## トポロジー火曜セミナー

過去の記録 ～09/24｜次回の予定｜今後の予定 09/25～

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

担当者 | 河野 俊丈, 河澄 響矢, 北山 貴裕, 逆井卓也 |

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

備考 | Tea: 16:30 - 17:00 コモンルーム |

**過去の記録**

### 2010年10月26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

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

Tea: 16:40 - 17:00 コモンルーム

**葉廣 和夫 氏**(京都大学数理解析研究所)Quantum fundamental groups and quantum representation varieties for 3-manifolds (JAPANESE)

[ 講演概要 ]

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年10月19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

Optimistic limits of colored Jones invariants (ENGLISH)

Tea: 16:00 - 16:30 コモンルーム

**Jinseok Cho 氏**(早稲田大学)Optimistic limits of colored Jones invariants (ENGLISH)

[ 講演概要 ]

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年10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

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

Tea: 16:00 - 16:30 コモンルーム

**Andrei Pajitnov 氏**(Univ. de Nantes, 東京大学大学院数理科学研究科)Asymptotics of Morse numbers of finite coverings of manifolds (ENGLISH)

[ 講演概要 ]

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年07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

Quandle homology and complex volume

(Joint work with Yuichi Kabaya) (JAPANESE)

Tea: 16:00 - 16:30 コモンルーム

**井上 歩 氏**(東京工業大学)Quandle homology and complex volume

(Joint work with Yuichi Kabaya) (JAPANESE)

[ 講演概要 ]

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年07月20日(火)

17:00-18:00 数理科学研究科棟(駒場) 056号室

Tea: 16:30 - 17:00 コモンルーム

A polynomial invariant of pseudo-Anosov maps (JAPANESE)

Tea: 16:30 - 17:00 コモンルーム

**川室 圭子 氏**(University of Iowa)A polynomial invariant of pseudo-Anosov maps (JAPANESE)

[ 講演概要 ]

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年07月13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 Distance Knots in closed 3-manifolds (ENGLISH)

Tea: 16:00 - 16:30 コモンルーム

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

[ 講演概要 ]

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年07月06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

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

Tea: 16:40 - 17:00 コモンルーム

**河野 明 氏**(京都大学大学院理学研究科)On the cohomology of free and twisted loop spaces (JAPANESE)

[ 講演概要 ]

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年06月29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

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

Tea: 16:00 - 16:30 コモンルーム

**北山 貴裕 氏**(東京大学大学院数理科学研究科)Non-commutative Reidemeister torsion and Morse-Novikov theory (JAPANESE)

[ 講演概要 ]

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年06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

On exceptional surgeries on Montesinos knots

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

Tea: 16:00 - 16:30 コモンルーム

**市原 一裕 氏**(日本大学文理学部)On exceptional surgeries on Montesinos knots

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

[ 講演概要 ]

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年06月01日(火)

17:00-18:30 数理科学研究科棟(駒場) 056号室

Tea: 16:30 - 17:00 コモンルーム

On Fatou-Julia decompositions (JAPANESE)

Tea: 16:30 - 17:00 コモンルーム

**足助 太郎 氏**(東京大学大学院数理科学研究科)On Fatou-Julia decompositions (JAPANESE)

[ 講演概要 ]

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年05月18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 roots of Dehn twists (JAPANESE)

Tea: 16:00 - 16:30 コモンルーム

**門田 直之 氏**(大阪大学大学院理学研究科)On roots of Dehn twists (JAPANESE)

[ 講演概要 ]

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年05月11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 logarithms of Dehn twists (JAPANESE)

Tea: 16:00 - 16:30 コモンルーム

**河澄 響矢 氏**(東京大学大学院数理科学研究科)The logarithms of Dehn twists (JAPANESE)

[ 講演概要 ]

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年04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

On the complex volume of hyperbolic knots (JAPANESE)

Tea: 16:00 - 16:30 コモンルーム

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

[ 講演概要 ]

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年04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

Homotopy of foliations in dimension 3. (ENGLISH)

Tea: 16:00 - 16:30 コモンルーム

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

[ 講演概要 ]

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年04月13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

Torsors in non-commutative geometry (ENGLISH)

Tea: 16:00 - 16:30 コモンルーム

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

[ 講演概要 ]

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年02月16日(火)

17:30-18:30 数理科学研究科棟(駒場) 056号室

Tea: 17:00 - 17:30 コモンルーム

Characteristic numbers of algebraic varieties

Tea: 17:00 - 17:30 コモンルーム

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

[ 講演概要 ]

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年02月02日(火)

16:30-18:00 数理科学研究科棟(駒場) 056号室

Lie群論・表現論セミナーと合同, Tea: 16:00 - 16:30 コモンルーム

Deformation of compact quotients of homogeneous spaces

Lie群論・表現論セミナーと合同, Tea: 16:00 - 16:30 コモンルーム

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

[ 講演概要 ]

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

### 2010年01月26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

On the (co)chain type levels of spaces

Tea: 16:40 - 17:00 コモンルーム

**栗林 勝彦 氏**(信州大学)On the (co)chain type levels of spaces

[ 講演概要 ]

Avramov, Buchweitz, Iyengar and Miller have introduced

the notion of the level for an object of a triangulated category.

