## Tuesday Seminar on Topology

Seminar information archive ～09/30｜Next seminar｜Future seminars 10/01～

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

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

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

**Seminar information archive**

### 2012/10/23

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

A geometric approach to the Johnson homomorphisms (JAPANESE)

**Nariya Kawazumi**(The University of Tokyo)A geometric approach to the Johnson homomorphisms (JAPANESE)

[ Abstract ]

We re-construct the Johnson homomorphisms as an embeddig of the Torelli

group

into the completed Goldman-Turaev Lie bialgebra. Then the image is

included in the

kernel of the Turaev cobracket. In the case where the boundary is

connected,

the Turaev cobracket clarifies a geometric meaning of the Morita traces.

Time permitting, we also discuss the case of holed discs.

This talk is based on a joint work with Yusuke Kuno (Tsuda College).

We re-construct the Johnson homomorphisms as an embeddig of the Torelli

group

into the completed Goldman-Turaev Lie bialgebra. Then the image is

included in the

kernel of the Turaev cobracket. In the case where the boundary is

connected,

the Turaev cobracket clarifies a geometric meaning of the Morita traces.

Time permitting, we also discuss the case of holed discs.

This talk is based on a joint work with Yusuke Kuno (Tsuda College).

### 2012/10/16

17:10-18:10 Room #056 (Graduate School of Math. Sci. Bldg.)

Analytic torsion of log-Enriques surfaces (JAPANESE)

**Ken-Ichi Yoshikawa**(Kyoto University)Analytic torsion of log-Enriques surfaces (JAPANESE)

[ Abstract ]

Log-Enriques surfaces are rational surfaces with nowhere vanishing

pluri-canonical forms. We report the recent progress on the computation

of analytic torsion of log-Enriques surfaces.

Log-Enriques surfaces are rational surfaces with nowhere vanishing

pluri-canonical forms. We report the recent progress on the computation

of analytic torsion of log-Enriques surfaces.

### 2012/10/09

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

The growth series of pure Artin groups of dihedral type (JAPANESE)

**Michihiko Fujii**(Kyoto University)The growth series of pure Artin groups of dihedral type (JAPANESE)

[ Abstract ]

In this talk, I consider the kernel of the natural projection from

the Artin group of dihedral type to the corresponding Coxeter group,

that we call a pure Artin group of dihedral type,

and present rational function expressions for both the spherical and

geodesic growth series

of the pure Artin group of dihedral type with respect to a natural

generating set.

Also, I show that their growth rates are Pisot numbers.

This talk is partially based on a joint work with Takao Satoh.

In this talk, I consider the kernel of the natural projection from

the Artin group of dihedral type to the corresponding Coxeter group,

that we call a pure Artin group of dihedral type,

and present rational function expressions for both the spherical and

geodesic growth series

of the pure Artin group of dihedral type with respect to a natural

generating set.

Also, I show that their growth rates are Pisot numbers.

This talk is partially based on a joint work with Takao Satoh.

### 2012/10/02

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

Geometric flows and their self-similar solutions

(JAPANESE)

**Akito Futaki**(The University of Tokyo)Geometric flows and their self-similar solutions

(JAPANESE)

[ Abstract ]

In the first half of this expository talk we consider the Ricci flow and its self-similar solutions,

namely the Ricci solitons. We then specialize in the K\\"ahler case and discuss on the K\\"ahler-Einstein

problem. In the second half of this talk we consider the mean curvature flow and its self-similar

solutions, and see common aspects of the two geometric flows.

In the first half of this expository talk we consider the Ricci flow and its self-similar solutions,

namely the Ricci solitons. We then specialize in the K\\"ahler case and discuss on the K\\"ahler-Einstein

problem. In the second half of this talk we consider the mean curvature flow and its self-similar

solutions, and see common aspects of the two geometric flows.

### 2012/09/04

17:00-18:00 Room #002 (Graduate School of Math. Sci. Bldg.)

