## Tuesday Seminar on Topology

Seminar information archive ～09/24｜Next seminar｜Future seminars 09/25～

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

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

**Seminar information archive**

### 2014/06/10

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

On relation between the Milnor's $¥mu$-invariant and HOMFLYPT

polynomial (JAPANESE)

**Yuka Kotorii**(The University of Tokyo)On relation between the Milnor's $¥mu$-invariant and HOMFLYPT

polynomial (JAPANESE)

[ Abstract ]

Milnor introduced a family of invariants for ordered oriented link,

called $¥bar{¥mu}$-invariants. Polyak showed a relation between the $¥

bar{¥mu}$-invariant of length 3 sequence and Conway polynomial.

Moreover, Habegger-Lin showed that Milnor's invariants are invariants of

string link, called $¥mu$-invariants. We show that any $¥mu$-invariant

of length $¥leq k$ can be represented as a combination of HOMFLYPT

polynomials if all $¥mu$-invariant of length $¥leq k-2$ vanish.

This result is an extension of Polyak's result.

Milnor introduced a family of invariants for ordered oriented link,

called $¥bar{¥mu}$-invariants. Polyak showed a relation between the $¥

bar{¥mu}$-invariant of length 3 sequence and Conway polynomial.

Moreover, Habegger-Lin showed that Milnor's invariants are invariants of

string link, called $¥mu$-invariants. We show that any $¥mu$-invariant

of length $¥leq k$ can be represented as a combination of HOMFLYPT

polynomials if all $¥mu$-invariant of length $¥leq k-2$ vanish.

This result is an extension of Polyak's result.

### 2014/06/03

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

Vector partition functions and the topology of multiple weight varieties

(JAPANESE)

**Tatsuru Takakura**(Chuo University)Vector partition functions and the topology of multiple weight varieties

(JAPANESE)

[ Abstract ]

A multiple weight variety is a symplectic quotient of a direct product

of several coadjoint orbits of a compact Lie group $G$, with respect to

the diagonal action of the maximal torus. Its geometry and topology are

closely related to the combinatorics concerned with the weight space

decomposition of a tensor product of irreducible representations of $G$.

For example, when considering the Riemann-Roch index, we are naturally

lead to the study of vector partition functions with multiplicities.

In this talk, we discuss some formulas for vector partition functions,

especially a generalization of the formula of Brion-Vergne. Then, by

using

them, we investigate the structure of the cohomology of certain multiple

weight varieties of type $A$ in detail.

A multiple weight variety is a symplectic quotient of a direct product

of several coadjoint orbits of a compact Lie group $G$, with respect to

the diagonal action of the maximal torus. Its geometry and topology are

closely related to the combinatorics concerned with the weight space

decomposition of a tensor product of irreducible representations of $G$.

For example, when considering the Riemann-Roch index, we are naturally

lead to the study of vector partition functions with multiplicities.

In this talk, we discuss some formulas for vector partition functions,

especially a generalization of the formula of Brion-Vergne. Then, by

using

them, we investigate the structure of the cohomology of certain multiple

weight varieties of type $A$ in detail.

### 2014/05/27

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

The Teichmuller space and the stable quasiconformal mapping class group for a Riemann surface of infinite type (JAPANESE)

**Ege Fujikawa**(Chiba University)The Teichmuller space and the stable quasiconformal mapping class group for a Riemann surface of infinite type (JAPANESE)

[ Abstract ]

We explain recent developments of the theory of infinite dimensional Teichmuller space. In particular, we observe the dynamics of the orbits by the action of the stable quasiconformal mapping class group on the Teichmuller space and consider the relationship with the asymptotic Teichmuller space. We also introduce the generalized fixed point theorem and the Nielsen realization theorem. Furthermore, we investigate the moduli space of Riemann surface of infinite type.

We explain recent developments of the theory of infinite dimensional Teichmuller space. In particular, we observe the dynamics of the orbits by the action of the stable quasiconformal mapping class group on the Teichmuller space and consider the relationship with the asymptotic Teichmuller space. We also introduce the generalized fixed point theorem and the Nielsen realization theorem. Furthermore, we investigate the moduli space of Riemann surface of infinite type.

