## Tuesday Seminar on Topology

Seminar information archive ～12/05｜Next seminar｜Future seminars 12/06～

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

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

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

**Seminar information archive**

### 2009/03/05

16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Extending surface automorphisms over 4-space

**Shicheng Wang**(Peking University)Extending surface automorphisms over 4-space

[ Abstract ]

Let $e: M^p\\to R^{p+2}$ be a co-dimensional 2 smooth embedding

from a closed orientable manifold to the Euclidean space and $E_e$ be the subgroup of ${\\cal M}_M$, the mapping class group

of $M$, whose elements extend over $R^{p+2}$ as self-diffeomorphisms. Then there is a spin structure

on $M$ derived from the embedding which is preserved by each $\\tau \\in E_e$.

Some applications: (1) the index $[{\\cal M}_{F_g}:E_e]\\geq 2^{2g-1}+2^{g-1}$ for any embedding $e:F_g\\to R^4$, where $F_g$

is the surface of genus $g$. (2) $[{\\cal M}_{T^p}:E_e]\\geq 2^p-1$ for any unknotted embedding

$e:T^p\\to R^{p+2}$, where $T^p$ is the $p$-dimensional torus. Those two lower bounds are known to be sharp.

This is a joint work of Ding-Liu-Wang-Yao.

Let $e: M^p\\to R^{p+2}$ be a co-dimensional 2 smooth embedding

from a closed orientable manifold to the Euclidean space and $E_e$ be the subgroup of ${\\cal M}_M$, the mapping class group

of $M$, whose elements extend over $R^{p+2}$ as self-diffeomorphisms. Then there is a spin structure

on $M$ derived from the embedding which is preserved by each $\\tau \\in E_e$.

Some applications: (1) the index $[{\\cal M}_{F_g}:E_e]\\geq 2^{2g-1}+2^{g-1}$ for any embedding $e:F_g\\to R^4$, where $F_g$

is the surface of genus $g$. (2) $[{\\cal M}_{T^p}:E_e]\\geq 2^p-1$ for any unknotted embedding

$e:T^p\\to R^{p+2}$, where $T^p$ is the $p$-dimensional torus. Those two lower bounds are known to be sharp.

This is a joint work of Ding-Liu-Wang-Yao.

### 2009/01/27

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

Lagrangian Floer homology and quasi homomorphism

from the group of Hamiltonian diffeomorphism

**深谷 賢治**(京都大学大学院理学研究科)Lagrangian Floer homology and quasi homomorphism

from the group of Hamiltonian diffeomorphism

[ Abstract ]

Entov-Polterovich constructed quasi homomorphism

from the group of Hamiltonian diffeomorphisms using

spectral invariant due to Oh etc.

In this talk I will explain a way to study this

quasi homomorphism by using Lagrangian Floer homology.

I will also explain its generalization to use quantum

cohomology with bulk deformation.

When applied to the case of toric manifold, it

gives an example where (infinitely) many quasi homomorphism

exists.

(Joint work with Oh-Ohta-Ono).

Entov-Polterovich constructed quasi homomorphism

from the group of Hamiltonian diffeomorphisms using

spectral invariant due to Oh etc.

In this talk I will explain a way to study this

quasi homomorphism by using Lagrangian Floer homology.

I will also explain its generalization to use quantum

cohomology with bulk deformation.

When applied to the case of toric manifold, it

gives an example where (infinitely) many quasi homomorphism

exists.

(Joint work with Oh-Ohta-Ono).

### 2009/01/20

16:30-17:30 Room #002 (Graduate School of Math. Sci. Bldg.)

Five dimensional $K$-contact manifolds of rank 2

**野澤 啓**(東京大学大学院数理科学研究科)Five dimensional $K$-contact manifolds of rank 2

[ Abstract ]

A $K$-contact manifold is an odd dimensional manifold $M$ with a contact form $\\alpha$ whose Reeb flow preserves a Riemannian metric on $M$. For examples, the underlying manifold with the underlying contact form of a Sasakian manifold is $K$-contact. In this talk, we will state our results on classification up to surgeries, the existence of compatible Sasakian metrics and a sufficient condition to be toric for closed $5$-dimensional $K$-contact manifolds with a $T^2$ action given by the closure of the Reeb flow, which are obtained by the application of Morse theory to the contact moment map for the $T^2$ action.

