constant which only

depends on the surface.

(2) The conjugacy class having the minimal dilatation reaches the minimal volume.

On the observation (1), in case of the mapping class group of a once--punctured

torus, we give a concrete lower bound of the ratio.

We concern two invariants of pseudo-Anosov mapping classes.

One is the dilatation of pseudo--Anosov maps and the other is the volume

of mapping tori. To study how two invariants are related, fixing a surface

we represent a mapping class by using the standard generator set and compute

these two for all pseudo--Anosov mapping classes with up to some word length.

In the talk, we observe two properties:

(1) The ratio of the topological entropy (i.e. logarithm of the dilatation) to

the volume is bounded from below by some positive constant which only

depends on the surface.

(2) The conjugacy class having the minimal dilatation reaches the minimal volume.

On the observation (1), in case of the mapping class group of a once--punctured

torus, we give a concrete lower bound of the ratio.

### 2007年06月12日(火)

16:30-18:00 数理科学研究科棟(駒場) 056号室

Tea: 16:00 - 16:30 コモンルーム

The Kodaira dimension of symplectic 4-manifolds

Tea: 16:00 - 16:30 コモンルーム

**Tian-Jun Li 氏**(University of Minnesota)The Kodaira dimension of symplectic 4-manifolds

[ 講演概要 ]

Various results and questions about symplectic4-manifolds can be

formulated in terms of the notion of the Kodaira dimension. In particular,

we will discuss the classification and the geography problems. It is interesting

to understand how it behaves undersome basic constructions.Time permitting

we will discuss the symplectic birational aspect of this notion and speculate

how to extend it to higher dimensional manifolds.

Various results and questions about symplectic4-manifolds can be

formulated in terms of the notion of the Kodaira dimension. In particular,

we will discuss the classification and the geography problems. It is interesting

to understand how it behaves undersome basic constructions.Time permitting

we will discuss the symplectic birational aspect of this notion and speculate

how to extend it to higher dimensional manifolds.

### 2007年06月05日(火)

17:00-18:30 数理科学研究科棟(駒場) 117号室

Tea: 16:40 - 17:00 コモンルーム

Symplectic mapping classes and fillings

Tea: 16:40 - 17:00 コモンルーム

**Emmanuel Giroux 氏**(ENS Lyon)Symplectic mapping classes and fillings

[ 講演概要 ]

We will describe a joint work in progress with Paul Biran in

which contact geometry is combined with properties of Lagrangian manifolds

in subcritical Stein domains to obtain nontrivaility results for symplectic

mapping classes.

We will describe a joint work in progress with Paul Biran in

which contact geometry is combined with properties of Lagrangian manifolds

in subcritical Stein domains to obtain nontrivaility results for symplectic

mapping classes.

### 2007年05月15日(火)

16:30-18:00 数理科学研究科棟(駒場) 056号室

Tea: 16:00 - 16:30 コモンルーム

Kontsevich's characteristic classes for higher dimensional homology sphere bundles

Tea: 16:00 - 16:30 コモンルーム

**渡邉 忠之 氏**(京都大学数理解析研究所)Kontsevich's characteristic classes for higher dimensional homology sphere bundles

[ 講演概要 ]

As an analogue of the perturbative Chern-Simons theory, Maxim Kontsevich

constructed universal characteristic classes of smooth fiber bundles with fiber

diffeomorphic to a singularly framed odd dimensional homology sphere.

In this talk, I will give a sketch proof of our result on non-triviality of the

Kontsevich classes for 7-dimensional homology sphere bundles.

As an analogue of the perturbative Chern-Simons theory, Maxim Kontsevich

constructed universal characteristic classes of smooth fiber bundles with fiber

diffeomorphic to a singularly framed odd dimensional homology sphere.

In this talk, I will give a sketch proof of our result on non-triviality of the

Kontsevich classes for 7-dimensional homology sphere bundles.