Poincare inequalities, rigid groups and applications (ENGLISH)

**Piotr Nowak**(the Institute of Mathematics, Polish Academy of Sciences)Poincare inequalities, rigid groups and applications (ENGLISH)

[ Abstract ]

Kazhdan’s property (T) for a group G can be expressed as a

fixed point property for affine isometric actions of G on a Hilbert

space. This definition generalizes naturally to other normed spaces. In

this talk we will focus on the spectral (aka geometric) method for

proving property (T), based on the work of Garland and studied earlier

by Pansu, Zuk, Ballmann-Swiatkowski, Dymara-Januszkiewicz

(“lambda_1>1/2” conditions) and we generalize it to to the setting of

all reflexive Banach spaces.

As applications we will show estimates of the conformal dimension of the

boundary of random hyperbolic groups in the Gromov density model and

present progress on Shalom’s conjecture on vanishing of 1-cohomology

with coefficients in uniformly bounded representations on Hilbert spaces.

Kazhdan’s property (T) for a group G can be expressed as a

fixed point property for affine isometric actions of G on a Hilbert

space. This definition generalizes naturally to other normed spaces. In

this talk we will focus on the spectral (aka geometric) method for

proving property (T), based on the work of Garland and studied earlier

by Pansu, Zuk, Ballmann-Swiatkowski, Dymara-Januszkiewicz

(“lambda_1>1/2” conditions) and we generalize it to to the setting of

all reflexive Banach spaces.

As applications we will show estimates of the conformal dimension of the

boundary of random hyperbolic groups in the Gromov density model and

present progress on Shalom’s conjecture on vanishing of 1-cohomology

with coefficients in uniformly bounded representations on Hilbert spaces.

### 2012/07/24

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

Orthospectra and identities (ENGLISH)

**Greg McShane**(Institut Fourier, Grenoble)Orthospectra and identities (ENGLISH)

[ Abstract ]

The orthospectra of a hyperbolic manifold with geodesic

boundary consists of the lengths of all geodesics perpendicular to the

boundary.

We discuss the properties of the orthospectra, asymptotics, multiplicity

and identities due to Basmajian, Bridgeman and Calegari. We will give

a proof that the identities of Bridgeman and Calegari are the same.

The orthospectra of a hyperbolic manifold with geodesic

boundary consists of the lengths of all geodesics perpendicular to the

boundary.

We discuss the properties of the orthospectra, asymptotics, multiplicity

and identities due to Basmajian, Bridgeman and Calegari. We will give

a proof that the identities of Bridgeman and Calegari are the same.

### 2012/07/17

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

Contact structure of mixed links (JAPANESE)

**Mutsuo Oka**(Tokyo University of Science)Contact structure of mixed links (JAPANESE)

[ Abstract ]

A strongly non-degenerate mixed function has a Milnor open book

structures on a sufficiently small sphere. We introduce the notion of

{\\em a holomorphic-like} mixed function

and we will show that a link defined by such a mixed function has a

canonical contact structure.

Then we will show that this contact structure for a certain

holomorphic-like mixed function

is carried by the Milnor open book.

A strongly non-degenerate mixed function has a Milnor open book

structures on a sufficiently small sphere. We introduce the notion of

{\\em a holomorphic-like} mixed function

and we will show that a link defined by such a mixed function has a

canonical contact structure.

Then we will show that this contact structure for a certain

holomorphic-like mixed function

is carried by the Milnor open book.

### 2012/07/10

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

Topology in Gravitational Lensing (ENGLISH)

**Marcus Werner**(Kavli IPMU)Topology in Gravitational Lensing (ENGLISH)

[ Abstract ]

General relativity implies that light is deflected by masses

due to the curvature of spacetime. The ensuing gravitational

lensing effect is an important tool in modern astronomy, and

topology plays a significant role in its properties. In this

talk, I will review topological aspects of gravitational lensing

theory: the connection of image numbers with Morse theory; the

interpretation of certain invariant sums of the signed image

magnification in terms of Lefschetz fixed point theory; and,

finally, a new partially topological perspective on gravitational

light deflection that emerges from the concept of optical geometry

and applications of the Gauss-Bonnet theorem.

