## 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**

### 2011/01/25

16:30-17:30 Room #056 (Graduate School of Math. Sci. Bldg.)

On unknotting of surface-knots with small sheet numbers

(JAPANESE)

**Chikara Haruta**(Graduate School of Mathematical Sciences, the University of Tokyo )On unknotting of surface-knots with small sheet numbers

(JAPANESE)

[ Abstract ]

A connected surface smoothly embedded in ${\\mathbb R}^4$ is called a surface-knot. In particular, if a surface-knot $F$ is homeomorphic to the $2$-sphere or the torus, then it is called an $S^2$-knot or a $T^2$-knot, respectively. The sheet number of a surface-knot is an invariant analogous to the crossing number of a $1$-knot. M. Saito and S. Satoh proved some results concerning the sheet number of an $S^2$-knot. In particular, it is known that an $S^2$-knot is trivial if and only if its sheet number is $1$, and there is no $S^2$-knot whose sheet number is $2$. In this talk, we show that there is no $S^2$-knot whose sheet number is $3$, and a $T^2$-knot is trivial if and only if its sheet number is $1$.

A connected surface smoothly embedded in ${\\mathbb R}^4$ is called a surface-knot. In particular, if a surface-knot $F$ is homeomorphic to the $2$-sphere or the torus, then it is called an $S^2$-knot or a $T^2$-knot, respectively. The sheet number of a surface-knot is an invariant analogous to the crossing number of a $1$-knot. M. Saito and S. Satoh proved some results concerning the sheet number of an $S^2$-knot. In particular, it is known that an $S^2$-knot is trivial if and only if its sheet number is $1$, and there is no $S^2$-knot whose sheet number is $2$. In this talk, we show that there is no $S^2$-knot whose sheet number is $3$, and a $T^2$-knot is trivial if and only if its sheet number is $1$.

### 2011/01/11

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

The Chas-Sullivan conjecture for a surface of infinite genus (JAPANESE)

**Nariya Kawazumi**(The University of Tokyo)The Chas-Sullivan conjecture for a surface of infinite genus (JAPANESE)

[ Abstract ]

Let \\Sigma_{\\infty,1} be the inductive limit of compact

oriented surfaces with one boundary component. We prove the

center of the Goldman Lie algebra of the surface \\Sigma_{\\infty,1}

is spanned by the constant loop.

A similar statement for a closed oriented surface was conjectured

by Chas and Sullivan, and proved by Etingof. Our result is deduced

from a computation of the center of the Lie algebra of oriented chord

diagrams.

If time permits, the Lie bracket on the space of linear chord diagrams

will be discussed. This talk is based on a joint work with Yusuke Kuno

(Hiroshima U./JSPS).

Let \\Sigma_{\\infty,1} be the inductive limit of compact

oriented surfaces with one boundary component. We prove the

center of the Goldman Lie algebra of the surface \\Sigma_{\\infty,1}

is spanned by the constant loop.

A similar statement for a closed oriented surface was conjectured

by Chas and Sullivan, and proved by Etingof. Our result is deduced

from a computation of the center of the Lie algebra of oriented chord

diagrams.

If time permits, the Lie bracket on the space of linear chord diagrams

will be discussed. This talk is based on a joint work with Yusuke Kuno

(Hiroshima U./JSPS).

### 2010/12/14

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

On the coarse geometry of Teichmueller space (ENGLISH)

**Kenneth Schackleton**(IPMU)On the coarse geometry of Teichmueller space (ENGLISH)

[ Abstract ]

We discuss the synthetic geometry of the pants graph in

comparison with the Weil-Petersson metric, whose geometry the

pants graph coarsely models following work of Brock’s. We also

restrict our attention to the 5-holed sphere, studying the Gromov

bordification of the pants graph and the dynamics of pseudo-Anosov

mapping classes.

We discuss the synthetic geometry of the pants graph in

comparison with the Weil-Petersson metric, whose geometry the

pants graph coarsely models following work of Brock’s. We also

restrict our attention to the 5-holed sphere, studying the Gromov

bordification of the pants graph and the dynamics of pseudo-Anosov

mapping classes.