### 2007年05月08日(火)

16:30-18:00 数理科学研究科棟(駒場) 056号室

Tea: 16:00 - 16:30 コモンルーム

On the vanishing of the Rohlin invariant

Tea: 16:00 - 16:30 コモンルーム

**森山 哲裕 氏**(東京大学大学院数理科学研究科)On the vanishing of the Rohlin invariant

[ 講演概要 ]

The vanishing of the Rohlin invariant of an amphichiral integral

homology $3$-sphere $M$ (i.e. $M \\cong -M$) is a natural consequence

of some elementary properties of the Casson invariant. In this talk, we

give a new direct (and more elementary) proof of this vanishing

property. The main idea comes from the definition of the degree 1

part of the Kontsevich-Kuperberg-Thurston invariant, and we progress

by constructing some $7$-dimensional manifolds in which $M$ is embedded.

The vanishing of the Rohlin invariant of an amphichiral integral

homology $3$-sphere $M$ (i.e. $M \\cong -M$) is a natural consequence

of some elementary properties of the Casson invariant. In this talk, we

give a new direct (and more elementary) proof of this vanishing

property. The main idea comes from the definition of the degree 1

part of the Kontsevich-Kuperberg-Thurston invariant, and we progress

by constructing some $7$-dimensional manifolds in which $M$ is embedded.

### 2007年04月24日(火)

16:30-18:00 数理科学研究科棟(駒場) 056号室

Tea: 16:00 - 16:30 コモンルーム

Realization of twisted $K$-theory and

finite-dimensional approximation of Fredholm operators

Tea: 16:00 - 16:30 コモンルーム

**五味 清紀 氏**(東京大学大学院数理科学研究科)Realization of twisted $K$-theory and

finite-dimensional approximation of Fredholm operators

[ 講演概要 ]

A problem in twisted $K$-theory is to realize twisted $K$-groups generally by means of finite-dimensional geometric objects, like vector bundles. I would like to talk about an approach toward the problem by means of Mikio Furuta's generalized vector bundles. By using a twisted version of the generalized vector bundle and a finite-dimensional approximation of Fredholm operators, I construct a group into which there exists a natural injection from the twisted $K$-group twisted by any third integral cohomology class.

A problem in twisted $K$-theory is to realize twisted $K$-groups generally by means of finite-dimensional geometric objects, like vector bundles. I would like to talk about an approach toward the problem by means of Mikio Furuta's generalized vector bundles. By using a twisted version of the generalized vector bundle and a finite-dimensional approximation of Fredholm operators, I construct a group into which there exists a natural injection from the twisted $K$-group twisted by any third integral cohomology class.

### 2007年04月17日(火)

16:30-18:00 数理科学研究科棟(駒場) 056号室

Tea: 16:00 - 16:30 コモンルーム

Existence Problem of Compact Locally Symmetric Spaces

Tea: 16:00 - 16:30 コモンルーム

**小林 俊行 氏**(東京大学大学院数理科学研究科)Existence Problem of Compact Locally Symmetric Spaces

[ 講演概要 ]

The local to global study of geometries was a major trend of 20th century geometry, with remarkable developments achieved particularly in Riemannian geometry. In contrast, in areas such as Lorentz geometry, familiar to us as the space-time of relativity theory, and more generally in pseudo-Riemannian geometry, as well as in various other kinds of geometry (symplectic, complex geometry, ...), surprising little is known about global properties of the geometry even if we impose a locally homogeneous structure.

In this talk, I will give a survey on the recent developments on the question about how the local geometric structure affects the global nature of non-Riemannian manifolds with emphasis on the existence problem of compact models of locally symmetric spaces.

The local to global study of geometries was a major trend of 20th century geometry, with remarkable developments achieved particularly in Riemannian geometry. In contrast, in areas such as Lorentz geometry, familiar to us as the space-time of relativity theory, and more generally in pseudo-Riemannian geometry, as well as in various other kinds of geometry (symplectic, complex geometry, ...), surprising little is known about global properties of the geometry even if we impose a locally homogeneous structure.