A $K$-contact manifold is an odd dimensional manifold $M$ with a contact form $\\alpha$ whose Reeb flow preserves a Riemannian metric on $M$. For examples, the underlying manifold with the underlying contact form of a Sasakian manifold is $K$-contact. In this talk, we will state our results on classification up to surgeries, the existence of compatible Sasakian metrics and a sufficient condition to be toric for closed $5$-dimensional $K$-contact manifolds with a $T^2$ action given by the closure of the Reeb flow, which are obtained by the application of Morse theory to the contact moment map for the $T^2$ action.

### 2009/01/20

17:30-18:30 Room #002 (Graduate School of Math. Sci. Bldg.)

Surface links which are coverings of a trivial torus knot (JAPANESE)

**中村 伊南沙**(東京大学大学院数理科学研究科)Surface links which are coverings of a trivial torus knot (JAPANESE)

[ Abstract ]

We consider surface links which are in the form of coverings of a

trivial torus knot, which we will call torus-covering-links.

By definition, torus-covering-links include

spun $T^2$-knots, turned spun $T^2$-knots, and symmetry-spun tori.

We see some properties of torus-covering-links.

We consider surface links which are in the form of coverings of a

trivial torus knot, which we will call torus-covering-links.

By definition, torus-covering-links include

spun $T^2$-knots, turned spun $T^2$-knots, and symmetry-spun tori.

We see some properties of torus-covering-links.

### 2009/01/13

16:30-17:30 Room #002 (Graduate School of Math. Sci. Bldg.)

Compactification of the homeomorphism group of a graph

**山下 温**(東京大学大学院数理科学研究科)Compactification of the homeomorphism group of a graph

[ Abstract ]

Topological properties of homeomorphism groups, especially of finite-dimensional manifolds,

have been of interest in the area of infinite-dimensional manifold topology.

For a locally finite graph $\\Gamma$ with countably many components,

the homeomorphism group $\\mathcal{H}(\\Gamma)$

and its identity component $\\mathcal{H}_+(\\Gamma)$ are topological groups

with respect to the compact-open topology. I will define natural compactifications

$\\overline{\\mathcal{H}}(\\Gamma)$ and

$\\overline{\\mathcal{H}}_+(\\Gamma)$ of these groups and describe the

topological type of the pair $(\\overline{\\mathcal{H}}_+(\\Gamma), \\mathcal{H}_+(\\Gamma))$

using the data of $\\Gamma$. I will also discuss the topological structure of

$\\overline{\\mathcal{H}}(\\Gamma)$ where $\\Gamma$ is the circle.

Topological properties of homeomorphism groups, especially of finite-dimensional manifolds,

have been of interest in the area of infinite-dimensional manifold topology.

For a locally finite graph $\\Gamma$ with countably many components,

the homeomorphism group $\\mathcal{H}(\\Gamma)$

and its identity component $\\mathcal{H}_+(\\Gamma)$ are topological groups

with respect to the compact-open topology. I will define natural compactifications

$\\overline{\\mathcal{H}}(\\Gamma)$ and

$\\overline{\\mathcal{H}}_+(\\Gamma)$ of these groups and describe the

topological type of the pair $(\\overline{\\mathcal{H}}_+(\\Gamma), \\mathcal{H}_+(\\Gamma))$

using the data of $\\Gamma$. I will also discuss the topological structure of

$\\overline{\\mathcal{H}}(\\Gamma)$ where $\\Gamma$ is the circle.

### 2008/12/09

16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Tits alternative in $Diff^1(S^1)$

**Bertrand Deroin**(CNRS, Orsay, Universit\'e Paris-Sud 11)Tits alternative in $Diff^1(S^1)$

[ Abstract ]

The following form of Tits alternative for subgroups of

homeomorphisms of the circle has been proved by Margulis: or the group

preserve a probability measure on the circle, or it contains a free

subgroup on two generators. We will prove that if the group acts by diffeomorphisms of

class $C^1$ and does not preserve a probability measure on the circle, then

in fact it contains a subgroup topologically conjugated to a Schottky group.

This is a joint work with V. Kleptsyn and A. Navas.