General relativity implies that light is deflected by masses

due to the curvature of spacetime. The ensuing gravitational

lensing effect is an important tool in modern astronomy, and

topology plays a significant role in its properties. In this

talk, I will review topological aspects of gravitational lensing

theory: the connection of image numbers with Morse theory; the

interpretation of certain invariant sums of the signed image

magnification in terms of Lefschetz fixed point theory; and,

finally, a new partially topological perspective on gravitational

light deflection that emerges from the concept of optical geometry

and applications of the Gauss-Bonnet theorem.

### 2012/06/19

17:10-18:10 Room #056 (Graduate School of Math. Sci. Bldg.)

On the universal degenerating family of Riemann surfaces

over the D-M compactification of moduli space (JAPANESE)

**Yukio Matsumoto**(Gakushuin University)On the universal degenerating family of Riemann surfaces

over the D-M compactification of moduli space (JAPANESE)

[ Abstract ]

It is usually understood that over the Deligne-

Mumford compactification of moduli space of Riemann surfaces of

genus > 1, there is a family of stable curves. However, if one tries to

construct this family precisely, he/she must first take a disjoint union

of various types of smooth families of stable curves, and then divide

them by their automorphisms to paste them together. In this talk we will

show that once the smooth families are divided, the resulting quotient

family contains not only stable curves but virtually all types of

degeneration of Riemann surfaces, becoming a kind of universal

degenerating family of Riemann surfaces.

It is usually understood that over the Deligne-

Mumford compactification of moduli space of Riemann surfaces of

genus > 1, there is a family of stable curves. However, if one tries to

construct this family precisely, he/she must first take a disjoint union

of various types of smooth families of stable curves, and then divide

them by their automorphisms to paste them together. In this talk we will

show that once the smooth families are divided, the resulting quotient

family contains not only stable curves but virtually all types of

degeneration of Riemann surfaces, becoming a kind of universal

degenerating family of Riemann surfaces.

### 2012/06/12

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

Topological interpretation of the quandle cocycle invariants of links (JAPANESE)

**Takefumi Nosaka**(RIMS, Kyoto University, JSPS)Topological interpretation of the quandle cocycle invariants of links (JAPANESE)

[ Abstract ]

Carter et al. introduced many quandle cocycle invariants

combinatorially constructed from link-diagrams. For connected quandles of

finite order, we give a topological meaning of the invariants, without

some torsion parts. Precisely, this invariant equals a sum of "knot

colouring polynomial" and of a Z-equivariant part of the Dijkgraaf-Witten

invariant. Moreover, our approach involves applications to compute "good"

torsion subgroups of the 3-rd quandle homologies and the 2-nd homotopy

groups of rack spaces.

Carter et al. introduced many quandle cocycle invariants

combinatorially constructed from link-diagrams. For connected quandles of

finite order, we give a topological meaning of the invariants, without

some torsion parts. Precisely, this invariant equals a sum of "knot

colouring polynomial" and of a Z-equivariant part of the Dijkgraaf-Witten

invariant. Moreover, our approach involves applications to compute "good"

torsion subgroups of the 3-rd quandle homologies and the 2-nd homotopy

groups of rack spaces.

### 2012/06/05

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

A generalization of Dehn twists (JAPANESE)

**Yusuke Kuno**(Tsuda College)A generalization of Dehn twists (JAPANESE)

[ Abstract ]

We introduce a generalization

of Dehn twists for loops which are not

necessarily simple loops on an oriented surface.

Our generalization is an element of a certain

enlargement of the mapping class group of the surface.

A natural question is whether a generalized Dehn twist is

in the mapping class group. We show some results related to this question.

This talk is partially based on a joint work

with Nariya Kawazumi (Univ. Tokyo).

We introduce a generalization

of Dehn twists for loops which are not

necessarily simple loops on an oriented surface.

Our generalization is an element of a certain

enlargement of the mapping class group of the surface.

A natural question is whether a generalized Dehn twist is

in the mapping class group. We show some results related to this question.

This talk is partially based on a joint work

with Nariya Kawazumi (Univ. Tokyo).