In this talk, I will give a survey on the recent developments on the question about how the local geometric structure affects the global nature of non-Riemannian manifolds with emphasis on the existence problem of compact models of locally symmetric spaces.

### 2007年01月30日(火)

16:30-18:00 数理科学研究科棟(駒場) 056号室

Tea: 16:00 - 16:30 コモンルーム

Elliptic genera and some finite groups

Tea: 16:00 - 16:30 コモンルーム

**John F. Duncan 氏**(Harvard University)Elliptic genera and some finite groups

[ 講演概要 ]

Recent developments in the representation theory of sporadic groups

suggest new formulations of `moonshine' in which Jacobi forms take on the

role played by modular forms in the monstrous case. On the other hand,

Jacobi forms arise naturally in the study of elliptic genera. We review

the use of vertex algebra as a tool for constructing the elliptic genus of

a suitable vector bundle, and illustrate connections between this and

vertex algebraic representations of certain sporadic simple groups.

Recent developments in the representation theory of sporadic groups

suggest new formulations of `moonshine' in which Jacobi forms take on the

role played by modular forms in the monstrous case. On the other hand,

Jacobi forms arise naturally in the study of elliptic genera. We review

the use of vertex algebra as a tool for constructing the elliptic genus of

a suitable vector bundle, and illustrate connections between this and

vertex algebraic representations of certain sporadic simple groups.

### 2007年01月23日(火)

16:30-18:30 数理科学研究科棟(駒場) 056号室

Tea: 16:00 - 16:30 コモンルーム

The twistor correspondence for self-dual Zollfrei metrics

----their singularities and reduction

On the homology group of $Out(F_n)$

Tea: 16:00 - 16:30 コモンルーム

**中田 文憲 氏**(東京大学大学院数理科学研究科) 16:30-17:30The twistor correspondence for self-dual Zollfrei metrics

----their singularities and reduction

[ 講演概要 ]

C. LeBrun and L. J. Mason investigated a twistor-type correspondence

between indefinite conformal structures of signature (2,2) with some properties

and totally real embeddings from RP^3 to CP^3.

In this talk, two themes following LeBrun and Mason are explained.

First we consider an additional structure:

the conformal structure is equipped with a null surface foliation

which has some singularity.

We establish a global twistor correspondence for such structures,

and we show that a low dimensional correspondence

between some quotient spaces is induced from this twistor correspondence.

Next we formulate a certain singularity for the conformal structures.

We generalize the formulation of LeBrun and Mason's theorem

admitting the singularity, and we show explicit examples.

C. LeBrun and L. J. Mason investigated a twistor-type correspondence

between indefinite conformal structures of signature (2,2) with some properties

and totally real embeddings from RP^3 to CP^3.

In this talk, two themes following LeBrun and Mason are explained.

First we consider an additional structure:

the conformal structure is equipped with a null surface foliation

which has some singularity.

We establish a global twistor correspondence for such structures,

and we show that a low dimensional correspondence

between some quotient spaces is induced from this twistor correspondence.

Next we formulate a certain singularity for the conformal structures.

We generalize the formulation of LeBrun and Mason's theorem

admitting the singularity, and we show explicit examples.

**大橋 了 氏**(東京大学大学院数理科学研究科) 17:30-18:30On the homology group of $Out(F_n)$

[ 講演概要 ]

Suppose $F_n$ is the free group of rank $n$,

$Out(F_n) = Aut(F_n)/Inn(F_n)$ the outer automorphism group of $F_n$.

We compute $H_*(Out(F_n);\\mathbb{Q})$ for $n\\leq 6$ and conclude

that non-trivial classes in this range are generated

by Morita classes $\\mu_i \\in H_{4i}(Out(F_{2i+2});\\mathbb{Q})$.

Also we compute odd primary part of $H^*(Out(F_4);\\mathbb{Z})$.