### 2014/05/20

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

An application of torus graphs to characterize torus manifolds

with extended actions (JAPANESE)

**Shintaro Kuroki**(The Univeristy of Tokyo)An application of torus graphs to characterize torus manifolds

with extended actions (JAPANESE)

[ Abstract ]

A torus manifold is a compact, oriented 2n-dimensional T^n-

manifolds with fixed points. This notion is introduced by Hattori and

Masuda as a topological generalization of toric manifolds. For a given

torus manifold, we can define a labelled graph called a torus graph (

this may be regarded as a generalization of some class of GKM graphs).

It is known that the equivariant cohomology ring of some nice class of

torus manifolds can be computed by using a combinatorial data of torus

graphs. In this talk, we study which torus action of torus manifolds can

be extended to a non-abelian compact connected Lie group. To do this, we

introduce root systems of (abstract) torus graphs and characterize

extended actions of torus manifolds. This is a joint work with Mikiya

Masuda.

A torus manifold is a compact, oriented 2n-dimensional T^n-

manifolds with fixed points. This notion is introduced by Hattori and

Masuda as a topological generalization of toric manifolds. For a given

torus manifold, we can define a labelled graph called a torus graph (

this may be regarded as a generalization of some class of GKM graphs).

It is known that the equivariant cohomology ring of some nice class of

torus manifolds can be computed by using a combinatorial data of torus

graphs. In this talk, we study which torus action of torus manifolds can

be extended to a non-abelian compact connected Lie group. To do this, we

introduce root systems of (abstract) torus graphs and characterize

extended actions of torus manifolds. This is a joint work with Mikiya

Masuda.

### 2014/05/13

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

Transverse projective structures of foliations and deformations of the Godbillon-Vey class (JAPANESE)

**Taro Asuke**(The University of Tokyo)Transverse projective structures of foliations and deformations of the Godbillon-Vey class (JAPANESE)

[ Abstract ]

Given a smooth family of foliations, we can define the derivative of the Godbillon-Vey class

with respect to the family. The derivative is known to be represented in terms of the projective

Schwarzians of holonomy maps. In this talk, we will study transverse projective structures

and connections, and show that the derivative is in fact determined by the projective structure

and the family.

Given a smooth family of foliations, we can define the derivative of the Godbillon-Vey class

with respect to the family. The derivative is known to be represented in terms of the projective

Schwarzians of holonomy maps. In this talk, we will study transverse projective structures

and connections, and show that the derivative is in fact determined by the projective structure

and the family.

### 2014/04/15

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

On the rational string operations of classifying spaces and the

Hochschild cohomology (JAPANESE)

**Takahito Naito**(The University of Tokyo)On the rational string operations of classifying spaces and the

Hochschild cohomology (JAPANESE)

[ Abstract ]

Chataur and Menichi initiated the theory of string topology of

classifying spaces.

In particular, the cohomology of the free loop space of a classifying

space is endowed with a product

called the dual loop coproduct. In this talk, I will discuss the

algebraic structure and relate the rational dual loop coproduct to the

cup product on the Hochschild cohomology via the Van den Bergh isomorphism.

Chataur and Menichi initiated the theory of string topology of

classifying spaces.

In particular, the cohomology of the free loop space of a classifying

space is endowed with a product

called the dual loop coproduct. In this talk, I will discuss the

algebraic structure and relate the rational dual loop coproduct to the

cup product on the Hochschild cohomology via the Van den Bergh isomorphism.

### 2014/04/08

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

On the number of commensurable fibrations on a hyperbolic 3-manifold. (JAPANESE)

**Hidetoshi Masai**(The University of Tokyo)On the number of commensurable fibrations on a hyperbolic 3-manifold. (JAPANESE)

[ Abstract ]

By work of Thurston, it is known that if a hyperbolic fibred

$3$-manifold $M$ has Betti number greater than 1, then

$M$ admits infinitely many distinct fibrations.

For any fibration $\\omega$ on a hyperbolic $3$-manifold $M$,

the number of fibrations on $M$ that are commensurable in the sense of

Calegari-Sun-Wang to $\\omega$ is known to be finite.

