## Tuesday Seminar on Topology

Seminar information archive ～12/05｜Next seminar｜Future seminars 12/06～

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

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

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

**Seminar information archive**

### 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.

### 2011/11/08

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

Fiberwise bordism groups and related topics (JAPANESE)

**Shoji Yokura**(Kagoshima University )Fiberwise bordism groups and related topics (JAPANESE)

[ Abstract ]

We have recently introduced the notion of fiberwise bordism. In this talk, after a quick review of some of the classical (co)bordism theories, we will explain motivations of considering fiberwise bordism and some results and connections with other known works, such as M. Kreck's bordism groups of orientation preserving diffeomorphisms and Emerson-Meyer's bivariant K-theory etc. An essential motivation is our recent work towards constructing a bivariant-theoretic analogue (in the sense of Fulton-MacPherson) of Levine-Morel's or Levine-Pandharipande's algebraic cobordism.

We have recently introduced the notion of fiberwise bordism. In this talk, after a quick review of some of the classical (co)bordism theories, we will explain motivations of considering fiberwise bordism and some results and connections with other known works, such as M. Kreck's bordism groups of orientation preserving diffeomorphisms and Emerson-Meyer's bivariant K-theory etc. An essential motivation is our recent work towards constructing a bivariant-theoretic analogue (in the sense of Fulton-MacPherson) of Levine-Morel's or Levine-Pandharipande's algebraic cobordism.

### 2011/11/01

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

Motivic Milnor fibers and Jordan normal forms of monodromies (JAPANESE)

**Kiyoshi Takeuchi**(University of Tsukuba)Motivic Milnor fibers and Jordan normal forms of monodromies (JAPANESE)

[ Abstract ]

We introduce a method to calculate the equivariant

Hodge-Deligne numbers of toric hypersurfaces.

Then we apply it to motivic Milnor

fibers introduced by Denef-Loeser and study the Jordan

normal forms of the local and global monodromies

of polynomials maps in various situations.

Especially we focus our attention on monodromies

at infinity studied by many people. The results will be

explicitly described by the ``convexity" of

the Newton polyhedra of polynomials. This is a joint work

with Y. Matsui and A. Esterov.

We introduce a method to calculate the equivariant

Hodge-Deligne numbers of toric hypersurfaces.

Then we apply it to motivic Milnor

fibers introduced by Denef-Loeser and study the Jordan

normal forms of the local and global monodromies

of polynomials maps in various situations.

Especially we focus our attention on monodromies

at infinity studied by many people. The results will be

explicitly described by the ``convexity" of

the Newton polyhedra of polynomials. This is a joint work

with Y. Matsui and A. Esterov.

### 2011/10/25

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

Circle-valued Morse theory for complex hyperplane arrangements (ENGLISH)

**Andrei Pajitnov**(Univ. de Nantes, Univ. of Tokyo)Circle-valued Morse theory for complex hyperplane arrangements (ENGLISH)

[ Abstract ]

Let A be a complex hyperplane arrangement

in an n-dimensional complex vector space V.

Denote by H the union of the hyperplanes

and by M the complement to H in V.

We develop the real-valued and circle-valued Morse

theory on M. We prove that if A is essential then

M has the homotopy type of a space

obtained from a finite n-dimensional

CW complex fibered over a circle,

by attaching several cells of dimension n.

We compute the Novikov homology of M and show

that its structure is similar to the

homology with generic local coefficients:

it vanishes for all dimensions except n.

This is a joint work with Toshitake Kohno.

Let A be a complex hyperplane arrangement

in an n-dimensional complex vector space V.

Denote by H the union of the hyperplanes

and by M the complement to H in V.

We develop the real-valued and circle-valued Morse

theory on M. We prove that if A is essential then

M has the homotopy type of a space

obtained from a finite n-dimensional

CW complex fibered over a circle,

by attaching several cells of dimension n.

We compute the Novikov homology of M and show

that its structure is similar to the

homology with generic local coefficients:

it vanishes for all dimensions except n.

This is a joint work with Toshitake Kohno.

### 2011/10/11

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

Making foliations of codimension one,

thirty years after Thurston's works

(ENGLISH)

**Gael Meigniez**(Univ. de Bretagne-Sud, Chuo Univ.)Making foliations of codimension one,

thirty years after Thurston's works

(ENGLISH)

[ Abstract ]

In 1976 Thurston proved that every closed manifold M whose

Euler characteristic is null carries a smooth foliation F of codimension

one. He actually established a h-principle allowing the regularization of

Haefliger structures through homotopy. I shall give some accounts of a new,

simpler proof of Thurston's result, not using Mather's homology equivalence; and also show that this proof allows to make F have dense leaves if dim M is at least 4. The emphasis will be put on the high dimensions.