### 2010/12/07

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

Diffeomorphism-invariant geometries and noncommutative geometry (ENGLISH)

**Raphael Ponge**(The University of Tokyo)Diffeomorphism-invariant geometries and noncommutative geometry (ENGLISH)

[ Abstract ]

In many geometric situations we may encounter the action of

a group $G$ on a manifold $M$, e.g., in the context of foliations. If

the action is free and proper, then the quotient $M/G$ is a smooth

manifold. However, in general the quotient $M/G$ need not even be

Hausdorff. Furthermore, it is well-known that a manifold has structure

invariant under the full group of diffeomorphisms except the

differentiable structure itself. Under these conditions how can one do

diffeomorphism-invariant geometry?

Noncommutative geometry provides us with the solution of trading the

ill-behaved space $M/G$ for a non-commutative algebra which

essentially plays the role of the algebra of smooth functions on that

space. The local index formula of Atiyah-Singer ultimately holds in

the setting of noncommutative geometry. Using this framework Connes

and Moscovici then obtained in the 90s a striking reformulation of the

local index formula in diffeomorphism-invariant geometry.

An important part the talk will be devoted to reviewing noncommutative

geometry and Connes-Moscovici's index formula. We will then hint to on-

going projects about reformulating the local index formula in two new

geometric settings: biholomorphism-invariant geometry of strictly

pseudo-convex domains and contactomorphism-invariant geometry.

In many geometric situations we may encounter the action of

a group $G$ on a manifold $M$, e.g., in the context of foliations. If

the action is free and proper, then the quotient $M/G$ is a smooth

manifold. However, in general the quotient $M/G$ need not even be

Hausdorff. Furthermore, it is well-known that a manifold has structure

invariant under the full group of diffeomorphisms except the

differentiable structure itself. Under these conditions how can one do

diffeomorphism-invariant geometry?

Noncommutative geometry provides us with the solution of trading the

ill-behaved space $M/G$ for a non-commutative algebra which

essentially plays the role of the algebra of smooth functions on that

space. The local index formula of Atiyah-Singer ultimately holds in

the setting of noncommutative geometry. Using this framework Connes

and Moscovici then obtained in the 90s a striking reformulation of the

local index formula in diffeomorphism-invariant geometry.

An important part the talk will be devoted to reviewing noncommutative

geometry and Connes-Moscovici's index formula. We will then hint to on-

going projects about reformulating the local index formula in two new

geometric settings: biholomorphism-invariant geometry of strictly

pseudo-convex domains and contactomorphism-invariant geometry.

### 2010/11/30

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

Pin^-(2)-monopole equations and intersection forms with local coefficients of 4-manifolds (JAPANESE)

**Nobuhiro Nakamura**(The University of Tokyo)Pin^-(2)-monopole equations and intersection forms with local coefficients of 4-manifolds (JAPANESE)

[ Abstract ]

We introduce a variant of the Seiberg-Witten equations, Pin^-(2)-monopole equations, and explain its applications to intersection forms with local coefficients of 4-manifolds.

The first application is an analogue of Froyshov's results on 4-manifolds which have definite forms with local coefficients.

The second one is a local coefficient version of Furuta's 10/8-inequality.

As a corollary, we construct nonsmoothable spin 4-manifolds satisfying Rohlin's theorem and the 10/8-inequality.

We introduce a variant of the Seiberg-Witten equations, Pin^-(2)-monopole equations, and explain its applications to intersection forms with local coefficients of 4-manifolds.

The first application is an analogue of Froyshov's results on 4-manifolds which have definite forms with local coefficients.

The second one is a local coefficient version of Furuta's 10/8-inequality.

As a corollary, we construct nonsmoothable spin 4-manifolds satisfying Rohlin's theorem and the 10/8-inequality.

### 2010/11/16

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

On a colored Khovanov bicomplex (JAPANESE)

**Noboru Ito**(Waseda University)On a colored Khovanov bicomplex (JAPANESE)

[ Abstract ]

We discuss the existence of a bicomplex which is a Khovanov-type

complex associated with categorification of a colored Jones polynomial.

This is an answer to the question proposed by A. Beliakova and S. Wehrli.

Then the second term of the spectral sequence of the bicomplex corresponds

to the Khovanov-type homology group. In this talk, we explain how to define

the bicomplex. If time permits, we also define a colored Rasmussen invariant

by using another spectral sequence of the colored Jones polynomial.