In this talk, we prove that the number can be arbitrarily large.

By work of Thurston, it is known that if a hyperbolic fibred

$3$-manifold $M$ has Betti number greater than 1, then

$M$ admits infinitely many distinct fibrations.

For any fibration $\\omega$ on a hyperbolic $3$-manifold $M$,

the number of fibrations on $M$ that are commensurable in the sense of

Calegari-Sun-Wang to $\\omega$ is known to be finite.

In this talk, we prove that the number can be arbitrarily large.

### 2014/01/21

16:30-17:30 Room #056 (Graduate School of Math. Sci. Bldg.)

On contact submanifolds of the odd dimensional Euclidean space (JAPANESE)

**Naohiko Kasuya**(The University of Tokyo)On contact submanifolds of the odd dimensional Euclidean space (JAPANESE)

[ Abstract ]

We prove that the Chern class of a closed contact manifold is an

obstruction for codimension two contact embeddings in the odd

dimensional Euclidean space.

By Gromov's h-principle,

for any closed contact $3$-manifold with trivial first Chern class,

there is a contact structure on $\\mathbb{R}^5$ which admits a contact

embedding.

We prove that the Chern class of a closed contact manifold is an

obstruction for codimension two contact embeddings in the odd

dimensional Euclidean space.

By Gromov's h-principle,

for any closed contact $3$-manifold with trivial first Chern class,

there is a contact structure on $\\mathbb{R}^5$ which admits a contact

embedding.

### 2014/01/21

17:30-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Weak eigenvalues in homoclinic classes: perturbations from saddles

with small angles (ENGLISH)

**Xiaolong Li**(The University of Tokyo)Weak eigenvalues in homoclinic classes: perturbations from saddles

with small angles (ENGLISH)

[ Abstract ]

For 3-dimensional homoclinic classes of saddles with index 2, a

new sufficient condition for creating weak contracting eigenvalues is

provided. Our perturbation makes use of small angles between stable and

unstable subspaces of saddles. In particular, by recovering the unstable

eigenvector, we can designate that the newly created weak eigenvalue is

contracting. As applications, we obtain C^1-generic non-trivial index-

intervals of homoclinic classes and the C^1-approximation of robust

heterodimensional cycles. In particular, this sufficient condition is

satisfied by a substantial class of saddles with homoclinic tangencies.

For 3-dimensional homoclinic classes of saddles with index 2, a

new sufficient condition for creating weak contracting eigenvalues is

provided. Our perturbation makes use of small angles between stable and

unstable subspaces of saddles. In particular, by recovering the unstable

eigenvector, we can designate that the newly created weak eigenvalue is

contracting. As applications, we obtain C^1-generic non-trivial index-

intervals of homoclinic classes and the C^1-approximation of robust

heterodimensional cycles. In particular, this sufficient condition is

satisfied by a substantial class of saddles with homoclinic tangencies.

### 2014/01/14

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

State-integral partition functions on shaped triangulations (ENGLISH)

**Rinat Kashaev**(University of Geneva)State-integral partition functions on shaped triangulations (ENGLISH)

[ Abstract ]

Quantum Teichm\\"uller theory can be promoted to a

generalized TQFT within the combinatorial framework of shaped

triangulations with the tetrahedral weight functions given in

terms of the Weil-Gelfand-Zak transformation of Faddeev.FN"s

quantum dilogarithm. By using simple examples, I will

illustrate the connection of this theory with the hyperbolic

geometry in three dimensions.

Quantum Teichm\\"uller theory can be promoted to a

generalized TQFT within the combinatorial framework of shaped

triangulations with the tetrahedral weight functions given in

terms of the Weil-Gelfand-Zak transformation of Faddeev.FN"s

quantum dilogarithm. By using simple examples, I will

illustrate the connection of this theory with the hyperbolic

geometry in three dimensions.