The following form of Tits alternative for subgroups of

homeomorphisms of the circle has been proved by Margulis: or the group

preserve a probability measure on the circle, or it contains a free

subgroup on two generators. We will prove that if the group acts by diffeomorphisms of

class $C^1$ and does not preserve a probability measure on the circle, then

in fact it contains a subgroup topologically conjugated to a Schottky group.

This is a joint work with V. Kleptsyn and A. Navas.

### 2008/12/02

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

Vanishing and Rigidity

**金井 雅彦**(名古屋大学多元数理科学研究科)Vanishing and Rigidity

[ Abstract ]

The aim of my talk is to reveal an unforeseen link between

the classical vanishing theorems of Matsushima and Weil, on the one hand,

andrigidity of the Weyl chamber flow, a dynamical system arising from a higher-rank

noncompact Lie group, on the other. The connection is established via

"transverse extension theorems": Roughly speaking, they claim that a tangential 1- form of the

orbit foliation of the Weyl chamber flow that is tangentially closed

(and satisfies a certain mild additional condition) can be extended to a closed 1- form on the

whole space in a canonical manner. In particular, infinitesimal

rigidity of the orbit foliation of the Weyl chamber flow is proved as an application.

The aim of my talk is to reveal an unforeseen link between

the classical vanishing theorems of Matsushima and Weil, on the one hand,

andrigidity of the Weyl chamber flow, a dynamical system arising from a higher-rank

noncompact Lie group, on the other. The connection is established via

"transverse extension theorems": Roughly speaking, they claim that a tangential 1- form of the

orbit foliation of the Weyl chamber flow that is tangentially closed

(and satisfies a certain mild additional condition) can be extended to a closed 1- form on the

whole space in a canonical manner. In particular, infinitesimal

rigidity of the orbit foliation of the Weyl chamber flow is proved as an application.

### 2008/11/25

16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Circle-valued Morse theory for knots and links

**Andrei Pajitnov**(Univ. de Nantes)Circle-valued Morse theory for knots and links

[ Abstract ]

We will discuss several recent developments in

this theory. In the first part of the talk we prove that the Morse-Novikov number of a knot is less than or equal to twice the tunnel number of the knot, and present consequences of this result. In the second part we report on our joint project with Hiroshi Goda on the half-transversal Morse-Novikov theory for 3-manifolds.

We will discuss several recent developments in

this theory. In the first part of the talk we prove that the Morse-Novikov number of a knot is less than or equal to twice the tunnel number of the knot, and present consequences of this result. In the second part we report on our joint project with Hiroshi Goda on the half-transversal Morse-Novikov theory for 3-manifolds.

### 2008/11/18

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

複素力学系のくりこみと剛性

**宍倉 光広**(京都大学大学院理学研究科)複素力学系のくりこみと剛性

[ Abstract ]

無限に分岐が集積しているような

ある種の力学系においては、相空間やパラメータ空間の

微細構造が取り上げる個々の力学系の族に寄らない

普遍的構造をもったりすることが知られており、

ある意味で剛性の問題とつながっている。この現象の説明には、

ある部分集合への再帰写像の構成から得られる、あるクラスの

力学系全体の空間で定義されるメタ力学系としての

くりこみ作用素の考え方が重要である。これに関して、

講演者と稲生啓行による中立的不動点をもつ複素力学系の

近放物型くりこみの話を中心に解説する。

無限に分岐が集積しているような

ある種の力学系においては、相空間やパラメータ空間の

微細構造が取り上げる個々の力学系の族に寄らない

普遍的構造をもったりすることが知られており、

ある意味で剛性の問題とつながっている。この現象の説明には、

ある部分集合への再帰写像の構成から得られる、あるクラスの

力学系全体の空間で定義されるメタ力学系としての

くりこみ作用素の考え方が重要である。これに関して、

講演者と稲生啓行による中立的不動点をもつ複素力学系の

近放物型くりこみの話を中心に解説する。

### 2008/11/11

16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

The second rational homology group of the moduli space of curves

with level structures

**Thomas Andrew Putman**(MIT)The second rational homology group of the moduli space of curves

with level structures

[ Abstract ]

Let $\\Gamma$ be a finite-index subgroup of the mapping class

group of a closed genus $g$ surface that contains the Torelli group. For

instance, $\\Gamma$ can be the level $L$ subgroup or the spin mapping class

group. We show that $H_2(\\Gamma;\\Q) \\cong \\Q$ for $g \\geq 5$. A corollary

of this is that the rational Picard groups of the associated finite covers

of the moduli space of curves are equal to $\\Q$. We also prove analogous

results for surface with punctures and boundary components.