We discuss the existence of a bicomplex which is a Khovanov-type

complex associated with categorification of a colored Jones polynomial.

This is an answer to the question proposed by A. Beliakova and S. Wehrli.

Then the second term of the spectral sequence of the bicomplex corresponds

to the Khovanov-type homology group. In this talk, we explain how to define

the bicomplex. If time permits, we also define a colored Rasmussen invariant

by using another spectral sequence of the colored Jones polynomial.

### 2010/11/09

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

Characterising bumping points on deformation spaces of Kleinian groups (JAPANESE)

**Ken'ichi Ohshika**(Osaka University)Characterising bumping points on deformation spaces of Kleinian groups (JAPANESE)

[ Abstract ]

It is known that components of the interior of a deformation space of a Kleinian group can bump, and a component of the interior can bump itself, on the boundary of the deformation space.

Anderson-Canary-McCullogh gave a necessary and sufficient condition for two components to bump.

In this talk, I shall give a criterion for points on the boundary to be bumping points.

It is known that components of the interior of a deformation space of a Kleinian group can bump, and a component of the interior can bump itself, on the boundary of the deformation space.

Anderson-Canary-McCullogh gave a necessary and sufficient condition for two components to bump.

In this talk, I shall give a criterion for points on the boundary to be bumping points.

### 2010/11/02

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

Periodic-end manifolds and SW theory (ENGLISH)

**Daniel Ruberman**(Brandeis University)Periodic-end manifolds and SW theory (ENGLISH)

[ Abstract ]

We study an extension of Seiberg-Witten invariants to

4-manifolds with the homology of S^1 \\times S^3. This extension has

many potential applications in low-dimensional topology, including the

study of the homology cobordism group. Because b_2^+ =0, the usual

strategy for defining invariants does not work--one cannot disregard

reducible solutions. In fact, the count of solutions can jump in a

family of metrics or perturbations. To remedy this, we define an

index-theoretic counter-term that jumps by the same amount. The

counterterm is the index of the Dirac operator on a manifold with a

periodic end modeled at infinity by the infinite cyclic cover of the

manifold. This is joint work with Tomasz Mrowka and Nikolai Saveliev.

We study an extension of Seiberg-Witten invariants to

4-manifolds with the homology of S^1 \\times S^3. This extension has

many potential applications in low-dimensional topology, including the

study of the homology cobordism group. Because b_2^+ =0, the usual

strategy for defining invariants does not work--one cannot disregard

reducible solutions. In fact, the count of solutions can jump in a

family of metrics or perturbations. To remedy this, we define an

index-theoretic counter-term that jumps by the same amount. The

counterterm is the index of the Dirac operator on a manifold with a

periodic end modeled at infinity by the infinite cyclic cover of the

manifold. This is joint work with Tomasz Mrowka and Nikolai Saveliev.

### 2010/10/26

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

Quantum fundamental groups and quantum representation varieties for 3-manifolds (JAPANESE)

**Kazuo Habiro**(RIMS, Kyoto University)Quantum fundamental groups and quantum representation varieties for 3-manifolds (JAPANESE)

[ Abstract ]

We define a refinement of the fundamental groups of 3-manifolds and

a generalization of representation variety of the fundamental group

of 3-manifolds. We consider the category $H$ whose morphisms are

nonnegative integers, where $n$ corresponds to a genus $n$ handlebody

equipped with an embedding of a disc into the boundary, and whose

morphisms are the isotopy classes of embeddings of handlebodies

compatible with the embeddings of the disc into the boundaries. For

each 3-manifold $M$ with an embedding of a disc into the boundary, we

can construct a contravariant functor from $H$ to the category of

sets, where the object $n$ of $H$ is mapped to the set of isotopy

classes of embedding of the genus $n$ handlebody into $M$, compatible

with the embeddings of the disc into the boundaries. This functor can

be regarded as a refinement of the fundamental group of $M$, and we

call it the quantum fundamental group of $M$. Using this invariant, we

can construct for each co-ribbon Hopf algebra $A$ an invariant of

3-manifolds which may be regarded as (the space of regular functions

on) the representation variety of $M$ with respect to $A$.