### 2013/12/24

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

Stable homotopy type for monopole Floer homology (ENGLISH)

**Tirasan Khandhawit**(Kavli IPMU)Stable homotopy type for monopole Floer homology (ENGLISH)

[ Abstract ]

In this talk, I will try to give an overview of the

construction of stable homotopy type for monopole Floer homology. The

construction associates a stable homotopy object to 3-manifolds, which

will recover the Floer groups by appropriate homology theory. The main

ingredients are finite dimensional approximation technique and Conley

index theory. In addition, I will demonstrate construction for certain

3-manifolds such as the 3-torus.

In this talk, I will try to give an overview of the

construction of stable homotopy type for monopole Floer homology. The

construction associates a stable homotopy object to 3-manifolds, which

will recover the Floer groups by appropriate homology theory. The main

ingredients are finite dimensional approximation technique and Conley

index theory. In addition, I will demonstrate construction for certain

3-manifolds such as the 3-torus.

### 2013/12/17

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

Satellites of an oriented surface link and their local moves (JAPANESE)

**Inasa Nakamura**(The University of Tokyo)Satellites of an oriented surface link and their local moves (JAPANESE)

[ Abstract ]

For an oriented surface link $F$ in $\\mathbb{R}^4$,

we consider a satellite construction of a surface link, called a

2-dimensional braid over $F$, which is in the form of a covering over

$F$. We introduce the notion of an $m$-chart on a surface diagram

$p(F)\\subset \\mathbb{R}^3$ of $F$, which is a finite graph on $p(F)$

satisfying certain conditions and is an extended notion of an

$m$-chart on a 2-disk presenting a surface braid.

A 2-dimensional braid over $F$ is presented by an $m$-chart on $p(F)$.

It is known that two surface links are equivalent if and only if their

surface diagrams are related by a finite sequence of ambient isotopies

of $\\mathbb{R}^3$ and local moves called Roseman moves.

We show that Roseman moves for surface diagrams with $m$-charts can be

well-defined. Further, we give some applications.

For an oriented surface link $F$ in $\\mathbb{R}^4$,

we consider a satellite construction of a surface link, called a

2-dimensional braid over $F$, which is in the form of a covering over

$F$. We introduce the notion of an $m$-chart on a surface diagram

$p(F)\\subset \\mathbb{R}^3$ of $F$, which is a finite graph on $p(F)$

satisfying certain conditions and is an extended notion of an

$m$-chart on a 2-disk presenting a surface braid.

A 2-dimensional braid over $F$ is presented by an $m$-chart on $p(F)$.

It is known that two surface links are equivalent if and only if their

surface diagrams are related by a finite sequence of ambient isotopies

of $\\mathbb{R}^3$ and local moves called Roseman moves.

We show that Roseman moves for surface diagrams with $m$-charts can be

well-defined. Further, we give some applications.

### 2013/12/10

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

Corks, plugs, and local moves of 4-manifolds. (JAPANESE)

**Motoo Tange**(University of Tsukuba)Corks, plugs, and local moves of 4-manifolds. (JAPANESE)

[ Abstract ]

Akbulut and Yasui defined cork, and plug

to produce many exotic pairs.

In this talk, we introduce a plug

with respect to Fintushel-Stern's knot surgery

or more 4-dimensional local moves and

and argue by using Heegaard Fleor theory.

Akbulut and Yasui defined cork, and plug

to produce many exotic pairs.

In this talk, we introduce a plug

with respect to Fintushel-Stern's knot surgery

or more 4-dimensional local moves and

and argue by using Heegaard Fleor theory.

### 2013/12/03

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

Hyperbolic four-manifolds with one cusp (cancelled) (JAPANESE)

**Bruno Martelli**(Univ. di Pisa)Hyperbolic four-manifolds with one cusp (cancelled) (JAPANESE)

[ Abstract ]

(joint work with A. Kolpakov)

We introduce a simple algorithm which transforms every

four-dimensional cubulation into a cusped finite-volume hyperbolic

four-manifold. Combinatorially distinct cubulations give rise to

topologically distinct manifolds. Using this algorithm we construct

the first examples of finite-volume hyperbolic four-manifolds with one

cusp. More generally, we show that the number of k-cusped hyperbolic

four-manifolds with volume smaller than V grows like C^{V log V} for

any fixed k. As a corollary, we deduce that the 3-torus bounds

geometrically a hyperbolic manifold.