In 1976 Thurston proved that every closed manifold M whose

Euler characteristic is null carries a smooth foliation F of codimension

one. He actually established a h-principle allowing the regularization of

Haefliger structures through homotopy. I shall give some accounts of a new,

simpler proof of Thurston's result, not using Mather's homology equivalence; and also show that this proof allows to make F have dense leaves if dim M is at least 4. The emphasis will be put on the high dimensions.

### 2011/10/04

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

Relatively quasiconvex subgroups of relatively hyperbolic groups (JAPANESE)

**Yoshifumi Matsuda**(The University of Tokyo)Relatively quasiconvex subgroups of relatively hyperbolic groups (JAPANESE)

[ Abstract ]

Relative hyperbolicity of groups was introduced by Gromov as a

generalization of word hyperbolicity. Motivating examples of relatively

hyperbolic groups are fundamental groups of noncompact complete

hyperbolic manifolds of finite volume. The class of relatively

quasiconvex subgroups of a realtively hyperbolic group is defined as a

genaralization of that of quasicovex subgroups of a word hyperbolic

group. The notion of hyperbolically embedded subgroups of a relatively

hyperbolic group was introduced by Osin and such groups are

characterized as relatively quasiconvex subgroups with additional

algebraic properties. In this talk I will present an introduction to

relatively quasiconvex subgroups and discuss recent joint work with Shin

-ichi Oguni and Saeko Yamagata on hyperbolically embedded subgroups.

Relative hyperbolicity of groups was introduced by Gromov as a

generalization of word hyperbolicity. Motivating examples of relatively

hyperbolic groups are fundamental groups of noncompact complete

hyperbolic manifolds of finite volume. The class of relatively

quasiconvex subgroups of a realtively hyperbolic group is defined as a

genaralization of that of quasicovex subgroups of a word hyperbolic

group. The notion of hyperbolically embedded subgroups of a relatively

hyperbolic group was introduced by Osin and such groups are

characterized as relatively quasiconvex subgroups with additional

algebraic properties. In this talk I will present an introduction to

relatively quasiconvex subgroups and discuss recent joint work with Shin

-ichi Oguni and Saeko Yamagata on hyperbolically embedded subgroups.

### 2011/09/20

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

Functorial semi-norms on singular homology (ENGLISH)

**Clara Loeh**(Univ. Regensburg)Functorial semi-norms on singular homology (ENGLISH)

[ Abstract ]

Functorial semi-norms on singular homology add metric information to

homology classes that is compatible with continuous maps. In particular,

functorial semi-norms give rise to degree theorems for certain classes

of manifolds; an invariant fitting into this context is Gromov's

simplicial volume. On the other hand, knowledge about mapping degrees

allows to construct functorial semi-norms with interesting properties;

for example, so-called inflexible simply connected manifolds give rise

to functorial semi-norms that are non-trivial on certain simply connected

spaces.

Functorial semi-norms on singular homology add metric information to

homology classes that is compatible with continuous maps. In particular,

functorial semi-norms give rise to degree theorems for certain classes

of manifolds; an invariant fitting into this context is Gromov's

simplicial volume. On the other hand, knowledge about mapping degrees

allows to construct functorial semi-norms with interesting properties;

for example, so-called inflexible simply connected manifolds give rise

to functorial semi-norms that are non-trivial on certain simply connected

spaces.

### 2011/07/12

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

The self linking number and planar open book decomposition (ENGLISH)

**Keiko Kawamuro**(University of Iowa)The self linking number and planar open book decomposition (ENGLISH)

[ Abstract ]

I will show a self linking number formula, in language of

braids, for transverse knots in contact manifolds that admit planar

open book decompositions. Our formula extends the Bennequin's for

the standar contact 3-sphere.

I will show a self linking number formula, in language of

braids, for transverse knots in contact manifolds that admit planar

open book decompositions. Our formula extends the Bennequin's for

the standar contact 3-sphere.