We define a refinement of the fundamental groups of 3-manifolds and

a generalization of representation variety of the fundamental group

of 3-manifolds. We consider the category $H$ whose morphisms are

nonnegative integers, where $n$ corresponds to a genus $n$ handlebody

equipped with an embedding of a disc into the boundary, and whose

morphisms are the isotopy classes of embeddings of handlebodies

compatible with the embeddings of the disc into the boundaries. For

each 3-manifold $M$ with an embedding of a disc into the boundary, we

can construct a contravariant functor from $H$ to the category of

sets, where the object $n$ of $H$ is mapped to the set of isotopy

classes of embedding of the genus $n$ handlebody into $M$, compatible

with the embeddings of the disc into the boundaries. This functor can

be regarded as a refinement of the fundamental group of $M$, and we

call it the quantum fundamental group of $M$. Using this invariant, we

can construct for each co-ribbon Hopf algebra $A$ an invariant of

3-manifolds which may be regarded as (the space of regular functions

on) the representation variety of $M$ with respect to $A$.

### 2010/10/19

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

Optimistic limits of colored Jones invariants (ENGLISH)

**Jinseok Cho**(Waseda University)Optimistic limits of colored Jones invariants (ENGLISH)

[ Abstract ]

Yokota made a wonderful theory on the optimistic limit of Kashaev

invariant of a hyperbolic knot

that the limit determines the hyperbolic volume and the Chern-Simons

invariant of the knot.

Especially, his theory enables us to calculate the volume of a knot

combinatorially from its diagram for many cases.

We will briefly discuss Yokota theory, and then move to the optimistic

limit of colored Jones invariant.

We will explain a parallel version of Yokota theory based on the

optimistic limit of colored Jones invariant.

Especially, we will show the optimistic limit of colored Jones

invariant coincides with that of Kashaev invariant modulo 2\\pi^2.

This implies the optimistic limit of colored Jones invariant also

determines the volume and Chern-Simons invariant of the knot, and

probably more information.

This is a joint-work with Jun Murakami of Waseda University.

Yokota made a wonderful theory on the optimistic limit of Kashaev

invariant of a hyperbolic knot

that the limit determines the hyperbolic volume and the Chern-Simons

invariant of the knot.

Especially, his theory enables us to calculate the volume of a knot

combinatorially from its diagram for many cases.

We will briefly discuss Yokota theory, and then move to the optimistic

limit of colored Jones invariant.

We will explain a parallel version of Yokota theory based on the

optimistic limit of colored Jones invariant.

Especially, we will show the optimistic limit of colored Jones

invariant coincides with that of Kashaev invariant modulo 2\\pi^2.

This implies the optimistic limit of colored Jones invariant also

determines the volume and Chern-Simons invariant of the knot, and

probably more information.

This is a joint-work with Jun Murakami of Waseda University.

### 2010/10/12

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

Asymptotics of Morse numbers of finite coverings of manifolds (ENGLISH)

**Andrei Pajitnov**(Univ. de Nantes, The University of Tokyo)Asymptotics of Morse numbers of finite coverings of manifolds (ENGLISH)

[ Abstract ]

Let X be a closed manifold;

denote by m(X) the Morse number of X

(that is, the minimal number of critical

points of a Morse function on X).

Let Y be a finite covering of X of degree d.

In this survey talk we will address the following question

posed by M. Gromov: What are the asymptotic properties

of m(N) as d goes to infinity?

It turns out that for high-dimensional manifolds with

free abelian fundamental group the asymptotics of

the number m(N)/d is directly related to the Novikov homology

of N. We prove this theorem and discuss related results.

Let X be a closed manifold;

denote by m(X) the Morse number of X

(that is, the minimal number of critical

points of a Morse function on X).

Let Y be a finite covering of X of degree d.

In this survey talk we will address the following question

posed by M. Gromov: What are the asymptotic properties

of m(N) as d goes to infinity?

It turns out that for high-dimensional manifolds with

free abelian fundamental group the asymptotics of

the number m(N)/d is directly related to the Novikov homology

of N. We prove this theorem and discuss related results.