(joint work with A. Kolpakov)

We introduce a simple algorithm which transforms every

four-dimensional cubulation into a cusped finite-volume hyperbolic

four-manifold. Combinatorially distinct cubulations give rise to

topologically distinct manifolds. Using this algorithm we construct

the first examples of finite-volume hyperbolic four-manifolds with one

cusp. More generally, we show that the number of k-cusped hyperbolic

four-manifolds with volume smaller than V grows like C^{V log V} for

any fixed k. As a corollary, we deduce that the 3-torus bounds

geometrically a hyperbolic manifold.

### 2013/11/26

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

Rational elliptic surfaces and certain line-conic arrangements (JAPANESE)

**Hiroo Tokunaga**(Tokyo Metropolitan University)Rational elliptic surfaces and certain line-conic arrangements (JAPANESE)

[ Abstract ]

Let S be a rational elliptic surface. The generic

fiber of S can be considered as an elliptic curve over

the rational function field of one variable. We can make

use of its group structure in order to cook up a curve C_2 on

S from a given section C_1.

In this talk, we consider certain line-conic arrangements of

degree 7 based on this method.

Let S be a rational elliptic surface. The generic

fiber of S can be considered as an elliptic curve over

the rational function field of one variable. We can make

use of its group structure in order to cook up a curve C_2 on

S from a given section C_1.

In this talk, we consider certain line-conic arrangements of

degree 7 based on this method.

### 2013/11/19

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

Minimal $C^1$-diffeomorphisms of the circle which admit

measurable fundamental domains (JAPANESE)

**Hiroki Kodama**(The University of Tokyo)Minimal $C^1$-diffeomorphisms of the circle which admit

measurable fundamental domains (JAPANESE)

[ Abstract ]

We construct, for each irrational number $\\alpha$, a minimal

$C^1$-diffeomorphism of the circle with rotation number $\\alpha$

which admits a measurable fundamental domain with respect to

the Lebesgue measure.

This is a joint work with Shigenori Matsumoto (Nihon University).

We construct, for each irrational number $\\alpha$, a minimal

$C^1$-diffeomorphism of the circle with rotation number $\\alpha$

which admits a measurable fundamental domain with respect to

the Lebesgue measure.

This is a joint work with Shigenori Matsumoto (Nihon University).

### 2013/11/12

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

The Batalin-Vilkovisky Formalism and Cohomology of Moduli Spaces (ENGLISH)

**Alexander Voronov**(University of Minnesota)The Batalin-Vilkovisky Formalism and Cohomology of Moduli Spaces (ENGLISH)

[ Abstract ]

We use the Batalin-Vilkovisky formalism to give a new proof of Costello's theorem on the existence and uniqueness of solution to the Quantum Master Equation. We also make a physically motivated conjecture on the rational homology of moduli spaces. This is a joint work with Domenico D'Alessandro.

We use the Batalin-Vilkovisky formalism to give a new proof of Costello's theorem on the existence and uniqueness of solution to the Quantum Master Equation. We also make a physically motivated conjecture on the rational homology of moduli spaces. This is a joint work with Domenico D'Alessandro.

### 2013/11/05

16:30-18:00 Room #123 (Graduate School of Math. Sci. Bldg.)

The isotopy problem of non-singular closed 1-forms. (ENGLISH)

**Carlos Moraga Ferrandiz**(The University of Tokyo, JSPS)The isotopy problem of non-singular closed 1-forms. (ENGLISH)

[ Abstract ]

Given alpha_0, alpha_1 two cohomologous non-singular closed 1-forms of a compact manifold M, are they always isotopic? We expect a negative answer to this question, at least in high dimensions by the work of Laudenbach, as well as an obstruction living in the algebraic K-theory of the Novikov ring associated to the underlying cohomology class.

A similar problem for functions N x [0,1] --> [0,1] without critical points was treated by Hatcher and Wagoner in the 70s.