The invariant measures the number of steps to build the given object

out of some fixed object with triangles.

Using this notion in the derived category of modules over a (co)chain

algebra,

we define a new topological invariant, which is called

the (co)chain type level of a space.

In this talk, after explaining fundamental properties of the invariant,

I describe the chain type level of the Borel construction

of a homogeneous space as a computational example.

I will also relate the chain type level of a space to algebraic

approximations of the L.-S. category due to Kahl and to

the original L.-S. category of a map.

Avramov, Buchweitz, Iyengar and Miller have introduced

the notion of the level for an object of a triangulated category.

The invariant measures the number of steps to build the given object

out of some fixed object with triangles.

Using this notion in the derived category of modules over a (co)chain

algebra,

we define a new topological invariant, which is called

the (co)chain type level of a space.

In this talk, after explaining fundamental properties of the invariant,

I describe the chain type level of the Borel construction

of a homogeneous space as a computational example.

I will also relate the chain type level of a space to algebraic

approximations of the L.-S. category due to Kahl and to

the original L.-S. category of a map.

### 2010年01月19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

Localization via group action and its application to

the period condition of algebraic minimal surfaces

Tea: 16:40 - 17:00 コモンルーム

**小林 亮一 氏**(名古屋大学)Localization via group action and its application to

the period condition of algebraic minimal surfaces

[ 講演概要 ]

The optimal estimate for the number of exceptional

values of the Gauss map of algebraic minimal surfaces is a long

standing problem. In this lecture, I will introduce new ideas

toward the solution of this problem. The ``collective Cohn-Vossen

inequality" is the key idea. From this we have effective

Nevanlinna's lemma on logarithmic derivative for a certain class

of meromorphic functions on the disk. On the other hand, we can

construct a family holomorphic functions on the disk from the

Weierstrass data of the algebraic minimal surface under

consideration, which encodes the period condition.

Applying effective Lemma on logarithmic derivative to these

functions, we can extract an intriguing inequality.

The optimal estimate for the number of exceptional

values of the Gauss map of algebraic minimal surfaces is a long

standing problem. In this lecture, I will introduce new ideas

toward the solution of this problem. The ``collective Cohn-Vossen

inequality" is the key idea. From this we have effective

Nevanlinna's lemma on logarithmic derivative for a certain class

of meromorphic functions on the disk. On the other hand, we can

construct a family holomorphic functions on the disk from the

Weierstrass data of the algebraic minimal surface under

consideration, which encodes the period condition.

Applying effective Lemma on logarithmic derivative to these

functions, we can extract an intriguing inequality.

### 2010年01月12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

Index problem for generically-wild homoclinic classes in dimension three

On a generalized suspension theorem for directed Fukaya categories

Tea: 16:00 - 16:30 コモンルーム

**篠原 克寿 氏**(東京大学大学院数理科学研究科) 16:30-17:30Index problem for generically-wild homoclinic classes in dimension three

[ 講演概要 ]

In the sphere of non-hyperbolic differentiable dynamical systems, one can construct an example of a homolinic class which does not admit any kind of dominated splittings (a weak form of hyperbolicity) in a robust way. In this talk, we discuss the index (dimension of the unstable manifold) of the periodic points inside such homoclinic classes from a $C^1$-generic viewpoint.

In the sphere of non-hyperbolic differentiable dynamical systems, one can construct an example of a homolinic class which does not admit any kind of dominated splittings (a weak form of hyperbolicity) in a robust way. In this talk, we discuss the index (dimension of the unstable manifold) of the periodic points inside such homoclinic classes from a $C^1$-generic viewpoint.

**二木 昌宏 氏**(東京大学大学院数理科学研究科) 17:30-18:30On a generalized suspension theorem for directed Fukaya categories

[ 講演概要 ]

The directed Fukaya category $\\mathrm{Fuk} W$ of exact Lefschetz

fibration $W : X \\to \\mathbb{C}$ proposed by Kontsevich is a

categorification of the Milnor lattice of $W$. This is defined as the

directed $A_\\infty$-category $\\mathrm{Fuk} W = \\mathrm{Fuk}^\\to

\\mathbb{V}$ generated by a distinguished basis $\\mathbb{V}$ of

vanishing cycles.