### 2010/07/27

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

Quandle homology and complex volume

(Joint work with Yuichi Kabaya) (JAPANESE)

**Ayumu Inoue**(Tokyo Institute of Technology)Quandle homology and complex volume

(Joint work with Yuichi Kabaya) (JAPANESE)

[ Abstract ]

For a hyperbolic 3-manifold M, the complex value (Vol(M) + i CS(M)) is called the complex volume of M. Here, Vol(M) denotes the volume of M, and CS(M) the Chern-Simons invariant of M.

In 2004, Neumann defined the extended Bloch group, and showed that there is an element of the extended Bloch group corresponding to a hyperbolic 3-manifold.

He further showed that we can compute the complex volume of the manifold by evaluating the element of the extended Bloch group.

To obtain an element of the extended Bloch group corresponding to a hyperbolic 3-manifold, we have to find an ideal triangulation of the manifold and to solve several equations.

On the other hand, we show that the element of the extended Bloch group corresponding to the exterior of a hyperbolic link is obtained from a quandle shadow coloring.

It means that we can compute the complex volume combinatorially from a link diagram.

For a hyperbolic 3-manifold M, the complex value (Vol(M) + i CS(M)) is called the complex volume of M. Here, Vol(M) denotes the volume of M, and CS(M) the Chern-Simons invariant of M.

In 2004, Neumann defined the extended Bloch group, and showed that there is an element of the extended Bloch group corresponding to a hyperbolic 3-manifold.

He further showed that we can compute the complex volume of the manifold by evaluating the element of the extended Bloch group.

To obtain an element of the extended Bloch group corresponding to a hyperbolic 3-manifold, we have to find an ideal triangulation of the manifold and to solve several equations.

On the other hand, we show that the element of the extended Bloch group corresponding to the exterior of a hyperbolic link is obtained from a quandle shadow coloring.

It means that we can compute the complex volume combinatorially from a link diagram.

### 2010/07/20

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

A polynomial invariant of pseudo-Anosov maps (JAPANESE)

**Keiko Kawamuro**(University of Iowa)A polynomial invariant of pseudo-Anosov maps (JAPANESE)

[ Abstract ]

Thurston-Nielsen classified surface homomorphism into three classes. Among them, the pseudo-Anosov class is the most interesting since there is strong connection to the hyperbolic manifolds. As an invariant, the dilatation number has been known. In this talk, I will introduce a new invariant of pseudo-Anosov maps. It is an integer coefficient polynomial, which contains the dilatation as the largest real root and is often reducible. I will show properties of the polynomials, examples, and some application to knot theory. (This is a joint work with Joan Birman and Peter Brinkmann.)

Thurston-Nielsen classified surface homomorphism into three classes. Among them, the pseudo-Anosov class is the most interesting since there is strong connection to the hyperbolic manifolds. As an invariant, the dilatation number has been known. In this talk, I will introduce a new invariant of pseudo-Anosov maps. It is an integer coefficient polynomial, which contains the dilatation as the largest real root and is often reducible. I will show properties of the polynomials, examples, and some application to knot theory. (This is a joint work with Joan Birman and Peter Brinkmann.)

### 2010/07/13

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

High Distance Knots in closed 3-manifolds (ENGLISH)

**Marion Moore**(University of California, Davis)High Distance Knots in closed 3-manifolds (ENGLISH)

[ Abstract ]

Let M be a closed 3-manifold with a given Heegaard splitting.

We show that after a single stabilization, some core of the

stabilized splitting has arbitrarily high distance with respect

to the splitting surface. This generalizes a result of Minsky,

Moriah, and Schleimer for knots in S^3. We also show that in the

complex of curves, handlebody sets are either coarsely distinct

or identical. We define the coarse mapping class group of a

Heeegaard splitting, and show that if (S,V,W) is a Heegaard

splitting of genus greater than or equal to 2, then the coarse

mapping class group of (S,V,W) is isomorphic to the mapping class

group of (S, V, W). This is joint work with Matt Rathbun.

Let M be a closed 3-manifold with a given Heegaard splitting.