Let $\\Gamma$ be a finite-index subgroup of the mapping class

group of a closed genus $g$ surface that contains the Torelli group. For

instance, $\\Gamma$ can be the level $L$ subgroup or the spin mapping class

group. We show that $H_2(\\Gamma;\\Q) \\cong \\Q$ for $g \\geq 5$. A corollary

of this is that the rational Picard groups of the associated finite covers

of the moduli space of curves are equal to $\\Q$. We also prove analogous

results for surface with punctures and boundary components.

### 2008/11/04

16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Lefschetz SL(2)-action and cohomology of Kaehler manifolds

**Misha Verbitsky**(ITEP, Moscow)Lefschetz SL(2)-action and cohomology of Kaehler manifolds

[ Abstract ]

Let M be compact Kaehler manifold. It is well

known that any Kaehler form generates a Lefschetz SL(2)-triple

acting on cohomology of M. This action can be used to compute

cohomology of M. If M is a hyperkaehler manifold, of real

dimension 4n, then the subalgebra of its cohomology generated by

the second cohomology is isomorphic to a polynomial algebra,

up to the middle degree.

Let M be compact Kaehler manifold. It is well

known that any Kaehler form generates a Lefschetz SL(2)-triple

acting on cohomology of M. This action can be used to compute

cohomology of M. If M is a hyperkaehler manifold, of real

dimension 4n, then the subalgebra of its cohomology generated by

the second cohomology is isomorphic to a polynomial algebra,

up to the middle degree.

### 2008/10/28

16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Nonsmoothable group actions on spin 4-manifolds

**清野 和彦**(東京大学大学院数理科学研究科)Nonsmoothable group actions on spin 4-manifolds

[ Abstract ]

We call a locally linear group action on a topological manifold nonsmoothable

if the action is not smooth with respect to any possible smooth structure.

We show in this lecture that every closed, simply connected, spin topological 4-manifold

not homeomorphic to neither S^2\\times S^2 nor S^4 allows a nonsmoothable

group action of any cyclic group with sufficiently large prime order

which depends on the manifold.

We call a locally linear group action on a topological manifold nonsmoothable

if the action is not smooth with respect to any possible smooth structure.

We show in this lecture that every closed, simply connected, spin topological 4-manifold

not homeomorphic to neither S^2\\times S^2 nor S^4 allows a nonsmoothable

group action of any cyclic group with sufficiently large prime order

which depends on the manifold.

### 2008/10/21

16:30-18:00 Room #002 (Graduate School of Math. Sci. Bldg.)

On embeddings of 3-manifolds in 6-manifolds

**森山 哲裕**(東京大学大学院数理科学研究科)On embeddings of 3-manifolds in 6-manifolds

[ Abstract ]

In this talk, we give a simple axiomatic definition of an invariant of

smooth embeddings of 3-manifolds in 6-manifolds.

The axiom is expressed in terms of some cobordisms of pairs of manifolds of

dimensions 6 and 3 (equipped with some cohomology class of the complement) and

the signature of 4-manifolds.

We then show that our invariant gives a unified framework for some classical

invariants in low-dimensions (Haefliger invariant, Milnor's triple

linking number, Rokhlin invariant, Casson invariant,

Takase's invariant, Skopenkov's invariants).

In this talk, we give a simple axiomatic definition of an invariant of

smooth embeddings of 3-manifolds in 6-manifolds.

The axiom is expressed in terms of some cobordisms of pairs of manifolds of

dimensions 6 and 3 (equipped with some cohomology class of the complement) and

the signature of 4-manifolds.

We then show that our invariant gives a unified framework for some classical

invariants in low-dimensions (Haefliger invariant, Milnor's triple

linking number, Rokhlin invariant, Casson invariant,

Takase's invariant, Skopenkov's invariants).

