## Tuesday Seminar on Topology

Seminar information archive ～04/19｜Next seminar｜Future seminars 04/20～

Date, time & place | Tuesday 17:00 - 18:30 056Room #056 (Graduate School of Math. Sci. Bldg.) |
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Organizer(s) | KAWAZUMI Nariya, KITAYAMA Takahiro, SAKASAI Takuya |

**Seminar information archive**

### 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

### 2010/01/26

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

On the (co)chain type levels of spaces

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

[ Abstract ]

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.

### 2010/01/19

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

Localization via group action and its application to

the period condition of algebraic minimal surfaces

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

the period condition of algebraic minimal surfaces

[ Abstract ]

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.

### 2010/01/12

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

Index problem for generically-wild homoclinic classes in dimension three

On a generalized suspension theorem for directed Fukaya categories

**篠原 克寿**(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

[ Abstract ]

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**(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

[ Abstract ]

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.

### 2010/01/05

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

The volume growth of hyperkaehler manifolds of type $A_{\\infty}$

**服部 広大**(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 $A_{\\infty}$

[ Abstract ]

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**(東京大学大学院数理科学研究科) 17:30-18:30

On the Runge theorem for instantons

[ Abstract ]

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.

### 2009/12/22

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

Relative DG-category, mixed elliptic motives and elliptic polylog

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

[ Abstract ]

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.

### 2009/12/15

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

Open Problems in Discrete Geometric Analysis

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

[ Abstract ]

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.

### 2009/12/01

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

Non-Abelian Novikov homology

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

[ Abstract ]

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.

### 2009/11/24

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

A topological approach to left orderable groups

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

[ Abstract ]

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.

### 2009/11/17

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

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

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

[ Abstract ]

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.

### 2009/11/10

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

Resurgent analysis of the Witten Laplacian in one dimension

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

[ Abstract ]

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.

### 2009/10/27

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

A new appearance of the Morita-Penner cocycle

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

[ Abstract ]

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.

### 2009/10/20

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

Torus fibrations and localization of index

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

[ Abstract ]

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.

### 2009/10/13

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

Instanton Floer homology for lens spaces

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

[ Abstract ]

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.

### 2009/09/29

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

Symbol of the Conway polynomial and Drinfeld associator

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

[ Abstract ]

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.

### 2009/07/14

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

The Cannon-Thurston maps and the canonical decompositions

of punctured-torus bundles over the circle.

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

of punctured-torus bundles over the circle.

[ Abstract ]

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.

### 2009/06/30

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

Torsion volume forms and twisted Alexander functions on

character varieties of knots

**北山 貴裕**(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

[ Abstract ]

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.

### 2009/06/23

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

The Meyer functions for projective varieties and their applications

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

[ Abstract ]

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

### 2009/06/16

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

The abelianization of the level 2 mapping class group

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

[ Abstract ]

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.