Recently Seidel has proved that this is stable under the suspension $W

+ u^2$ as a consequence of his foundational work on the directed

Fukaya category. We generalize his suspension theorem to the $W + u^d$

case by considering partial tensor product $\\mathrm{Fuk} W \\otimes'

\\mathcal{A}_{d-1}$, where $\\mathcal{A}_{d-1}$ is the category

corresponding to the $A_n$-type quiver. This also generalizes a recent

work by the author with Kazushi Ueda.

The directed Fukaya category $\\mathrm{Fuk} W$ of exact Lefschetz

fibration $W : X \\to \\mathbb{C}$ proposed by Kontsevich is a

categorification of the Milnor lattice of $W$. This is defined as the

directed $A_\\infty$-category $\\mathrm{Fuk} W = \\mathrm{Fuk}^\\to

\\mathbb{V}$ generated by a distinguished basis $\\mathbb{V}$ of

vanishing cycles.

Recently Seidel has proved that this is stable under the suspension $W

+ u^2$ as a consequence of his foundational work on the directed

Fukaya category. We generalize his suspension theorem to the $W + u^d$

case by considering partial tensor product $\\mathrm{Fuk} W \\otimes'

\\mathcal{A}_{d-1}$, where $\\mathcal{A}_{d-1}$ is the category

corresponding to the $A_n$-type quiver. This also generalizes a recent

work by the author with Kazushi Ueda.

### 2010年01月05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 volume growth of hyperkaehler manifolds of type $A_{\\infty}$

Tea: 16:00 - 16:30 コモンルーム

**服部 広大 氏**(東京大学大学院数理科学研究科) 16:30-17:30The volume growth of hyperkaehler manifolds of type $A_{\\infty}$

[ 講演概要 ]

Hyperkaehler manifolds of type $A_{\\infty}$ were constructed due to Anderson-Kronheimer-LeBrun and Goto. These manifolds are 4-demensional, noncompact and their homology groups are infinitely generated. We focus on the volume growth of these hyperkaehler metrics. Here, the volume growth is asymptotic behavior of the volume of a ball of radius $r0$ with the center fixed. There are known examples of hyperkaehler manifolds whose volume growth is $r^4$ (ALE space) or $r^3$ (Taub-NUT space). In this talk we show that there exists a hyperkaehler manifold of type $A_{\\infty}$ whose volume growth is $r^c$ for a given $3

On the Runge theorem for instantons

Hyperkaehler manifolds of type $A_{\\infty}$ were constructed due to Anderson-Kronheimer-LeBrun and Goto. These manifolds are 4-demensional, noncompact and their homology groups are infinitely generated. We focus on the volume growth of these hyperkaehler metrics. Here, the volume growth is asymptotic behavior of the volume of a ball of radius $r0$ with the center fixed. There are known examples of hyperkaehler manifolds whose volume growth is $r^4$ (ALE space) or $r^3$ (Taub-NUT space). In this talk we show that there exists a hyperkaehler manifold of type $A_{\\infty}$ whose volume growth is $r^c$ for a given $3

**松尾 信一郎 氏**(東京大学大学院数理科学研究科) 17:30-18:30

On the Runge theorem for instantons

[ 講演概要 ]

A classical theorem of Runge in complex analysis asserts that a

meromorphic function on a domain in the Riemann sphere can be

approximated, over compact subsets, by rational functions, that is,

meromorphic functions on the Riemann sphere.

This theorem can be paraphrased by saying that any solution of the

Cauchy-Riemann equations on a domain in the Riemann sphere can be

approximated, over compact subsets, by global solutions.

In this talk we will present an analogous result in which the

Cauchy-Riemann equations on Riemann surfaces are replaced by the

Yang-Mills instanton equations on oriented 4-manifolds.

We will also mention that the Runge theorem for instantons can be

applied to develop Yang-Mills gauge theory on open 4-manifolds.

A classical theorem of Runge in complex analysis asserts that a

meromorphic function on a domain in the Riemann sphere can be

approximated, over compact subsets, by rational functions, that is,

meromorphic functions on the Riemann sphere.

This theorem can be paraphrased by saying that any solution of the

Cauchy-Riemann equations on a domain in the Riemann sphere can be

approximated, over compact subsets, by global solutions.

In this talk we will present an analogous result in which the

Cauchy-Riemann equations on Riemann surfaces are replaced by the

Yang-Mills instanton equations on oriented 4-manifolds.

We will also mention that the Runge theorem for instantons can be

applied to develop Yang-Mills gauge theory on open 4-manifolds.