Suppose $F_n$ is the free group of rank $n$,

$Out(F_n) = Aut(F_n)/Inn(F_n)$ the outer automorphism group of $F_n$.

We compute $H_*(Out(F_n);\\mathbb{Q})$ for $n\\leq 6$ and conclude

that non-trivial classes in this range are generated

by Morita classes $\\mu_i \\in H_{4i}(Out(F_{2i+2});\\mathbb{Q})$.

Also we compute odd primary part of $H^*(Out(F_4);\\mathbb{Z})$.

### 2007年01月16日(火)

16:30-18:30 数理科学研究科棟(駒場) 056号室

Tea: 16:00 - 16:30 コモンルーム

An $SO(3)$-version of $2$-torsion instanton invariants

On the non-acyclic Reidemeister torsion for knots

Tea: 16:00 - 16:30 コモンルーム

**笹平 裕史 氏**(東京大学大学院数理科学研究科) 16:30-17:30An $SO(3)$-version of $2$-torsion instanton invariants

[ 講演概要 ]

We construct invariants for simply connected, non-spin $4$-manifolds using torsion cohomology classes of moduli spaces of ASD connections on $SO(3)$-bundles. The invariants are $SO(3)$-version of Fintushel-Stern's $2$-torsion instanton invariants. We show that this $SO(3)$-torsion invariant of $2CP^2 \\# -CP^2$ is non-trivial, while it is known that any invariants of $2CP^2 \\# - CP^2$ coming from the Seiberg-Witten theory are trivial

since $2CP^2 \\# -CP^2$ has a positive scalar curvature metric.

We construct invariants for simply connected, non-spin $4$-manifolds using torsion cohomology classes of moduli spaces of ASD connections on $SO(3)$-bundles. The invariants are $SO(3)$-version of Fintushel-Stern's $2$-torsion instanton invariants. We show that this $SO(3)$-torsion invariant of $2CP^2 \\# -CP^2$ is non-trivial, while it is known that any invariants of $2CP^2 \\# - CP^2$ coming from the Seiberg-Witten theory are trivial

since $2CP^2 \\# -CP^2$ has a positive scalar curvature metric.

**山口 祥司 氏**(東京大学大学院数理科学研究科) 17:30-18:30On the non-acyclic Reidemeister torsion for knots

[ 講演概要 ]

The Reidemeister torsion is an invariant of a CW-complex and a representation of its fundamental group. We consider the Reidemeister torsion for a knot exterior in a homology three sphere and a representation given by the composition of an SL(2, C)- (or SU(2)-) representation of the knot group and the adjoint action to the Lie algebra.

We will see that this invariant is expressed by the differential coefficient of the twisted Alexander invariant of the knot and investigate some properties of the invariant by using this relation.

The Reidemeister torsion is an invariant of a CW-complex and a representation of its fundamental group. We consider the Reidemeister torsion for a knot exterior in a homology three sphere and a representation given by the composition of an SL(2, C)- (or SU(2)-) representation of the knot group and the adjoint action to the Lie algebra.

We will see that this invariant is expressed by the differential coefficient of the twisted Alexander invariant of the knot and investigate some properties of the invariant by using this relation.

### 2006年12月19日(火)

16:30-18:30 数理科学研究科棟(駒場) 056号室

Tea: 16:00 - 16:30 コモンルーム

Poisson structures on the homology of the spaces of knots

On projections of pseudo-ribbon sphere-links

Tea: 16:00 - 16:30 コモンルーム

**境 圭一 氏**(東京大学大学院数理科学研究科) 16:30-17:30Poisson structures on the homology of the spaces of knots

[ 講演概要 ]

We study the homological properties of the space $K$ of (framed) long knots in $\\R^n$, $n>3$, in particular its Poisson algebra structures.

We had known two kinds of Poisson structures, both of which are based on the action of little disks operad. One definition is via the action on the space $K$. Another comes from the action of chains of little disks on the Hochschild complex of an operad, which appears as $E^1$-term of certain spectral sequence converging to $H_* (K)$. The main result is that these two Poisson structures are the same.