### 2008/10/14

16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Pontrjagin-Thom maps and the Deligne-Mumford compactification

**Jeffrey Herschel Giansiracusa**(Oxford University)Pontrjagin-Thom maps and the Deligne-Mumford compactification

[ Abstract ]

An embedding f: M -> N produces, via a construction of Pontrjagin-Thom, a map from N to the Thom space of the normal bundle over M. If f is an arbitrary map then one instead gets a map from N to the infinite loop space of the Thom spectrum of the normal bundle of f. We extend this Pontrjagin-Thom construction of wrong-way maps to differentiable stacks and use it to produce interesting maps from the Deligne-Mumford compactification of the moduli space of curves to certain infinite loop spaces. We show that these maps are surjective on mod p homology in a range of degrees. We thus produce large new families of torsion cohomology classes on the Deligne-Mumford compactification.

An embedding f: M -> N produces, via a construction of Pontrjagin-Thom, a map from N to the Thom space of the normal bundle over M. If f is an arbitrary map then one instead gets a map from N to the infinite loop space of the Thom spectrum of the normal bundle of f. We extend this Pontrjagin-Thom construction of wrong-way maps to differentiable stacks and use it to produce interesting maps from the Deligne-Mumford compactification of the moduli space of curves to certain infinite loop spaces. We show that these maps are surjective on mod p homology in a range of degrees. We thus produce large new families of torsion cohomology classes on the Deligne-Mumford compactification.

### 2008/07/15

16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

One-step Markov Theorem on exchange classes

**松田 浩**(広島大学大学院理学研究科)One-step Markov Theorem on exchange classes

[ Abstract ]

The main theorem in this talk claims the following.

``Let L_A and L_B denote a closed a-braid and a closed b-braid,

respectively, that represent one link type.

After at most (a^2 b^2)/4 exchange moves on L_A,

we can 'see' the pair of closed braids."

In this talk, we explain the main theorem in details, and

we present some applications.

In particular, we propose a strategy to construct an algorithm

that determines whether two links are ambient isotopic.

The main theorem in this talk claims the following.

``Let L_A and L_B denote a closed a-braid and a closed b-braid,

respectively, that represent one link type.

After at most (a^2 b^2)/4 exchange moves on L_A,

we can 'see' the pair of closed braids."

In this talk, we explain the main theorem in details, and

we present some applications.

In particular, we propose a strategy to construct an algorithm

that determines whether two links are ambient isotopic.

### 2008/07/08

16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Contact homology of left-handed stabilizations and connected sums

**Otto van Koert**(北海道大学大学院理学研究科, JSPS)Contact homology of left-handed stabilizations and connected sums

[ Abstract ]

In this talk, we will give a brief introduction to contact homology, an invariant of contact manifolds defined by counting holomorphic curves in the symplectization of a contact manifold. We shall show that certain left-handed stabilizations of contact open books have vanishing contact homology.

This is done by finding restrictions on the behavior of holomorphic curves in connected sums. An additional application of this behavior of holomorphic curves is a long exact sequence for linearized contact homology.

In this talk, we will give a brief introduction to contact homology, an invariant of contact manifolds defined by counting holomorphic curves in the symplectization of a contact manifold. We shall show that certain left-handed stabilizations of contact open books have vanishing contact homology.

This is done by finding restrictions on the behavior of holomorphic curves in connected sums. An additional application of this behavior of holomorphic curves is a long exact sequence for linearized contact homology.

### 2008/07/01

16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

On the Johnson homomorphisms of the automorphism group of

a free metabelian group

**佐藤 隆夫**(大阪大学大学院理学研究科, JSPS)On the Johnson homomorphisms of the automorphism group of

a free metabelian group

[ Abstract ]

The main object of our research is the automorphism group of a

free group. To be brief, the Johnson homomorphisms are studied in order to

describe one by one approximations of the automorphism group of a free group

. They play important roles on the study of the homology groups of the autom

orphism group of a free group. In general, to determine their images are ver

y difficult problem. In this talk, we define the Johnson homomorphisms of th

e automorphism group of a free metabelian group, and determine their images.

Using these results, we can give a lower bound on the image of the Johnson

homomorphisms of the automorphism group of a free group.