We show that after a single stabilization, some core of the

stabilized splitting has arbitrarily high distance with respect

to the splitting surface. This generalizes a result of Minsky,

Moriah, and Schleimer for knots in S^3. We also show that in the

complex of curves, handlebody sets are either coarsely distinct

or identical. We define the coarse mapping class group of a

Heeegaard splitting, and show that if (S,V,W) is a Heegaard

splitting of genus greater than or equal to 2, then the coarse

mapping class group of (S,V,W) is isomorphic to the mapping class

group of (S, V, W). This is joint work with Matt Rathbun.

### 2010/07/06

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

On the cohomology of free and twisted loop spaces (JAPANESE)

**Akira Kono**(Kyoto University)On the cohomology of free and twisted loop spaces (JAPANESE)

[ Abstract ]

A natural extension of cohomology suspension to a free loop space is

constructed from the evaluation map and is shown to have a good

properties in cohomology calculation. This map is generalized to a

twisted loop space.

As an application, the cohomology of free and twisted loop space of

classifying spaces of compact Lie groups, including certain finite

Chevalley groups is calculated.

A natural extension of cohomology suspension to a free loop space is

constructed from the evaluation map and is shown to have a good

properties in cohomology calculation. This map is generalized to a

twisted loop space.

As an application, the cohomology of free and twisted loop space of

classifying spaces of compact Lie groups, including certain finite

Chevalley groups is calculated.

### 2010/06/29

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

Non-commutative Reidemeister torsion and Morse-Novikov theory (JAPANESE)

**Takahiro Kitayama**(The University of Tokyo)Non-commutative Reidemeister torsion and Morse-Novikov theory (JAPANESE)

[ Abstract ]

For a circle-valued Morse function of a closed oriented manifold, we

show that Reidemeister torsion over a non-commutative formal Laurent

polynomial ring equals the product of a certain non-commutative

Lefschetz-type zeta function and the algebraic torsion of the Novikov

complex over the ring. This gives a generalization of the results of

Hutchings-Lee and Pazhitnov on abelian coefficients. As a consequence we

obtain Morse theoretical and dynamical descriptions of the higher-order

Alexander polynomials.

For a circle-valued Morse function of a closed oriented manifold, we

show that Reidemeister torsion over a non-commutative formal Laurent

polynomial ring equals the product of a certain non-commutative

Lefschetz-type zeta function and the algebraic torsion of the Novikov

complex over the ring. This gives a generalization of the results of

Hutchings-Lee and Pazhitnov on abelian coefficients. As a consequence we

obtain Morse theoretical and dynamical descriptions of the higher-order

Alexander polynomials.

### 2010/06/15

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

On exceptional surgeries on Montesinos knots

(joint works with In Dae Jong and Shigeru Mizushima) (JAPANESE)

**Kazuhiro Ichihara**(Nihon University)On exceptional surgeries on Montesinos knots

(joint works with In Dae Jong and Shigeru Mizushima) (JAPANESE)

[ Abstract ]

I will report recent progresses of the study on exceptional

surgeries on Montesinos knots.

In particular, we will focus on how homological invariants (e.g.

khovanov homology,

knot Floer homology) on knots can be used in the study of Dehn surgery.

I will report recent progresses of the study on exceptional

surgeries on Montesinos knots.

In particular, we will focus on how homological invariants (e.g.

khovanov homology,

knot Floer homology) on knots can be used in the study of Dehn surgery.

### 2010/06/01

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

On Fatou-Julia decompositions (JAPANESE)

**Taro Asuke**(The University of Tokyo)On Fatou-Julia decompositions (JAPANESE)

[ Abstract ]

We will explain that Fatou-Julia decompositions can be

introduced in a unified manner to several kinds of one-dimensional

complex dynamical systems, which include the action of Kleinian groups,

iteration of holomorphic mappings and complex codimension-one foliations.

In this talk we will restrict ourselves mostly to the cases where the

dynamical systems have a certain compactness, however, we will mention

how to deal with dynamical systems without compactness.

We will explain that Fatou-Julia decompositions can be

introduced in a unified manner to several kinds of one-dimensional

complex dynamical systems, which include the action of Kleinian groups,

iteration of holomorphic mappings and complex codimension-one foliations.

In this talk we will restrict ourselves mostly to the cases where the

dynamical systems have a certain compactness, however, we will mention

how to deal with dynamical systems without compactness.