We compute the first non-trivial example of the Poisson bracket. We show that this gives a first example of the homology class of $K$ which does not directly correspond to any chord diagrams.

We study the homological properties of the space $K$ of (framed) long knots in $\\R^n$, $n>3$, in particular its Poisson algebra structures.

We had known two kinds of Poisson structures, both of which are based on the action of little disks operad. One definition is via the action on the space $K$. Another comes from the action of chains of little disks on the Hochschild complex of an operad, which appears as $E^1$-term of certain spectral sequence converging to $H_* (K)$. The main result is that these two Poisson structures are the same.

We compute the first non-trivial example of the Poisson bracket. We show that this gives a first example of the homology class of $K$ which does not directly correspond to any chord diagrams.

**吉田 享平 氏**(東京大学大学院数理科学研究科) 17:30-18:30On projections of pseudo-ribbon sphere-links

[ 講演概要 ]

Suppose $F$ is an embedded closed surface in $R^4$.

We call $F$ a pseudo-ribbon surface link

if its projection is an immersion of $F$ into $R^3$

whose self-intersection set $\\Gamma(F)$ consists of disjointly embedded circles.

H. Aiso classified pseudo-ribbon sphere-knots ($F$ is a sphere.)

when $\\Gamma(F)$ consists of less than 6 circles.

We classify pseudo-ribbon sphere-links

when $F$ is two spheres and $\\Gamma(F)$ consists of less than 7 circles.

Suppose $F$ is an embedded closed surface in $R^4$.

We call $F$ a pseudo-ribbon surface link

if its projection is an immersion of $F$ into $R^3$

whose self-intersection set $\\Gamma(F)$ consists of disjointly embedded circles.

H. Aiso classified pseudo-ribbon sphere-knots ($F$ is a sphere.)

when $\\Gamma(F)$ consists of less than 6 circles.

We classify pseudo-ribbon sphere-links

when $F$ is two spheres and $\\Gamma(F)$ consists of less than 7 circles.

### 2006年12月12日(火)

16:30-18:00 数理科学研究科棟(駒場) 056号室

Tea: 16:00 - 16:30 コモンルーム

Thom polynomials for maps of curves with isolated singularities

(joint with S. Lando)

Tea: 16:00 - 16:30 コモンルーム

**Maxim Kazarian 氏**(Steklov Math. Institute)Thom polynomials for maps of curves with isolated singularities

(joint with S. Lando)

[ 講演概要 ]

Thom (residual) polynomials in characteristic classes are used in

the analysis of geometry of functional spaces. They serve as a

tool in description of classes Poincar\\'e dual to subvarieties of

functions of prescribed types. We give explicit universal

expressions for residual polynomials in spaces of functions on

complex curves having isolated singularities and

multisingularities, in terms of few characteristic classes. These

expressions lead to a partial explicit description of a

stratification of Hurwitz spaces.

Thom (residual) polynomials in characteristic classes are used in

the analysis of geometry of functional spaces. They serve as a

tool in description of classes Poincar\\'e dual to subvarieties of

functions of prescribed types. We give explicit universal

expressions for residual polynomials in spaces of functions on

complex curves having isolated singularities and

multisingularities, in terms of few characteristic classes. These

expressions lead to a partial explicit description of a

stratification of Hurwitz spaces.

### 2006年11月28日(火)

17:00-18:00 数理科学研究科棟(駒場) 056号室

Tea: 16:40 - 17:00 コモンルーム

The Yamabe constants of infinite coverings and a positive mass theorem

Tea: 16:40 - 17:00 コモンルーム

**芥川 和雄 氏**(東京理科大学理工学部)The Yamabe constants of infinite coverings and a positive mass theorem

[ 講演概要 ]

The {\\it Yamabe constant} $Y(M, C)$ of a given closed conformal manifold

$(M, C)$ is defined by the infimum of

the normalized total-scalar-curavarure functional $E$

among all metrics in $C$.