The main object of our research is the automorphism group of a

free group. To be brief, the Johnson homomorphisms are studied in order to

describe one by one approximations of the automorphism group of a free group

. They play important roles on the study of the homology groups of the autom

orphism group of a free group. In general, to determine their images are ver

y difficult problem. In this talk, we define the Johnson homomorphisms of th

e automorphism group of a free metabelian group, and determine their images.

Using these results, we can give a lower bound on the image of the Johnson

homomorphisms of the automorphism group of a free group.

### 2008/06/24

16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

On computing distances in the pants complex

http://www.is.titech.ac.jp/~kjshack5/FYEO.pdf

**Kenneth Shackleton**(東京工業大学, JSPS)On computing distances in the pants complex

[ Abstract ]

The pants complex is an accurate combinatorial

model for the Weil-Petersson metric (WP) on Teichmueller space

(Brock). One hopes that many of the geometric properties

of WP are accurately replicated in the pants complex, and

this is the source of many open questions. We compare these

in general, and then focus on the 5-holed sphere and the

2-holed torus, the first non-trivial surfaces. We arrive at

an algorithm for computing distances in the (1-skeleton of the)

pants complex of either surface.

[ Reference URL ]The pants complex is an accurate combinatorial

model for the Weil-Petersson metric (WP) on Teichmueller space

(Brock). One hopes that many of the geometric properties

of WP are accurately replicated in the pants complex, and

this is the source of many open questions. We compare these

in general, and then focus on the 5-holed sphere and the

2-holed torus, the first non-trivial surfaces. We arrive at

an algorithm for computing distances in the (1-skeleton of the)

pants complex of either surface.

http://www.is.titech.ac.jp/~kjshack5/FYEO.pdf

### 2008/06/17

16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Multiplier ideal sheaves and Futaki invariant on toric Fano manifolds.

**佐野 友二**(東京大学IPMU)Multiplier ideal sheaves and Futaki invariant on toric Fano manifolds.

[ Abstract ]

I would like to discuss the subvarieties cut off by the multiplier

ideal sheaves (MIS) and Futaki invariant on toric Fano manifolds.

Futaki invariant is one of the necessary conditions for the existence

of Kahler-Einstein metrics on Fano manifolds,

on the other hand MIS is one of the sufficient conditions introduced by Nadel.

Especially I would like to focus on the MIS related to the Monge-Ampere equation

for Kahler-Einstein metrics on non-KE toric Fano manifolds.

The motivation of this work comes from the investigation of the

relationship with slope stability

of polarized manifolds introduced by Ross and Thomas.

This talk will be based on a part of the joint work with Akito Futaki

(arXiv:0711.0614).

I would like to discuss the subvarieties cut off by the multiplier

ideal sheaves (MIS) and Futaki invariant on toric Fano manifolds.

Futaki invariant is one of the necessary conditions for the existence

of Kahler-Einstein metrics on Fano manifolds,

on the other hand MIS is one of the sufficient conditions introduced by Nadel.

Especially I would like to focus on the MIS related to the Monge-Ampere equation

for Kahler-Einstein metrics on non-KE toric Fano manifolds.

The motivation of this work comes from the investigation of the

relationship with slope stability

of polarized manifolds introduced by Ross and Thomas.

This talk will be based on a part of the joint work with Akito Futaki

(arXiv:0711.0614).

### 2008/06/03

16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

On the geometry of certain slices of the character variety of a knot group

**山口 祥司**(東京大学大学院数理科学研究科)On the geometry of certain slices of the character variety of a knot group

[ Abstract ]

joint work with Fumikazu Nagasato (Meijo University)

This talk is concerned with certain subsets in the character variety of a knot group.

These subsets are called '"slices", which are defined as a level set of a regular function associated to a meridian of a knot.

They are related to character varieties for branched covers along the knot.

Some investigations indicate that an equivariant theory for a knot is connected to a theory for branched covers via slices, for example, the equivariant signature of a knot and the equivariant Casson invariant.

In this talk, we will construct a map from slices into the character varieties for branched covers and investigate the properties.

In particular, we focus on slices called "trace-free", which are used to define the Casson-Lin invariant, and the relation to the character variety for two--fold branched cover.

joint work with Fumikazu Nagasato (Meijo University)

This talk is concerned with certain subsets in the character variety of a knot group.