### 2009年12月22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

Relative DG-category, mixed elliptic motives and elliptic polylog

Tea: 16:00 - 16:30 コモンルーム

**寺杣 友秀 氏**(東京大学大学院数理科学研究科)Relative DG-category, mixed elliptic motives and elliptic polylog

[ 講演概要 ]

We consider a full subcategory of

mixed motives generated by an elliptic curve

over a field, which is called the category of

mixed elliptic motives. We introduce a DG

Hopf algebra such that the categroy of

mixed elliptic motives is equal to that of

comodules over it. For the construction, we

use the notion of relative DG-category with

respect to GL(2). As an application, we construct

an mixed elliptic motif associated to

the elliptic polylog. It is a joint work with

Kenichiro Kimura.

We consider a full subcategory of

mixed motives generated by an elliptic curve

over a field, which is called the category of

mixed elliptic motives. We introduce a DG

Hopf algebra such that the categroy of

mixed elliptic motives is equal to that of

comodules over it. For the construction, we

use the notion of relative DG-category with

respect to GL(2). As an application, we construct

an mixed elliptic motif associated to

the elliptic polylog. It is a joint work with

Kenichiro Kimura.

### 2009年12月15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

Open Problems in Discrete Geometric Analysis

Tea: 16:40 - 17:00 コモンルーム

**砂田 利一 氏**(明治大学)Open Problems in Discrete Geometric Analysis

[ 講演概要 ]

Discrete geometric analysis is a hybrid field of several traditional disciplines: graph theory, geometry, theory of discrete groups, and probability. This field concerns solely analysis on graphs, a synonym of "1-dimensional cell complex". In this talk, I shall discuss several open problems related to the discrete Laplacian, a "protagonist" in discrete geometric analysis. Topics dealt with are 1. Ramanujan graphs, 2. Spectra of covering graphs, 3. Zeta functions of finitely generated groups.

Discrete geometric analysis is a hybrid field of several traditional disciplines: graph theory, geometry, theory of discrete groups, and probability. This field concerns solely analysis on graphs, a synonym of "1-dimensional cell complex". In this talk, I shall discuss several open problems related to the discrete Laplacian, a "protagonist" in discrete geometric analysis. Topics dealt with are 1. Ramanujan graphs, 2. Spectra of covering graphs, 3. Zeta functions of finitely generated groups.

### 2009年12月01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

Non-Abelian Novikov homology

Tea: 16:00 - 16:30 コモンルーム

**Andrei Pajitnov 氏**(Univ. de Nantes)Non-Abelian Novikov homology

[ 講演概要 ]

Classical construction of S.P. Novikov

associates to each circle-valued Morse map

a chain complex defined over a ring

of Laurent power series in one variable.

In this survey talk we shall explain several

results related to the construction and

properties of non-Abelian generalizations of the

Novikov complex.

Classical construction of S.P. Novikov

associates to each circle-valued Morse map

a chain complex defined over a ring

of Laurent power series in one variable.

In this survey talk we shall explain several

results related to the construction and

properties of non-Abelian generalizations of the

Novikov complex.

### 2009年11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

A topological approach to left orderable groups

Tea: 16:00 - 16:30 コモンルーム

**Adam Clay 氏**(University of British Columbia)A topological approach to left orderable groups

[ 講演概要 ]

A group G is said to be left orderable if there is a strict

total ordering of its elements such that gin G. Left orderable groups have been useful in solving many problems in topology, and now we find that topology is returning the favour: the set of all left orderings of a group is denoted by LO(G), and it admits a natural topology, under which LO(G) becomes a compact topological

space. In general, the structure of the space LO(G) is not well understood, although there are surprising results in a few special cases.

For example, the space of left orderings of the braid group B_n for n>2

contains isolated points (yet it is uncountable), while the space of left

orderings of the fundamental group of the Klein bottle is finite.

Twice in recent years, the space of left orderings has been used very

successfully to solve difficult open problems from the field of left

orderable groups, even though the connection between the topology of LO(G) and the algebraic properties of G was still unclear. I will explain the

newest understanding of this connection, and highlight some potential

applications of further advances.

A group G is said to be left orderable if there is a strict

total ordering of its elements such that g

space. In general, the structure of the space LO(G) is not well understood, although there are surprising results in a few special cases.

For example, the space of left orderings of the braid group B_n for n>2

contains isolated points (yet it is uncountable), while the space of left

orderings of the fundamental group of the Klein bottle is finite.

Twice in recent years, the space of left orderings has been used very

successfully to solve difficult open problems from the field of left

orderable groups, even though the connection between the topology of LO(G) and the algebraic properties of G was still unclear. I will explain the

newest understanding of this connection, and highlight some potential

applications of further advances.