### 2012/05/29

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

Triple linking numbers and triple point numbers

of torus-covering $T^2$-links

(JAPANESE)

**Inasa Nakamura**(Gakushuin University, JSPS)Triple linking numbers and triple point numbers

of torus-covering $T^2$-links

(JAPANESE)

[ Abstract ]

The triple linking number of an oriented surface link was defined as an

analogical notion of the linking number of a classical link. A

torus-covering $T^2$-link $\\mathcal{S}_m(a,b)$ is a surface link in the

form of an unbranched covering over the standard torus, determined from

two commutative $m$-braids $a$ and $b$.

In this talk, we consider $\\mathcal{S}_m(a,b)$ when $a$, $b$ are pure

$m$-braids ($m \\geq 3$), which is a surface link with $m$-components. We

present the triple linking number of $\\mathcal{S}_m(a,b)$ by using the

linking numbers of the closures of $a$ and $b$. This gives a lower bound

of the triple point number. In some cases, we can determine the triple

point numbers, each of which is a multiple of four.

The triple linking number of an oriented surface link was defined as an

analogical notion of the linking number of a classical link. A

torus-covering $T^2$-link $\\mathcal{S}_m(a,b)$ is a surface link in the

form of an unbranched covering over the standard torus, determined from

two commutative $m$-braids $a$ and $b$.

In this talk, we consider $\\mathcal{S}_m(a,b)$ when $a$, $b$ are pure

$m$-braids ($m \\geq 3$), which is a surface link with $m$-components. We

present the triple linking number of $\\mathcal{S}_m(a,b)$ by using the

linking numbers of the closures of $a$ and $b$. This gives a lower bound

of the triple point number. In some cases, we can determine the triple

point numbers, each of which is a multiple of four.

### 2012/05/22

17:10-18:10 Room #056 (Graduate School of Math. Sci. Bldg.)

Gamma Integral Structure in Gromov-Witten theory (JAPANESE)

**Hiroshi Iritani**(Kyoto University)Gamma Integral Structure in Gromov-Witten theory (JAPANESE)

[ Abstract ]

The quantum cohomology of a symplectic

manifold undelies a certain integral local system

defined by the Gamma characteristic class.

This local system originates from the natural integral

local sysmem on the B-side under mirror symmetry.

In this talk, I will explain its relationships to the problem

of analytic continuation of Gromov-Witten theoy (potentials),

including crepant resolution conjecture, LG/CY correspondence,

modularity in higher genus theory.

The quantum cohomology of a symplectic

manifold undelies a certain integral local system

defined by the Gamma characteristic class.

This local system originates from the natural integral

local sysmem on the B-side under mirror symmetry.

In this talk, I will explain its relationships to the problem

of analytic continuation of Gromov-Witten theoy (potentials),

including crepant resolution conjecture, LG/CY correspondence,

modularity in higher genus theory.

### 2012/05/08

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

Infinite examples of non-Garside monoids having fundamental elements (JAPANESE)

**Tadashi Ishibe**(The University of Tokyo, JSPS)Infinite examples of non-Garside monoids having fundamental elements (JAPANESE)

[ Abstract ]

The Garside group, as a generalization of Artin groups,

is defined as the group of fractions of a Garside monoid.

To understand the elliptic Artin groups, which are the fundamental

groups of the complement of discriminant divisors of the semi-versal

deformation of the simply elliptic singularities E_6~, E_7~ and E_8~,

we need to consider another generalization of Artin groups.

In this talk, we will study the presentations of fundamental groups

of the complement of complexified real affine line arrangements

and consider the associated monoids.

It turns out that, in some cases, they are not Garside monoids.

Nevertheless, we will show that they satisfy the cancellation condition

and carry certain particular elements similar to the fundamental elements

in Artin monoids.

As a result, we will show that the word problem can be solved

and the center of them are determined.

The Garside group, as a generalization of Artin groups,

is defined as the group of fractions of a Garside monoid.

To understand the elliptic Artin groups, which are the fundamental

groups of the complement of discriminant divisors of the semi-versal

deformation of the simply elliptic singularities E_6~, E_7~ and E_8~,

we need to consider another generalization of Artin groups.

In this talk, we will study the presentations of fundamental groups

of the complement of complexified real affine line arrangements

and consider the associated monoids.

It turns out that, in some cases, they are not Garside monoids.

Nevertheless, we will show that they satisfy the cancellation condition

and carry certain particular elements similar to the fundamental elements

in Artin monoids.