### 2011/07/05

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

The C*-algebra of codimension one foliations which

are almost without holonomy (ENGLISH)

**Catherine Oikonomides**(The University of Tokyo, JSPS)The C*-algebra of codimension one foliations which

are almost without holonomy (ENGLISH)

[ Abstract ]

Foliation C*-algebras have been defined abstractly by Alain Connes,

in the 1980s, as part of the theory of Noncommutative Geometry.

However, very few concrete examples of foliation C*-algebras

have been studied until now.

In this talk, we want to explain how to compute

the K-theory of the C*-algebra of codimension

one foliations which are "almost without holonomy",

meaning that the holonomy of all the noncompact leaves

of the foliation is trivial. Such foliations have a fairly

simple geometrical structure, which is well known thanks

to theorems by Imanishi, Hector and others. We will give some

concrete examples on 3-manifolds, in particular the 3-sphere

with the Reeb foliation, and also some slighty more

complicated examples.

Foliation C*-algebras have been defined abstractly by Alain Connes,

in the 1980s, as part of the theory of Noncommutative Geometry.

However, very few concrete examples of foliation C*-algebras

have been studied until now.

In this talk, we want to explain how to compute

the K-theory of the C*-algebra of codimension

one foliations which are "almost without holonomy",

meaning that the holonomy of all the noncompact leaves

of the foliation is trivial. Such foliations have a fairly

simple geometrical structure, which is well known thanks

to theorems by Imanishi, Hector and others. We will give some

concrete examples on 3-manifolds, in particular the 3-sphere

with the Reeb foliation, and also some slighty more

complicated examples.

### 2011/06/28

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

On a Sebastiani-Thom theorem for directed Fukaya categories (JAPANESE)

**Masahiro Futaki**(The University of Tokyo)On a Sebastiani-Thom theorem for directed Fukaya categories (JAPANESE)

[ Abstract ]

The directed Fukaya category defined by Seidel is a "

categorification" of the Milnor lattice of hypersurface singularities.

Sebastiani-Thom showed that the Milnor lattice and its monodromy behave

as tensor product for the sum of singularities. A directed Fukaya

category version of this theorem was conjectured by Auroux-Katzarkov-

Orlov (and checked for the Landau-Ginzburg mirror of P^1 \\times P^1). In

this talk I introduce the directed Fukaya category and show that a

Sebastiani-Thom type splitting holds in the case that one of the

potential is of complex dimension 1.

The directed Fukaya category defined by Seidel is a "

categorification" of the Milnor lattice of hypersurface singularities.

Sebastiani-Thom showed that the Milnor lattice and its monodromy behave

as tensor product for the sum of singularities. A directed Fukaya

category version of this theorem was conjectured by Auroux-Katzarkov-

Orlov (and checked for the Landau-Ginzburg mirror of P^1 \\times P^1). In

this talk I introduce the directed Fukaya category and show that a

Sebastiani-Thom type splitting holds in the case that one of the

potential is of complex dimension 1.

### 2011/06/14

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

Donaldson-Tian-Yau's Conjecture (JAPANESE)

**Toshiki Mabuchi**(Osaka University)Donaldson-Tian-Yau's Conjecture (JAPANESE)

[ Abstract ]

For polarized algebraic manifolds, the concept of K-stability

introduced by Tian and Donaldson is conjecturally strongly correlated

to the existence of constant scalar curvature metrics (or more

generally extremal K\\"ahler metrics) in the polarization class. This is

known as Donaldson-Tian-Yau's conjecture. Recently, a remarkable

progress has been made by many authors toward its solution. In this

talk, I'll discuss the topic mainly with emphasis on the existence

part of the conjecture.

For polarized algebraic manifolds, the concept of K-stability

introduced by Tian and Donaldson is conjecturally strongly correlated

to the existence of constant scalar curvature metrics (or more

generally extremal K\\"ahler metrics) in the polarization class. This is

known as Donaldson-Tian-Yau's conjecture. Recently, a remarkable

progress has been made by many authors toward its solution. In this

talk, I'll discuss the topic mainly with emphasis on the existence

part of the conjecture.

### 2011/06/07

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

Rigidity of group actions via invariant geometric structures (JAPANESE)

**Masahiko Kanai**(The University of Tokyo)Rigidity of group actions via invariant geometric structures (JAPANESE)

[ Abstract ]

It is a homomorphism into a FINITE dimensional Lie group that is concerned with in the classical RIGIDITY theorems such as those of Mostow and Margulis. In the meantime, differentiable GROUP ACTIONS for which we ask rigidity problems is a homomorphism into a diffeomorphism group, which is a typical example of INFINITE dimensional Lie groups. The purpose of the present talk is exhibiting several rigidity theorems for group actions in which I have been involved for years. Although quite a few fields of mathematics, such as ergodic theory, the theory of smooth dynamical systems, representation theory and so on, have made remarkable contributions to rigidity problems, I would rather emphasis geometric aspects: I would focus on those rigidity phenomenon for group actions that are observed by showing that the actions have INVARIANT GEOMETRIC STRUCTURES.