The first goal of this talk is to explain how we can carry a part of the strategy of Hatcher and Wagoner into the context of closed 1-forms and to indicate the main difficulties that appear by doing so. The second goal is to show the techniques to treat this difficulties and the progress in defining the expected obstruction.

Given alpha_0, alpha_1 two cohomologous non-singular closed 1-forms of a compact manifold M, are they always isotopic? We expect a negative answer to this question, at least in high dimensions by the work of Laudenbach, as well as an obstruction living in the algebraic K-theory of the Novikov ring associated to the underlying cohomology class.

A similar problem for functions N x [0,1] --> [0,1] without critical points was treated by Hatcher and Wagoner in the 70s.

The first goal of this talk is to explain how we can carry a part of the strategy of Hatcher and Wagoner into the context of closed 1-forms and to indicate the main difficulties that appear by doing so. The second goal is to show the techniques to treat this difficulties and the progress in defining the expected obstruction.

### 2013/10/29

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

Fundamental groups of algebraic varieties (ENGLISH)

**Daniel Matei**(IMAR, Bucharest)Fundamental groups of algebraic varieties (ENGLISH)

[ Abstract ]

We discuss restrictions imposed by the complex

structure on fundamental groups of quasi-projective

algebraic varieties with mild singularities.

We investigate quasi-projectivity of various geometric

classes of finitely presented groups.

We discuss restrictions imposed by the complex

structure on fundamental groups of quasi-projective

algebraic varieties with mild singularities.

We investigate quasi-projectivity of various geometric

classes of finitely presented groups.

### 2013/10/22

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

Cluster algebra and complex volume of knots (JAPANESE)

**Rei Inoue**(Chiba University)Cluster algebra and complex volume of knots (JAPANESE)

[ Abstract ]

The cluster algebra was introduced by Fomin and Zelevinsky around

2000. The characteristic operation in the algebra called `mutation' is

related to various notions in mathematics and mathematical physics. In

this talk I review a basics of the cluster algebra, and introduce its

application to study the complex volume of knot complements in S^3.

Here a mutation corresponds to an ideal tetrahedron.

This talk is based on joint work with Kazuhiro Hikami (Kyushu University).

The cluster algebra was introduced by Fomin and Zelevinsky around

2000. The characteristic operation in the algebra called `mutation' is

related to various notions in mathematics and mathematical physics. In

this talk I review a basics of the cluster algebra, and introduce its

application to study the complex volume of knot complements in S^3.

Here a mutation corresponds to an ideal tetrahedron.

This talk is based on joint work with Kazuhiro Hikami (Kyushu University).

### 2013/10/15

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

Desingularizing special generic maps (JAPANESE)

**Masamichi Takase**(Seikei University)Desingularizing special generic maps (JAPANESE)

[ Abstract ]

This is a joint work with Osamu Saeki (IMI, Kyushu University).

A special generic map is a generic map which has only definite

fold as its singularities.

We study the condition for a special generic map from a closed

n-manifold to the p-space (n+1>p), to factor through a codimension

one immersion (or an embedding). In particular, for the cases

where p = 1 and 2 we obtain complete results.

Our techniques are related to Smale-Hirsch theory,

topology of the space of immersions, relation between the space

of topological immersions and that of smooth immersions,

sphere eversions, differentiable structures of homotopy spheres,

diffeomorphism group of spheres, free group actions on the sphere, etc.

This is a joint work with Osamu Saeki (IMI, Kyushu University).

A special generic map is a generic map which has only definite

fold as its singularities.

We study the condition for a special generic map from a closed

n-manifold to the p-space (n+1>p), to factor through a codimension

one immersion (or an embedding). In particular, for the cases

where p = 1 and 2 we obtain complete results.

Our techniques are related to Smale-Hirsch theory,

topology of the space of immersions, relation between the space

of topological immersions and that of smooth immersions,

sphere eversions, differentiable structures of homotopy spheres,

diffeomorphism group of spheres, free group actions on the sphere, etc.

### 2013/10/08

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

An invariant of rational homology 3-spheres via vector fields. (JAPANESE)

**Tatsuro Shimizu**(The Univesity of Tokyo)An invariant of rational homology 3-spheres via vector fields. (JAPANESE)

[ Abstract ]

In this talk, we define an invariant of rational homology 3-spheres with

values in a space $\\mathcal A(\\emptyset)$ of Jacobi diagrams by using

vector fields.