As a result, we will show that the word problem can be solved

and the center of them are determined.

### 2012/05/01

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

Minimal models, formality and hard Lefschetz property of

solvmanifolds with local systems (JAPANESE)

**Hisashi Kasuya**(The University of Tokyo)Minimal models, formality and hard Lefschetz property of

solvmanifolds with local systems (JAPANESE)

### 2012/04/24

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

Combinatorial Heegaard Floer homology (ENGLISH)

**Dylan Thurston**(Columbia University)Combinatorial Heegaard Floer homology (ENGLISH)

[ Abstract ]

Heegaard Floer homology is a powerful invariant of 3- and 4-manifolds.

In 4 dimensions, Heegaard Floer homology (together with the

Seiberg-Witten and Donaldson equations, which are conjecturally

equivalent), provides essentially the only technique for

distinguishing smooth 4-manifolds. In 3 dimensions, it provides much

geometric information, like the simplest representatives of a given

homology class.

In this talk we will focus on recent progress in making Heegaard Floer

homology more computable, including a complete algorithm for computing

it for knots.

Heegaard Floer homology is a powerful invariant of 3- and 4-manifolds.

In 4 dimensions, Heegaard Floer homology (together with the

Seiberg-Witten and Donaldson equations, which are conjecturally

equivalent), provides essentially the only technique for

distinguishing smooth 4-manifolds. In 3 dimensions, it provides much

geometric information, like the simplest representatives of a given

homology class.

In this talk we will focus on recent progress in making Heegaard Floer

homology more computable, including a complete algorithm for computing

it for knots.

### 2012/04/17

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

Pseudo-Anosov mapping classes with small dilatation (ENGLISH)

**Eriko Hironaka**(Florida State University)Pseudo-Anosov mapping classes with small dilatation (ENGLISH)

[ Abstract ]

A mapping class is a homeomorphism of an oriented surface

to itself modulo isotopy. It is pseudo-Anosov if the lengths of essential

simple closed curves under iterations of the map have exponential growth

rate. The growth rate, an algebraic integer of degree bounded with

respect to the topology of the surface, is called the dilatation of the

mapping class. In this talk we will discuss the minimization problem

for dilatations of pseudo-Anosov mapping classes, and give two general

constructions of pseudo-Anosov mapping classes with small dilatation.

A mapping class is a homeomorphism of an oriented surface

to itself modulo isotopy. It is pseudo-Anosov if the lengths of essential

simple closed curves under iterations of the map have exponential growth

rate. The growth rate, an algebraic integer of degree bounded with

respect to the topology of the surface, is called the dilatation of the

mapping class. In this talk we will discuss the minimization problem

for dilatations of pseudo-Anosov mapping classes, and give two general

constructions of pseudo-Anosov mapping classes with small dilatation.

### 2012/04/10

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

On homology of symplectic derivation Lie algebras of

the free associative algebra and the free Lie algebra (JAPANESE)

**Takuya Sakasai**(The University of Tokyo)On homology of symplectic derivation Lie algebras of

the free associative algebra and the free Lie algebra (JAPANESE)

[ Abstract ]

We discuss homology of symplectic derivation Lie algebras of

the free associative algebra and the free Lie algebra

with particular stress on their abelianizations (degree 1 part).

Then, by using a theorem of Kontsevich,

we give some applications to rational cohomology of the moduli spaces of

Riemann surfaces and metric graphs.

This is a joint work with Shigeyuki Morita and Masaaki Suzuki.

We discuss homology of symplectic derivation Lie algebras of

the free associative algebra and the free Lie algebra

with particular stress on their abelianizations (degree 1 part).

Then, by using a theorem of Kontsevich,

we give some applications to rational cohomology of the moduli spaces of

Riemann surfaces and metric graphs.

This is a joint work with Shigeyuki Morita and Masaaki Suzuki.