It is a homomorphism into a FINITE dimensional Lie group that is concerned with in the classical RIGIDITY theorems such as those of Mostow and Margulis. In the meantime, differentiable GROUP ACTIONS for which we ask rigidity problems is a homomorphism into a diffeomorphism group, which is a typical example of INFINITE dimensional Lie groups. The purpose of the present talk is exhibiting several rigidity theorems for group actions in which I have been involved for years. Although quite a few fields of mathematics, such as ergodic theory, the theory of smooth dynamical systems, representation theory and so on, have made remarkable contributions to rigidity problems, I would rather emphasis geometric aspects: I would focus on those rigidity phenomenon for group actions that are observed by showing that the actions have INVARIANT GEOMETRIC STRUCTURES.

### 2011/05/31

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

Measures with maximum total exponent and generic properties of $C^

{1}$ expanding maps (JAPANESE)

**Takehiko Morita**(Osaka University)Measures with maximum total exponent and generic properties of $C^

{1}$ expanding maps (JAPANESE)

[ Abstract ]

This is a joint work with Yusuke Tokunaga. Let $M$ be an $N$

dimensional compact connected smooth Riemannian manifold without

boundary and let $\\mathcal{E}^{r}(M,M)$ be the space of $C^{r}$

expandig maps endowed with $C^{r}$ topology. We show that

each of the following properties for element $T$ in $\\mathcal{E}

^{1}(M,M)$ is generic.

\\begin{itemize}

\\item[(1)] $T$ has a unique measure with maximum total exponent.

\\item[(2)] Any measure with maximum total exponent for $T$ has

zero entropy.

\\item[(3)] Any measure with maximum total exponent for $T$ is

fully supported.

\\end{itemize}

On the contrary, we show that for $r\\ge 2$, a generic element

in $\\mathcal{E}^{r}(M,M)$ has no fully supported measures with

maximum total exponent.

This is a joint work with Yusuke Tokunaga. Let $M$ be an $N$

dimensional compact connected smooth Riemannian manifold without

boundary and let $\\mathcal{E}^{r}(M,M)$ be the space of $C^{r}$

expandig maps endowed with $C^{r}$ topology. We show that

each of the following properties for element $T$ in $\\mathcal{E}

^{1}(M,M)$ is generic.

\\begin{itemize}

\\item[(1)] $T$ has a unique measure with maximum total exponent.

\\item[(2)] Any measure with maximum total exponent for $T$ has

zero entropy.

\\item[(3)] Any measure with maximum total exponent for $T$ is

fully supported.

\\end{itemize}

On the contrary, we show that for $r\\ge 2$, a generic element

in $\\mathcal{E}^{r}(M,M)$ has no fully supported measures with

maximum total exponent.

### 2011/05/24

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

Minimal Stratifications for Line Arrangements (JAPANESE)

**Masahiko Yoshinaga**(Kyoto University)Minimal Stratifications for Line Arrangements (JAPANESE)

[ Abstract ]

The homotopy type of complements of complex

hyperplane arrangements have a special property,

so called minimality (Dimca-Papadima and Randell,

around 2000). Since then several approaches based

on (continuous, discrete) Morse theory have appeared.

In this talk, we introduce the "dual" object, which we

call minimal stratification for real two dimensional cases.

A merit is that the minimal stratification can be explicitly

described in terms of semi-algebraic sets.

We also see associated presentation of the fundamental group.

The homotopy type of complements of complex

hyperplane arrangements have a special property,

so called minimality (Dimca-Papadima and Randell,

around 2000). Since then several approaches based

on (continuous, discrete) Morse theory have appeared.

In this talk, we introduce the "dual" object, which we

call minimal stratification for real two dimensional cases.

A merit is that the minimal stratification can be explicitly

described in terms of semi-algebraic sets.

We also see associated presentation of the fundamental group.