The study of the second variation of this functional $E$ led O.Kobayashi and Schoen

to independently introduce a natural differential-topological invariant $Y(M)$,

which is obtained by taking the supremum of $Y(M, C)$ over the space of all conformal classes.

This invariant $Y(M)$ is called the {\\it Yamabe invariant} of $M$.

For the study of the Yamabe invariant,

the relationship between $Y(M, C)$ and those of its conformal coverings

is important, the case when $Y(M, C)> 0$ particularly.

When $Y(M, C) \\leq 0$, by the uniqueness of unit-volume constant scalar curvature metrics in $C$,

the desired relation is clear.

When $Y(M, C) > 0$, such a uniqueness does not hold.

However, Aubin proved that $Y(M, C)$ is strictly less than

the Yamabe constant of any of its non-trivial {\\it finite} conformal coverings,

called {\\it Aubin's Lemma}.

In this talk, we generalize this lemma to the one for the Yamabe constant of

any $(M_{\\infty}, C_{\\infty})$ of its {\\it infinite} conformal coverings,

under a certain topological condition on the relation between $\\pi_1(M)$ and $\\pi_1(M_{\\infty})$.

For the proof of this, we aslo establish a version of positive mass theorem

for a specific class of asymptotically flat manifolds with singularities.

The {\\it Yamabe constant} $Y(M, C)$ of a given closed conformal manifold

$(M, C)$ is defined by the infimum of

the normalized total-scalar-curavarure functional $E$

among all metrics in $C$.

The study of the second variation of this functional $E$ led O.Kobayashi and Schoen

to independently introduce a natural differential-topological invariant $Y(M)$,

which is obtained by taking the supremum of $Y(M, C)$ over the space of all conformal classes.

This invariant $Y(M)$ is called the {\\it Yamabe invariant} of $M$.

For the study of the Yamabe invariant,

the relationship between $Y(M, C)$ and those of its conformal coverings

is important, the case when $Y(M, C)> 0$ particularly.

When $Y(M, C) \\leq 0$, by the uniqueness of unit-volume constant scalar curvature metrics in $C$,

the desired relation is clear.

When $Y(M, C) > 0$, such a uniqueness does not hold.

However, Aubin proved that $Y(M, C)$ is strictly less than

the Yamabe constant of any of its non-trivial {\\it finite} conformal coverings,

called {\\it Aubin's Lemma}.

In this talk, we generalize this lemma to the one for the Yamabe constant of

any $(M_{\\infty}, C_{\\infty})$ of its {\\it infinite} conformal coverings,

under a certain topological condition on the relation between $\\pi_1(M)$ and $\\pi_1(M_{\\infty})$.

For the proof of this, we aslo establish a version of positive mass theorem

for a specific class of asymptotically flat manifolds with singularities.

### 2006年11月14日(火)

16:30-18:00 数理科学研究科棟(駒場) 056号室

Tea: 16:00 - 16:30 コモンルーム

High-codimensional knots spun about manifolds

Tea: 16:00 - 16:30 コモンルーム

**高瀬将道 氏**(信州大学理学部)High-codimensional knots spun about manifolds

[ 講演概要 ]

The spinning describes several methods of constructing higher-dimensional knots from lower-dimensional knots.

The original spinning (Emil Artin, 1925) has been generalized in various ways. Using one of the most generalized forms of spinning, called "deform-spinning about a submanifold" (Dennis Roseman, 1989), we analyze in a geometric way Haefliger's smoothly knotted (4k-1)-spheres in the 6k-sphere.

The spinning describes several methods of constructing higher-dimensional knots from lower-dimensional knots.

The original spinning (Emil Artin, 1925) has been generalized in various ways. Using one of the most generalized forms of spinning, called "deform-spinning about a submanifold" (Dennis Roseman, 1989), we analyze in a geometric way Haefliger's smoothly knotted (4k-1)-spheres in the 6k-sphere.