## トポロジー火曜セミナー

過去の記録 ～02/07｜次回の予定｜今後の予定 02/08～

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

担当者 | 河野 俊丈, 河澄 響矢, 北山 貴裕, 逆井卓也 |

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

備考 | Tea: 16:30 - 17:00 コモンルーム |

**過去の記録**

### 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.

### 2009年11月17日(火)

16:30-18:00 数理科学研究科棟(駒場) 056号室

Tea: 16:00 - 16:30 コモンルーム

On the $SO(N)$ and $Sp(N)$ free energy of a closed oriented 3-manifold

Tea: 16:00 - 16:30 コモンルーム

**高田 敏恵 氏**(新潟大学)On the $SO(N)$ and $Sp(N)$ free energy of a closed oriented 3-manifold

[ 講演概要 ]

We give an explicit formula of the $SO(N)$ and $Sp(N)$ free energy

of a lens space and show that the genus $g$ terms of it are analytic

in a neighborhood at zero, where we can choose the neighborhood

independently of $g$.

Moreover, it is proved that for any closed oriented 3-manifold $M$

and any $g$, the genus $g$ terms of $SO(N)$ and $Sp(N)$ free energy

of $M$ coincide up to sign.

We give an explicit formula of the $SO(N)$ and $Sp(N)$ free energy

of a lens space and show that the genus $g$ terms of it are analytic

in a neighborhood at zero, where we can choose the neighborhood

independently of $g$.

Moreover, it is proved that for any closed oriented 3-manifold $M$

and any $g$, the genus $g$ terms of $SO(N)$ and $Sp(N)$ free energy

of $M$ coincide up to sign.

### 2009年11月10日(火)

16:30-18:00 数理科学研究科棟(駒場) 056号室

Tea: 16:00 - 16:30 コモンルーム

Resurgent analysis of the Witten Laplacian in one dimension

Tea: 16:00 - 16:30 コモンルーム

**Alexander Getmanenko 氏**(IPMU)Resurgent analysis of the Witten Laplacian in one dimension

[ 講演概要 ]

I will recall Witten's approach to the Morse theory through properties of a certain differential operator. Then I will introduce resurgent analysis -- an asymptotic method used, in particular, for studying quantum-mechanical tunneling. In conclusion I will discuss how the methods of resurgent analysis can help us "see" pseudoholomorphic discs in the eigenfunctions of the Witten Laplacian.

I will recall Witten's approach to the Morse theory through properties of a certain differential operator. Then I will introduce resurgent analysis -- an asymptotic method used, in particular, for studying quantum-mechanical tunneling. In conclusion I will discuss how the methods of resurgent analysis can help us "see" pseudoholomorphic discs in the eigenfunctions of the Witten Laplacian.

### 2009年10月27日(火)

16:30-18:00 数理科学研究科棟(駒場) 056号室

Tea: 16:00 - 16:30 コモンルーム

A new appearance of the Morita-Penner cocycle

Tea: 16:00 - 16:30 コモンルーム

**Alex Bene 氏**(IPMU)A new appearance of the Morita-Penner cocycle

[ 講演概要 ]

In this talk, I will recall the Morita-Penner cocycle on the dual fatgraph complex for a surface with one boundary component. This cocycle, when restricted to paths representing elements of the mapping class group, represents the extended first Johnson homomorphism \\tau_1, thus can be viewed as a (in some specific sense canonical) "groupoid extension" of \\tau_1. There are now several different contexts in which this cocycle can be constructed, and in this talk I will briefly review several of them, including one discovered in the context of finite type invariants of homology cylinders in joint work with J.E. Andersen, J-B. Meilhan, and R.C. Penner.

In this talk, I will recall the Morita-Penner cocycle on the dual fatgraph complex for a surface with one boundary component. This cocycle, when restricted to paths representing elements of the mapping class group, represents the extended first Johnson homomorphism \\tau_1, thus can be viewed as a (in some specific sense canonical) "groupoid extension" of \\tau_1. There are now several different contexts in which this cocycle can be constructed, and in this talk I will briefly review several of them, including one discovered in the context of finite type invariants of homology cylinders in joint work with J.E. Andersen, J-B. Meilhan, and R.C. Penner.

### 2009年10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

Torus fibrations and localization of index

Tea: 16:00 - 16:30 コモンルーム

**吉田 尚彦 氏**(明治大学大学院理工学研究科)Torus fibrations and localization of index

[ 講演概要 ]

I will describe a localization of index of a Dirac type operator.