### 2012/02/21

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

Property (TT)/T and homomorphism superrigidity into mapping class groups (JAPANESE)

**Masato Mimura**(The University of Tokyo)Property (TT)/T and homomorphism superrigidity into mapping class groups (JAPANESE)

[ Abstract ]

Mapping class groups (MCG's), of compact oriented surfaces (possibly

with punctures), have many mysterious features: they behave not only

like higher rank lattices (namely, irreducible lattices in higher rank

algebraic groups); but also like rank one lattices. The following

theorem, the Farb--Kaimanovich--Masur superrigidity, states a rank one

phenomenon for MCG's: "every group homomorphism from higher rank

lattices (such as SL(3,Z) and cocompact lattices in SL(3,R)) into

MCG's has finite image."

In this talk, we show a generalization of the superrigidity above, to

the case where higher rank lattices are replaced with some

(non-arithmetic) matrix groups over general rings. Our main example of

such groups is called the "universal lattice", that is, the special

linear group over commutative finitely generated polynomial rings over

integers, (such as SL(3,Z[x])). To prove this, we introduce the notion

of "property (TT)/T" for groups, which is a strengthening of Kazhdan's

property (T).

We will explain these properties and relations to ordinary and bounded

cohomology of groups (with twisted unitary coefficients); and outline

the proof of our result.

Mapping class groups (MCG's), of compact oriented surfaces (possibly

with punctures), have many mysterious features: they behave not only

like higher rank lattices (namely, irreducible lattices in higher rank

algebraic groups); but also like rank one lattices. The following

theorem, the Farb--Kaimanovich--Masur superrigidity, states a rank one

phenomenon for MCG's: "every group homomorphism from higher rank

lattices (such as SL(3,Z) and cocompact lattices in SL(3,R)) into

MCG's has finite image."

In this talk, we show a generalization of the superrigidity above, to

the case where higher rank lattices are replaced with some

(non-arithmetic) matrix groups over general rings. Our main example of

such groups is called the "universal lattice", that is, the special

linear group over commutative finitely generated polynomial rings over

integers, (such as SL(3,Z[x])). To prove this, we introduce the notion

of "property (TT)/T" for groups, which is a strengthening of Kazhdan's

property (T).

We will explain these properties and relations to ordinary and bounded

cohomology of groups (with twisted unitary coefficients); and outline

the proof of our result.

### 2012/01/17

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

On the Johnson cokernels of the mapping class group of a surface (joint work with Naoya Enomoto) (JAPANESE)

**Takao Satoh**(Tokyo University of Science)On the Johnson cokernels of the mapping class group of a surface (joint work with Naoya Enomoto) (JAPANESE)

[ Abstract ]

In general, the Johnson homomorphisms of the mapping class group of a surface are used to investigate graded quotients of the Johnson filtration of the mapping class group. These graded quotients are considered as a sequence of approximations of the Torelli group. Now, there is a broad range of remarkable results for the Johnson homomorphisms.

In this talk, we concentrate our focus on the cokernels of the Johnson homomorphisms of the mapping class group. By a work of Shigeyuki Morita and Hiroaki Nakamura, it is known that an Sp-irreducible module [k] appears in the cokernel of the k-th Johnson homomorphism with multiplicity one if k=2m+1 for any positive integer m. In general, however, to determine Sp-structure of the cokernel is quite a difficult preblem.

Our goal is to show that we have detected new irreducible components in the cokernels. More precisely, we will show that there appears an Sp-irreducible module [1^k] in the cokernel of the k-th Johnson homomorphism with multiplicity one if k=4m+1 for any positive integer m.

In general, the Johnson homomorphisms of the mapping class group of a surface are used to investigate graded quotients of the Johnson filtration of the mapping class group. These graded quotients are considered as a sequence of approximations of the Torelli group. Now, there is a broad range of remarkable results for the Johnson homomorphisms.

In this talk, we concentrate our focus on the cokernels of the Johnson homomorphisms of the mapping class group. By a work of Shigeyuki Morita and Hiroaki Nakamura, it is known that an Sp-irreducible module [k] appears in the cokernel of the k-th Johnson homomorphism with multiplicity one if k=2m+1 for any positive integer m. In general, however, to determine Sp-structure of the cokernel is quite a difficult preblem.

Our goal is to show that we have detected new irreducible components in the cokernels. More precisely, we will show that there appears an Sp-irreducible module [1^k] in the cokernel of the k-th Johnson homomorphism with multiplicity one if k=4m+1 for any positive integer m.