### 2010/05/18

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

On roots of Dehn twists (JAPANESE)

**Naoyuki Monden**(Osaka University)On roots of Dehn twists (JAPANESE)

[ Abstract ]

Let $t_{c}$ be the Dehn twist about a nonseparating simple closed curve

$c$ in a closed orientable surface. If a mapping class $f$ satisfies

$t_{c}=f^{n}$ in mapping class group, we call $f$ a root of $t_{c}$ of

degree $n$. In 2009, Margalit and Schleimer constructed roots of $t_{c}$.

In this talk, I will explain the data set which determine a root of

$t_{c}$ up to conjugacy. Moreover, I will explain the minimal and the

maximal degree.

Let $t_{c}$ be the Dehn twist about a nonseparating simple closed curve

$c$ in a closed orientable surface. If a mapping class $f$ satisfies

$t_{c}=f^{n}$ in mapping class group, we call $f$ a root of $t_{c}$ of

degree $n$. In 2009, Margalit and Schleimer constructed roots of $t_{c}$.

In this talk, I will explain the data set which determine a root of

$t_{c}$ up to conjugacy. Moreover, I will explain the minimal and the

maximal degree.

### 2010/05/11

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

The logarithms of Dehn twists (JAPANESE)

**Nariya Kawazumi**(The University of Tokyo)The logarithms of Dehn twists (JAPANESE)

[ Abstract ]

We establish an explicit formula for the action of (non-separating and

separating) Dehn twists on the complete group ring of the fundamental group of a

surface. It generalizes the classical transvection formula on the first homology.

The proof is involved with a homological interpretation of the Goldman

Lie algebra. This talk is based on a jointwork with Yusuke Kuno (Hiroshima U./JSPS).

We establish an explicit formula for the action of (non-separating and

separating) Dehn twists on the complete group ring of the fundamental group of a

surface. It generalizes the classical transvection formula on the first homology.

The proof is involved with a homological interpretation of the Goldman

Lie algebra. This talk is based on a jointwork with Yusuke Kuno (Hiroshima U./JSPS).

### 2010/04/27

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

On the complex volume of hyperbolic knots (JAPANESE)

**横田 佳之**(首都大学東京)On the complex volume of hyperbolic knots (JAPANESE)

[ Abstract ]

In this talk, we give a formula of the volume and the Chern-Simons invariant of hyperbolic knot complements, which is closely related to the volume conjecture of hyperbolic knots.

We also discuss the volumes and the Chern-Simons invariants of closed 3-manifolds

obtained by Dehn surgeries on hyperbolic knots.

In this talk, we give a formula of the volume and the Chern-Simons invariant of hyperbolic knot complements, which is closely related to the volume conjecture of hyperbolic knots.

We also discuss the volumes and the Chern-Simons invariants of closed 3-manifolds

obtained by Dehn surgeries on hyperbolic knots.

### 2010/04/20

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

Homotopy of foliations in dimension 3. (ENGLISH)

**Helene Eynard-Bontemps**(東京大学大学院数理科学研究科, JSPS)Homotopy of foliations in dimension 3. (ENGLISH)

[ Abstract ]

We are interested in the connectedness of the space of

codimension one foliations on a closed 3-manifold. In 1969, J. Wood proved

the fundamental result:

Theorem: Every 2-plane field on a closed 3-manifold is homotopic to a

foliation.

W. R. gave a new proof of (and generalized) this result in 1973 using

local constructions. It is then natural to wonder if two foliations with

homotopic tangent plane fields can be linked by a continuous path of

foliations.

A. Larcanch\\'e gave a positive answer in the particular case of

"sufficiently close" taut foliations. We use the key construction of her

proof (among other tools) to show that this is actually always true,

provided one is not too picky about the regularity of the foliations of

the path:

Theorem: Two C^\\infty foliations with homotopic tangent plane fields can

be linked by a path of C^1 foliations.

We are interested in the connectedness of the space of

codimension one foliations on a closed 3-manifold. In 1969, J. Wood proved

the fundamental result:

Theorem: Every 2-plane field on a closed 3-manifold is homotopic to a

foliation.

W. R. gave a new proof of (and generalized) this result in 1973 using

local constructions. It is then natural to wonder if two foliations with

homotopic tangent plane fields can be linked by a continuous path of

foliations.