We make use of a structure of torus fibration, but the mechanism

of the localization does not rely on any group action. In the case of

Lagrangian fibration, we show that the index is described as a sum of

the contributions from Bohr-Sommerfeld fibers and singular fibers.

To show the localization we introduce a deformation of a Dirac type

operator for a manifold equipped with a fiber bundle structure which

satisfies a kind of acyclic condition. The deformation allows an

interpretation as an adiabatic limit or an infinite dimensional analogue

of Witten deformation.

Joint work with Hajime Fujita and Mikio Furuta.

I will describe a localization of index of a Dirac type operator.

We make use of a structure of torus fibration, but the mechanism

of the localization does not rely on any group action. In the case of

Lagrangian fibration, we show that the index is described as a sum of

the contributions from Bohr-Sommerfeld fibers and singular fibers.

To show the localization we introduce a deformation of a Dirac type

operator for a manifold equipped with a fiber bundle structure which

satisfies a kind of acyclic condition. The deformation allows an

interpretation as an adiabatic limit or an infinite dimensional analogue

of Witten deformation.

Joint work with Hajime Fujita and Mikio Furuta.

### 2009年10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

Instanton Floer homology for lens spaces

Tea: 16:00 - 16:30 コモンルーム

**笹平 裕史 氏**(東京大学大学院数理科学研究科)Instanton Floer homology for lens spaces

[ 講演概要 ]

Let Y be an oriented closed 3-manifold and P be an SU(2)-bundle on Y. Under a certain condition, instanton Floer homology for Y can be defined as the Morse homology of the Chern-Simons functional. The condition is that all flat connections on P are irreducible. When there is a reducible flat connection on P, instanton Floer homology is not defined in general.

Since the fundamental group of a lens sapce is commutative, all flat connections on the lens space are reducible. In this talk I will introduce instanton Floer homology for lens spaces. I also show calculations for some lens spaces.

Let Y be an oriented closed 3-manifold and P be an SU(2)-bundle on Y. Under a certain condition, instanton Floer homology for Y can be defined as the Morse homology of the Chern-Simons functional. The condition is that all flat connections on P are irreducible. When there is a reducible flat connection on P, instanton Floer homology is not defined in general.

Since the fundamental group of a lens sapce is commutative, all flat connections on the lens space are reducible. In this talk I will introduce instanton Floer homology for lens spaces. I also show calculations for some lens spaces.

### 2009年09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

Symbol of the Conway polynomial and Drinfeld associator

Tea: 16:00 - 16:30 コモンルーム

**Sergei Duzhin 氏**(Steklov Mathematical Institute, Petersburg Division)Symbol of the Conway polynomial and Drinfeld associator

[ 講演概要 ]

The Magnus expansion is a universal finite type invariant of pure braids

with values in the space of horizontal chord diagrams. The Conway polynomial

composed with the short circuit map from braids to knots gives rise to a

series of finite type invariants of pure braids and thus factors through

the Magnus map. We describe explicitly the resulting mapping from horizontal

chord diagrams on 3 strands to univariante polynomials and evaluate it on

the Drinfeld associator obtaining a beautiful generating function whose

coefficients are integer combinations of multple zeta values.

The Magnus expansion is a universal finite type invariant of pure braids

with values in the space of horizontal chord diagrams. The Conway polynomial

composed with the short circuit map from braids to knots gives rise to a

series of finite type invariants of pure braids and thus factors through

the Magnus map. We describe explicitly the resulting mapping from horizontal

chord diagrams on 3 strands to univariante polynomials and evaluate it on

the Drinfeld associator obtaining a beautiful generating function whose

coefficients are integer combinations of multple zeta values.

### 2009年07月14日(火)

17:00-18:00 数理科学研究科棟(駒場) 056号室

Tea: 16:40 - 17:00 コモンルーム

The Cannon-Thurston maps and the canonical decompositions

of punctured-torus bundles over the circle.

Tea: 16:40 - 17:00 コモンルーム

**作間 誠 氏**(広島大学)The Cannon-Thurston maps and the canonical decompositions

of punctured-torus bundles over the circle.