### 2011/12/20

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

Leafwise symplectic structures on Lawson's Foliation on the 5-sphere (JAPANESE)

**Yoshihiko Mitsumatsu**(Chuo University)Leafwise symplectic structures on Lawson's Foliation on the 5-sphere (JAPANESE)

[ Abstract ]

We are going to show that Lawson's foliation on the 5-sphere

admits a smooth leafwise symplectic sturcture. Historically, Lawson's

foliation is the first one among foliations of codimension one which are

constructed on the 5-sphere. It is obtained by modifying the Milnor

fibration associated with the Fermat type cubic polynominal in three

variables.

Alberto Verjovsky proposed a question whether if the Lawson's

foliation or slighty modified ones admit a leafwise smooth symplectic

structure and/or a leafwise complex structure. As Lawson's one has a

Kodaira-Thurston nil 4-manifold as a compact leaf, the question can not

be solved simultaneously both for the symplectic and the complex cases.

The main part of the construction is to show that the Fermat type

cubic surface admits an `end-periodic' symplectic structure, while the

natural one as an affine surface is conic at the end. Even though for

the other two families of the simple elliptic hypersurface singularities

almost the same construction works, at present, it seems very limited

where a Stein manifold admits an end-periodic symplectic structure. If

the time allows, we also discuss the existence of such structures on

globally convex symplectic manifolds.

We are going to show that Lawson's foliation on the 5-sphere

admits a smooth leafwise symplectic sturcture. Historically, Lawson's

foliation is the first one among foliations of codimension one which are

constructed on the 5-sphere. It is obtained by modifying the Milnor

fibration associated with the Fermat type cubic polynominal in three

variables.

Alberto Verjovsky proposed a question whether if the Lawson's

foliation or slighty modified ones admit a leafwise smooth symplectic

structure and/or a leafwise complex structure. As Lawson's one has a

Kodaira-Thurston nil 4-manifold as a compact leaf, the question can not

be solved simultaneously both for the symplectic and the complex cases.

The main part of the construction is to show that the Fermat type

cubic surface admits an `end-periodic' symplectic structure, while the

natural one as an affine surface is conic at the end. Even though for

the other two families of the simple elliptic hypersurface singularities

almost the same construction works, at present, it seems very limited

where a Stein manifold admits an end-periodic symplectic structure. If

the time allows, we also discuss the existence of such structures on

globally convex symplectic manifolds.

### 2011/12/13

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

Remarks on filtrations of the singular homology of real varieties. (ENGLISH)

**Mircea Voineagu**(IPMU, The University of Tokyo)Remarks on filtrations of the singular homology of real varieties. (ENGLISH)

[ Abstract ]

We discuss various conjectures about filtrations on the singular homology of real and complex varieties. We prove that a conjecture relating niveau filtration on Borel-Moore homology of real varieties and the image of generalized cycle maps from reduced Lawson homology is false. In the end, we discuss a certain decomposition of Borel-Haeflinger cycle map. This is a joint work with J.Heller.

We discuss various conjectures about filtrations on the singular homology of real and complex varieties. We prove that a conjecture relating niveau filtration on Borel-Moore homology of real varieties and the image of generalized cycle maps from reduced Lawson homology is false. In the end, we discuss a certain decomposition of Borel-Haeflinger cycle map. This is a joint work with J.Heller.

### 2011/11/29

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

Mapping class group actions (ENGLISH)

**Athanase Papadopoulos**(IRMA, Univ. de Strasbourg)Mapping class group actions (ENGLISH)

[ Abstract ]

I will describe and present some rigidity results on mapping

class group actions on spaces of foliations on surfaces, equipped with various topologies.

I will describe and present some rigidity results on mapping

class group actions on spaces of foliations on surfaces, equipped with various topologies.

### 2011/11/22

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

Quantum and homological representations of braid groups (JAPANESE)

**Toshitake Kohno**(The University of Tokyo)Quantum and homological representations of braid groups (JAPANESE)

[ Abstract ]

Homological representations of braid groups are defined as

the action of homeomorphisms of a punctured disk on

the homology of an abelian covering of its configuration space.