The construction of our invariant is a generalization of both that of

the Kontsevich-Kuperberg-Thurston invariant $z^{KKT}$

and that of Fukaya and Watanabe's Morse homotopy invariant $z^{FW}$.

As an application of our invariant, we prove that $z^{KKT}=z^{FW}$ for

integral homology 3-spheres.

In this talk, we define an invariant of rational homology 3-spheres with

values in a space $\\mathcal A(\\emptyset)$ of Jacobi diagrams by using

vector fields.

The construction of our invariant is a generalization of both that of

the Kontsevich-Kuperberg-Thurston invariant $z^{KKT}$

and that of Fukaya and Watanabe's Morse homotopy invariant $z^{FW}$.

As an application of our invariant, we prove that $z^{KKT}=z^{FW}$ for

integral homology 3-spheres.

### 2013/10/01

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

The geography problem of Lefschetz fibrations (JAPANESE)

**Naoyuki Monden**(Tokyo University of Science)The geography problem of Lefschetz fibrations (JAPANESE)

[ Abstract ]

To consider holomorphic fibrations complex surfaces over complex curves

and Lefschetz fibrations over surfaces is one method for the study of

complex surfaces of general type and symplectic 4-manifods, respectively.

In this talk, by comparing the geography problem of relatively minimal

holomorphic fibrations with that of relatively minimal Lefschetz

fibrations (i.e., the characterization of pairs $(x,y)$ of certain

invariants $x$ and $y$ corresponding to relatively minimal holomorphic

fibrations and relatively minimal Lefschetz fibrations), we observe the

difference between complex surfaces of general type and symplectic

4-manifolds. In particular, we construct Lefschetz fibrations violating

the ``slope inequality" which holds for any relatively minimal holomorphic

fibrations.

To consider holomorphic fibrations complex surfaces over complex curves

and Lefschetz fibrations over surfaces is one method for the study of

complex surfaces of general type and symplectic 4-manifods, respectively.

In this talk, by comparing the geography problem of relatively minimal

holomorphic fibrations with that of relatively minimal Lefschetz

fibrations (i.e., the characterization of pairs $(x,y)$ of certain

invariants $x$ and $y$ corresponding to relatively minimal holomorphic

fibrations and relatively minimal Lefschetz fibrations), we observe the

difference between complex surfaces of general type and symplectic

4-manifolds. In particular, we construct Lefschetz fibrations violating

the ``slope inequality" which holds for any relatively minimal holomorphic

fibrations.

### 2013/07/16

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

On new models of real hyperbolic spaces (JAPANESE)

**Sumio Yamada**(Gakushuin University)On new models of real hyperbolic spaces (JAPANESE)

[ Abstract ]

In this talk, I will introduce several new realization of the real hyperbolic spaces, using classical tools. The constructions will involve aspects of convex geometry as well as projective geometry, and they are interesting from the view point of the history of mathematics. This work belongs to a joint project with Athanase Papadopoulos.

In this talk, I will introduce several new realization of the real hyperbolic spaces, using classical tools. The constructions will involve aspects of convex geometry as well as projective geometry, and they are interesting from the view point of the history of mathematics. This work belongs to a joint project with Athanase Papadopoulos.

### 2013/07/09

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

Smooth 3-manifolds in the 4-sphere (ENGLISH)

**Ryan Budney**(University of Victoria)Smooth 3-manifolds in the 4-sphere (ENGLISH)

[ Abstract ]

Everyone who has studied topology knows the compact 2-manifolds that embed in the 3-sphere. One dimension up, the problem of which smooth 3-manifolds embed in the 4-sphere turns out to be much more involved with a handful of partial answers. I will describe what is known at the present moment.

Everyone who has studied topology knows the compact 2-manifolds that embed in the 3-sphere. One dimension up, the problem of which smooth 3-manifolds embed in the 4-sphere turns out to be much more involved with a handful of partial answers. I will describe what is known at the present moment.