A. Larcanch\\'e gave a positive answer in the particular case of

"sufficiently close" taut foliations. We use the key construction of her

proof (among other tools) to show that this is actually always true,

provided one is not too picky about the regularity of the foliations of

the path:

Theorem: Two C^\\infty foliations with homotopic tangent plane fields can

be linked by a path of C^1 foliations.

### 2010/04/13

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

Torsors in non-commutative geometry (ENGLISH)

**Christian Kassel**(CNRS, Univ. de Strasbourg)Torsors in non-commutative geometry (ENGLISH)

[ Abstract ]

G-torsors or principal homogeneous spaces are familiar objects in geometry. I'll present an extension of such objects to "non-commutative geometry". When the structural group G is finite, non-commutative G-torsors are governed by a group that has both an arithmetic component and a geometric one. The arithmetic part is given by a classical Galois cohomology group; the geometric input is encoded in a (not necessarily abelian) group that takes into account all normal abelian subgroups of G of central type. Various examples will be exhibited.

G-torsors or principal homogeneous spaces are familiar objects in geometry. I'll present an extension of such objects to "non-commutative geometry". When the structural group G is finite, non-commutative G-torsors are governed by a group that has both an arithmetic component and a geometric one. The arithmetic part is given by a classical Galois cohomology group; the geometric input is encoded in a (not necessarily abelian) group that takes into account all normal abelian subgroups of G of central type. Various examples will be exhibited.

### 2010/02/16

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

Characteristic numbers of algebraic varieties

**Dieter Kotschick**(Univ. M\"unchen)Characteristic numbers of algebraic varieties

[ Abstract ]

The Chern numbers of n-dimensional smooth projective varieties span a vector space whose dimension is the number of partitions of n. This vector space has many natural subspaces, some of which are quite small, for example the subspace spanned by Hirzebruch--Todd numbers, the subspace of topologically invariant combinations of Chern numbers, the subspace determined by the Hodge numbers, and the subspace of Chern numbers that can be bounded in terms of Betti numbers. I shall explain the relation between these subspaces, and characterize them in several ways. This leads in particular to the solution of a long-standing open problem originally formulated by Hirzebruch in the 1950s.

The Chern numbers of n-dimensional smooth projective varieties span a vector space whose dimension is the number of partitions of n. This vector space has many natural subspaces, some of which are quite small, for example the subspace spanned by Hirzebruch--Todd numbers, the subspace of topologically invariant combinations of Chern numbers, the subspace determined by the Hodge numbers, and the subspace of Chern numbers that can be bounded in terms of Betti numbers. I shall explain the relation between these subspaces, and characterize them in several ways. This leads in particular to the solution of a long-standing open problem originally formulated by Hirzebruch in the 1950s.

### 2010/02/02

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

Deformation of compact quotients of homogeneous spaces

**Fanny Kassel**(Univ. Paris-Sud, Orsay)Deformation of compact quotients of homogeneous spaces

[ Abstract ]

Let G/H be a reductive homogeneous space. In all known examples, if

G/H admits compact Clifford-Klein forms, then it admits "standard"

ones, by uniform lattices of some reductive subgroup L of G acting

properly on G/H. In order to obtain more generic Clifford-Klein forms,

we prove that for L of real rank 1, if one slightly deforms in G a

uniform lattice of L, then its action on G/H remains properly

discontinuous. As an application, we obtain compact quotients of

SO(2,2n)/U(1,n) by Zariski-dense discrete subgroups of SO(2,2n) acting

properly discontinuously.

https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar2010.html#20100202kassel

Let G/H be a reductive homogeneous space. In all known examples, if

G/H admits compact Clifford-Klein forms, then it admits "standard"

ones, by uniform lattices of some reductive subgroup L of G acting

properly on G/H. In order to obtain more generic Clifford-Klein forms,

we prove that for L of real rank 1, if one slightly deforms in G a

uniform lattice of L, then its action on G/H remains properly

discontinuous. As an application, we obtain compact quotients of

SO(2,2n)/U(1,n) by Zariski-dense discrete subgroups of SO(2,2n) acting

properly discontinuously.

https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar2010.html#20100202kassel