These representations were extensively studied by Lawrence,

Krammer and Bigelow. In this talk we show that specializations

of the homological representations of braid groups

are equivalent to the monodromy of the KZ equation with

values in the space of null vectors in the tensor product

of Verma modules when the parameters are generic.

To prove this we use representations of the solutions of the

KZ equation by hypergeometric integrals due to Schechtman,

Varchenko and others.

In the case of special parameters these representations

are extended to quantum representations of mapping

class groups. We describe the images of such representations

and show that the images of any Johnson subgroups

contain non-abelian free groups if the genus and the

level are sufficiently large. The last part is a joint

work with Louis Funar.

Homological representations of braid groups are defined as

the action of homeomorphisms of a punctured disk on

the homology of an abelian covering of its configuration space.

These representations were extensively studied by Lawrence,

Krammer and Bigelow. In this talk we show that specializations

of the homological representations of braid groups

are equivalent to the monodromy of the KZ equation with

values in the space of null vectors in the tensor product

of Verma modules when the parameters are generic.

To prove this we use representations of the solutions of the

KZ equation by hypergeometric integrals due to Schechtman,

Varchenko and others.

In the case of special parameters these representations

are extended to quantum representations of mapping

class groups. We describe the images of such representations

and show that the images of any Johnson subgroups

contain non-abelian free groups if the genus and the

level are sufficiently large. The last part is a joint

work with Louis Funar.

### 2011/11/15

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

Singular codimension-one foliations

and twisted open books in dimension 3.

(joint work with G. Meigniez)

(ENGLISH)

**Francois Laudenbach**(Univ. de Nantes)Singular codimension-one foliations

and twisted open books in dimension 3.

(joint work with G. Meigniez)

(ENGLISH)

[ Abstract ]

The allowed singularities are those of functions.

According to A. Haefliger (1958),

such structures on manifolds, called $\\Gamma_1$-structures,

are objects of a cohomological

theory with a classifying space $B\\Gamma_1$.

The problem of cancelling the singularities

(or regularization problem)

arise naturally.

For a closed manifold, it was solved by W.Thurston in a famous paper

(1976), with a proof relying on Mather's isomorphism (1971):

Diff$^\\infty(\\mathbb R)$ as a discrete group has the same homology

as the based loop space

$\\Omega B\\Gamma_1^+$.

For further extension to contact geometry, it is necessary

to solve the regularization problem

without using Mather's isomorphism.

That is what we have done in dimension 3. Our result is the following.

{\\it Every $\\Gamma_1$-structure $\\xi$ on a 3-manifold $M$ whose

normal bundle

embeds into the tangent bundle to $M$ is $\\Gamma_1$-homotopic

to a regular foliation

carried by a (possibily twisted) open book.}

The proof is elementary and relies on the dynamics of a (twisted)

pseudo-gradient of $\\xi$.

All the objects will be defined in the talk, in particular the notion

of twisted open book which is a central object in the reported paper.

The allowed singularities are those of functions.

According to A. Haefliger (1958),

such structures on manifolds, called $\\Gamma_1$-structures,

are objects of a cohomological

theory with a classifying space $B\\Gamma_1$.

The problem of cancelling the singularities

(or regularization problem)

arise naturally.

For a closed manifold, it was solved by W.Thurston in a famous paper

(1976), with a proof relying on Mather's isomorphism (1971):

Diff$^\\infty(\\mathbb R)$ as a discrete group has the same homology

as the based loop space

$\\Omega B\\Gamma_1^+$.

For further extension to contact geometry, it is necessary

to solve the regularization problem

without using Mather's isomorphism.

That is what we have done in dimension 3. Our result is the following.

{\\it Every $\\Gamma_1$-structure $\\xi$ on a 3-manifold $M$ whose

normal bundle

embeds into the tangent bundle to $M$ is $\\Gamma_1$-homotopic

to a regular foliation

carried by a (possibily twisted) open book.}

The proof is elementary and relies on the dynamics of a (twisted)

pseudo-gradient of $\\xi$.

All the objects will be defined in the talk, in particular the notion

of twisted open book which is a central object in the reported paper.