### 2011/05/17

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

Quandle colorings with non-commutative flows (JAPANESE)

**Atsushi Ishii**(University of Tsukuba)Quandle colorings with non-commutative flows (JAPANESE)

[ Abstract ]

This is a joint work with Masahide Iwakiri, Yeonhee Jang and Kanako Oshiro.

We introduce quandle coloring invariants and quandle cocycle invariants

with non-commutative flows for knots, spatial graphs, handlebody-knots,

where a handlebody-knot is a handlebody embedded in the $3$-sphere.

Two handlebody-knots are equivalent if one can be transformed into the

other by an isotopy of $S^3$.

The quandle coloring (resp. cocycle) invariant is a ``twisted'' quandle

coloring (resp. cocycle) invariant.

This is a joint work with Masahide Iwakiri, Yeonhee Jang and Kanako Oshiro.

We introduce quandle coloring invariants and quandle cocycle invariants

with non-commutative flows for knots, spatial graphs, handlebody-knots,

where a handlebody-knot is a handlebody embedded in the $3$-sphere.

Two handlebody-knots are equivalent if one can be transformed into the

other by an isotopy of $S^3$.

The quandle coloring (resp. cocycle) invariant is a ``twisted'' quandle

coloring (resp. cocycle) invariant.

### 2011/05/10

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

Isotated points in the space of group left orderings (JAPANESE)

**Tetsuya Ito**(The University of Tokyo)Isotated points in the space of group left orderings (JAPANESE)

[ Abstract ]

The set of all left orderings of a group G admits a natural

topology. In general the space of left orderings is homeomorphic to the

union of Cantor set and finitely many isolated points. In this talk I

will give a new method to construct left orderings corresponding to

isolated points, and will explain how such isolated orderings reflect

the structures of groups.

The set of all left orderings of a group G admits a natural

topology. In general the space of left orderings is homeomorphic to the

union of Cantor set and finitely many isolated points. In this talk I

will give a new method to construct left orderings corresponding to

isolated points, and will explain how such isolated orderings reflect

the structures of groups.

### 2011/04/26

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

Topological Blow-up (JAPANESE)

**Taro Yoshino**(The University of Tokyo)Topological Blow-up (JAPANESE)

[ Abstract ]

Suppose that a Lie group $G$ acts on a manifold

$M$. The quotient space $X:=G\\backslash M$ is locally compact,

but not Hausdorff in general. Our aim is to understand

such a non-Hausdorff space $X$.

The space $X$ has the crack $S$. Roughly speaking, $S$ is

the causal subset of non-Hausdorffness of $X$, and especially

$X\\setminus S$ is Hausdorff.

We introduce the concept of `topological blow-up' as a `repair'

of the crack. The `repaired' space $\\tilde{X}$ is

locally compact and Hausdorff space containing $X\\setminus S$

as its open subset. Moreover, the original space $X$ can be

recovered from the pair of $(\\tilde{X}, S)$.

Suppose that a Lie group $G$ acts on a manifold

$M$. The quotient space $X:=G\\backslash M$ is locally compact,

but not Hausdorff in general. Our aim is to understand

such a non-Hausdorff space $X$.

The space $X$ has the crack $S$. Roughly speaking, $S$ is

the causal subset of non-Hausdorffness of $X$, and especially

$X\\setminus S$ is Hausdorff.

We introduce the concept of `topological blow-up' as a `repair'

of the crack. The `repaired' space $\\tilde{X}$ is

locally compact and Hausdorff space containing $X\\setminus S$

as its open subset. Moreover, the original space $X$ can be

recovered from the pair of $(\\tilde{X}, S)$.

### 2011/04/12

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

On diffeomorphisms over non-orientable surfaces embedded in the 4-sphere (JAPANESE)

**Susumu Hirose**(Tokyo University of Science)On diffeomorphisms over non-orientable surfaces embedded in the 4-sphere (JAPANESE)

[ Abstract ]

For a closed orientable surface standardly embedded in the 4-sphere,

it was known that a diffeomorphism over this surface is extendable to

the 4-sphere if and only if this diffeomorphism preserves

the Rokhlin quadratic form of this surafce.

In this talk, we will explain an approach to the same kind of problem for

closed non-orientable surfaces.

For a closed orientable surface standardly embedded in the 4-sphere,

it was known that a diffeomorphism over this surface is extendable to

the 4-sphere if and only if this diffeomorphism preserves

the Rokhlin quadratic form of this surafce.

In this talk, we will explain an approach to the same kind of problem for

closed non-orientable surfaces.