[ 講演概要 ]

To each once-punctured-torus bundle over the circle

with pseudo-Anosov monodromy, there are associated two tessellations of the complex plane:

one is the triangulation of a horosphere induced by the canonical decomposition into ideal

tetrahedra, and the other is a fractal tessellation

given by the Cannon-Thurston map of the fiber group.

In this talk, I will explain the relation between these two tessellations

(joint work with Warren Dicks).

I will also explain the relation of the fractal tessellation and

the "circle chains" of double cusp groups converging to the fiber group

(joint work with Caroline Series).

If time permits, I would like to discuss possible generalization of these results

to higher-genus punctured surface bundles.

To each once-punctured-torus bundle over the circle

with pseudo-Anosov monodromy, there are associated two tessellations of the complex plane:

one is the triangulation of a horosphere induced by the canonical decomposition into ideal

tetrahedra, and the other is a fractal tessellation

given by the Cannon-Thurston map of the fiber group.

In this talk, I will explain the relation between these two tessellations

(joint work with Warren Dicks).

I will also explain the relation of the fractal tessellation and

the "circle chains" of double cusp groups converging to the fiber group

(joint work with Caroline Series).

If time permits, I would like to discuss possible generalization of these results

to higher-genus punctured surface bundles.

### 2009年06月30日(火)

16:30-18:00 数理科学研究科棟(駒場) 056号室

Tea: 16:00 - 16:30 コモンルーム

Torsion volume forms and twisted Alexander functions on

character varieties of knots

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)Torsion volume forms and twisted Alexander functions on

character varieties of knots

[ 講演概要 ]

Using non-acyclic Reidemeister torsion, we can canonically

construct a complex volume form on each component of the

lowest dimension of the $SL_2(\\mathbb{C})$-character

variety of a link group.

This volume form enjoys a certain compatibility with the

following natural transformations on the variety.

Two of them are involutions which come from the algebraic

structure of $SL_2(\\mathbb{C})$ and the other is the

action by the outer automorphism group of the link group.

Moreover, in the case of knots these results deduce a kind

of symmetry of the $SU_2$-twisted Alexander functions

which are globally described via the volume form.

Using non-acyclic Reidemeister torsion, we can canonically

construct a complex volume form on each component of the

lowest dimension of the $SL_2(\\mathbb{C})$-character

variety of a link group.

This volume form enjoys a certain compatibility with the

following natural transformations on the variety.

Two of them are involutions which come from the algebraic

structure of $SL_2(\\mathbb{C})$ and the other is the

action by the outer automorphism group of the link group.

Moreover, in the case of knots these results deduce a kind

of symmetry of the $SU_2$-twisted Alexander functions

which are globally described via the volume form.

### 2009年06月23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 Meyer functions for projective varieties and their applications

Tea: 16:00 - 16:30 コモンルーム

**久野 雄介 氏**(東京大学大学院数理科学研究科)The Meyer functions for projective varieties and their applications

[ 講演概要 ]

Meyer function is a kind of secondary invariant related to the signature

of surface bundles over surfaces. In this talk I will show there exist uniquely the Meyer function

for each smooth projective variety.

Our function is a class function on the fundamental group of some open algebraic variety.

I will also talk about its application to local signature for fibered 4-manifolds

Meyer function is a kind of secondary invariant related to the signature

of surface bundles over surfaces. In this talk I will show there exist uniquely the Meyer function

for each smooth projective variety.

Our function is a class function on the fundamental group of some open algebraic variety.

I will also talk about its application to local signature for fibered 4-manifolds

### 2009年06月16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 abelianization of the level 2 mapping class group

Tea: 16:00 - 16:30 コモンルーム

**佐藤 正寿 氏**(東京大学大学院数理科学研究科)The abelianization of the level 2 mapping class group

[ 講演概要 ]

The level d mapping class group is a finite index subgroup of the mapping class group of an orientable closed surface. For d greater than or equal to 3, the abelianization of this group is equal to the first homology group of the moduli space of nonsingular curves with level d structure.

In this talk, we determine the abelianization of the level d mapping class group for d=2 and odd d. For even d greater than 2, we also determine it up to a cyclic group of order 2.

The level d mapping class group is a finite index subgroup of the mapping class group of an orientable closed surface. For d greater than or equal to 3, the abelianization of this group is equal to the first homology group of the moduli space of nonsingular curves with level d structure.

In this talk, we determine the abelianization of the level d mapping class group for d=2 and odd d. For even d greater than 2, we also determine it up to a cyclic group of order 2.