These subsets are called '"slices", which are defined as a level set of a regular function associated to a meridian of a knot.

They are related to character varieties for branched covers along the knot.

Some investigations indicate that an equivariant theory for a knot is connected to a theory for branched covers via slices, for example, the equivariant signature of a knot and the equivariant Casson invariant.

In this talk, we will construct a map from slices into the character varieties for branched covers and investigate the properties.

In particular, we focus on slices called "trace-free", which are used to define the Casson-Lin invariant, and the relation to the character variety for two--fold branched cover.

### 2008/05/20

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

Turaev-Viro TQFT splitting.

**Jer\^ome Petit**(東京工業大学, JSPS)Turaev-Viro TQFT splitting.

[ Abstract ]

The Turaev-Viro invariant is a 3-manifolds invariant. It is obtained in this way :

1) we use a combinatorial description of 3-manifolds, in this case it is : triangulation / Pachner moves

2) we define a scalar thanks to a categorical data (spherical category) and a topological data (triangulation)

3)we verify that the scalar is invariant under Pachner moves, then we obtain a 3-manifolds invariant.

The Turaev-Viro invariant can also be extended to a TQFT. Roughly speaking a TQFT is a data which assigns a finite dimensional vector space to every closed surface and a linear application to every 3-manifold with boundary.

In this talk, we will give a decomposition of the Turaev-Viro TQFT. More precisely, we decompose it into blocks. These blocks are given by a group associated to the spherical category which was used to construct the Turaev-Viro invariant. We will show that these blocks define a HQFT (roughly speaking a TQFT with an "homotopical data"). This HQFT is obtained from an homotopical invariant, which is the homotopical version of the Turaev-Viro invariant. Moreover this invariant can be used to obtain the homological Turaev-Viro invariant defined by Yetter.

The Turaev-Viro invariant is a 3-manifolds invariant. It is obtained in this way :

1) we use a combinatorial description of 3-manifolds, in this case it is : triangulation / Pachner moves

2) we define a scalar thanks to a categorical data (spherical category) and a topological data (triangulation)

3)we verify that the scalar is invariant under Pachner moves, then we obtain a 3-manifolds invariant.

The Turaev-Viro invariant can also be extended to a TQFT. Roughly speaking a TQFT is a data which assigns a finite dimensional vector space to every closed surface and a linear application to every 3-manifold with boundary.

In this talk, we will give a decomposition of the Turaev-Viro TQFT. More precisely, we decompose it into blocks. These blocks are given by a group associated to the spherical category which was used to construct the Turaev-Viro invariant. We will show that these blocks define a HQFT (roughly speaking a TQFT with an "homotopical data"). This HQFT is obtained from an homotopical invariant, which is the homotopical version of the Turaev-Viro invariant. Moreover this invariant can be used to obtain the homological Turaev-Viro invariant defined by Yetter.

### 2008/05/13

16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

The problem of maximum Thurston--Bennequin number for knots

**Tamas Kalman**(東京大学大学院数理科学研究科, JSPS)The problem of maximum Thurston--Bennequin number for knots

[ Abstract ]

Legendrian submanifolds of contact 3-manifolds are

one-dimensional, just like knots. This ``coincidence'' gives rise to an

interesting and expanding intersection of contact and symplectic geometry

on the one hand and classical knot theory on the other. As an

illustration, we will survey recent results on maximizing the

Thurston--Bennequin number (which is a measure of the twisting of the

contact structure along a Legendrian) within a smooth knot type. In

particular, we will show how Kauffman's state circles can be used to solve

the maximization problem for so-called +adequate (among them, alternating

and positive) knots and links.

Legendrian submanifolds of contact 3-manifolds are

one-dimensional, just like knots. This ``coincidence'' gives rise to an

interesting and expanding intersection of contact and symplectic geometry

on the one hand and classical knot theory on the other. As an

illustration, we will survey recent results on maximizing the

Thurston--Bennequin number (which is a measure of the twisting of the

contact structure along a Legendrian) within a smooth knot type. In

particular, we will show how Kauffman's state circles can be used to solve

the maximization problem for so-called +adequate (among them, alternating

and positive) knots and links.