### 2009年06月09日(火)

16:30-18:00 数理科学研究科棟(駒場) 056号室

Tea: 16:00 - 16:30 コモンルーム

A finite-dimensional construction of the Chern character for

twisted K-theory

Tea: 16:00 - 16:30 コモンルーム

**五味 清紀 氏**(京都大学大学院理学研究科)A finite-dimensional construction of the Chern character for

twisted K-theory

[ 講演概要 ]

Twisted K-theory is a variant of topological K-theory, and

is attracting much interest due to applications to physics recently.

Usually, twisted K-theory is formulated infinite-dimensionally, and

hence known constructions of its Chern character are more or less

abstract. The aim of my talk is to explain a purely finite-dimensional

construction of the Chern character for twisted K-theory, which allows

us to compute examples concretely. The construction is based on

twisted version of Furuta's generalized vector bundle, and Quillen's

superconnection.

This is a joint work with Yuji Terashima.

Twisted K-theory is a variant of topological K-theory, and

is attracting much interest due to applications to physics recently.

Usually, twisted K-theory is formulated infinite-dimensionally, and

hence known constructions of its Chern character are more or less

abstract. The aim of my talk is to explain a purely finite-dimensional

construction of the Chern character for twisted K-theory, which allows

us to compute examples concretely. The construction is based on

twisted version of Furuta's generalized vector bundle, and Quillen's

superconnection.

This is a joint work with Yuji Terashima.

### 2009年06月02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

Graph homology: Koszul duality = Verdier duality

Tea: 16:00 - 16:30 コモンルーム

**Alexander Voronov 氏**(University of Minnesota)Graph homology: Koszul duality = Verdier duality

[ 講演概要 ]

Graph cohomology appears in computation of the cohomology of the moduli space of Riemann surfaces and the outer automorphism group of a free group. In the former case, it is graph cohomology of the commutative and Lie types, in the latter it is ribbon graph cohomology, that is to say, graph cohomology of the associative type. The presence of these three basic types of algebraic structures hints at a relation between Koszul duality for operads and Poincare-Lefschetz duality for manifolds. I will show how the more general Verdier duality for certain sheaves on the moduli spaces of graphs associated to Koszul operads corresponds to Koszul duality of operads. This is a joint work with Andrey Lazarev.

Graph cohomology appears in computation of the cohomology of the moduli space of Riemann surfaces and the outer automorphism group of a free group. In the former case, it is graph cohomology of the commutative and Lie types, in the latter it is ribbon graph cohomology, that is to say, graph cohomology of the associative type. The presence of these three basic types of algebraic structures hints at a relation between Koszul duality for operads and Poincare-Lefschetz duality for manifolds. I will show how the more general Verdier duality for certain sheaves on the moduli spaces of graphs associated to Koszul operads corresponds to Koszul duality of operads. This is a joint work with Andrey Lazarev.

### 2009年05月26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

Configuration space integrals and the cohomology of the space of long embeddings

Tea: 16:00 - 16:30 コモンルーム

**境 圭一 氏**(東京大学大学院数理科学研究科)Configuration space integrals and the cohomology of the space of long embeddings

[ 講演概要 ]

It is known that some non-trivial cohomology classes, such as finite type invariants for (long) 1-knots (Bott-Taubes, Kohno, ...) and invariants for codimension two, odd dimensional long embeddings (Bott, Cattaneo-Rossi, Watanabe) are given as configuration space integrals associated with trivalent graphs.

In this talk, I will describe more cohomology classes by means of configuration space integral, in particular those arising from non-trivalent graphs and a new formulation of the Haefliger invariant for long 3-embeddings in 6-space, in relation to Budney's little balls operad action and Roseman-Takase's deform-spinning.

This is in part a joint work with Tadayuki Watanabe.

It is known that some non-trivial cohomology classes, such as finite type invariants for (long) 1-knots (Bott-Taubes, Kohno, ...) and invariants for codimension two, odd dimensional long embeddings (Bott, Cattaneo-Rossi, Watanabe) are given as configuration space integrals associated with trivalent graphs.

In this talk, I will describe more cohomology classes by means of configuration space integral, in particular those arising from non-trivalent graphs and a new formulation of the Haefliger invariant for long 3-embeddings in 6-space, in relation to Budney's little balls operad action and Roseman-Takase's deform-spinning.

This is in part a joint work with Tadayuki Watanabe.