### 2008/04/22

16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Special fibers of pencils of hypersurfaces

**Sergey Yuzvinsky**(University of Oregon)Special fibers of pencils of hypersurfaces

[ Abstract ]

We consider pencils of hypersurfaces of degree $d>1$ in the complex

$n$-dimensional projective space subject to the condition that the

generic fiber is irreducible. We study the set of completely reducible

fibers, i.e., the unions of hyperplanes. The first surprinsing result is

that the cardinality of thie set has very strict uniformed upper bound

(not depending on $d$ or $n$). The other one gives a characterization

of this set in terms of either topology of its complement or combinatorics

of hyperplanes. We also include into consideration more general special

fibers are iimportant for characteristic varieties of the hyperplane

complements.

We consider pencils of hypersurfaces of degree $d>1$ in the complex

$n$-dimensional projective space subject to the condition that the

generic fiber is irreducible. We study the set of completely reducible

fibers, i.e., the unions of hyperplanes. The first surprinsing result is

that the cardinality of thie set has very strict uniformed upper bound

(not depending on $d$ or $n$). The other one gives a characterization

of this set in terms of either topology of its complement or combinatorics

of hyperplanes. We also include into consideration more general special

fibers are iimportant for characteristic varieties of the hyperplane

complements.

### 2008/04/15

16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

On the invariants of knots and 3-manifolds related to the restricted quantum group

**村上 順**(早稲田大学理工)On the invariants of knots and 3-manifolds related to the restricted quantum group

[ Abstract ]

I would like to talk about the colored Alexander invariant and the logarithmic

invariant of knots and links. They are constructed from the universal R-matrices

of the semi-resetricted and restricted quantum groups of sl(2) respectively,

and they are related to the hyperbolic volumes of the cone manifolds along

the knot. I also would like to explain an attempt to generalize these invariants to

a three manifold invariant which relates to the volume of the manifold actually.

I would like to talk about the colored Alexander invariant and the logarithmic

invariant of knots and links. They are constructed from the universal R-matrices

of the semi-resetricted and restricted quantum groups of sl(2) respectively,

and they are related to the hyperbolic volumes of the cone manifolds along

the knot. I also would like to explain an attempt to generalize these invariants to

a three manifold invariant which relates to the volume of the manifold actually.

### 2008/01/29

16:30-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

The rotation number function on groups of circle diffeomorphisms

A Diagrammatic Construction of Third Homology Classes of Knot Quandles

**松田 能文**(東京大学大学院数理科学研究科) 16:30-17:30The rotation number function on groups of circle diffeomorphisms

[ Abstract ]

ポアンカレは、円周の向きを保つ同相写像に対して、回転数の有理性と有限軌道の存在が

同値であることを示した。この講演では、この事実が円周の向きを保つ同相写像のなすあ

る種の群に対して一般化できることを説明する。特に、円周の向きを保つ実解析的微分同

相のなす非離散的な群に対して、回転数関数による像の有限性と有限軌道の存在が同値で

あることを示す。

ポアンカレは、円周の向きを保つ同相写像に対して、回転数の有理性と有限軌道の存在が

同値であることを示した。この講演では、この事実が円周の向きを保つ同相写像のなすあ

る種の群に対して一般化できることを説明する。特に、円周の向きを保つ実解析的微分同

相のなす非離散的な群に対して、回転数関数による像の有限性と有限軌道の存在が同値で

あることを示す。

**木村 康人**(東京大学大学院数理科学研究科) 17:30-18:30A Diagrammatic Construction of Third Homology Classes of Knot Quandles

[ Abstract ]

There exists a family of third (quandle / rack) homology classes,

called the shadow (fundamental / diagram) classes,

of the knot quandle, which are obtained from

the shadow colourings of knot diagrams.

We will show the construction of these homology classes,

and also show their relation to the shadow quandle cocycle

invariants of knots and that to other third homology classes.

There exists a family of third (quandle / rack) homology classes,

called the shadow (fundamental / diagram) classes,

of the knot quandle, which are obtained from

the shadow colourings of knot diagrams.

We will show the construction of these homology classes,

and also show their relation to the shadow quandle cocycle

invariants of knots and that to other third homology classes.