## Tuesday Seminar on Topology

Seminar information archive ～01/29｜Next seminar｜Future seminars 01/30～

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**

### 2013/03/19

16:30-18:00 Room #002 (Graduate School of Math. Sci. Bldg.)

Open book foliation and application to contact topology (ENGLISH)

**Keiko Kawamuro**(University of Iowa)Open book foliation and application to contact topology (ENGLISH)

[ Abstract ]

Open book foliation is a generalization of Birman and Menasco's braid foliation. Any 3-manifold admits open book decompositions. Open book foliation is a singular foliation on an embedded surface, and is define by the intersection of a surface and the pages of the open book decomposition. By Giroux's identification of open books and contact structures one can use open book foliation method to study contact structures. In this talk I define the open book foliation and show some applications to contact topology. This is joint work with Tetsuya Ito (University of British Columbia).

Open book foliation is a generalization of Birman and Menasco's braid foliation. Any 3-manifold admits open book decompositions. Open book foliation is a singular foliation on an embedded surface, and is define by the intersection of a surface and the pages of the open book decomposition. By Giroux's identification of open books and contact structures one can use open book foliation method to study contact structures. In this talk I define the open book foliation and show some applications to contact topology. This is joint work with Tetsuya Ito (University of British Columbia).

### 2013/02/19

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

On the ring of Fricke characters of free groups (JAPANESE)

**Eri Hatakenaka**(Tokyo University of Agriculture and Technology)On the ring of Fricke characters of free groups (JAPANESE)

[ Abstract ]

This is a joint work with Takao Satoh (Tokyo University of Science). We study a descending filtration of the ring of Fricke characters of a free group consisting of ideals on which the automorphism group of the free group naturally acts. Then by using it, we define a descending filtration of the automorphism group of a free group, and investigate a relation between it and the Andreadakis-Johnson filtration.

This is a joint work with Takao Satoh (Tokyo University of Science). We study a descending filtration of the ring of Fricke characters of a free group consisting of ideals on which the automorphism group of the free group naturally acts. Then by using it, we define a descending filtration of the automorphism group of a free group, and investigate a relation between it and the Andreadakis-Johnson filtration.

### 2013/01/22

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

On the autonomous metric of the area preserving diffeomorphism

of the two dimensional disc. (ENGLISH)

**Jarek Kedra**(University of Aberdeen)On the autonomous metric of the area preserving diffeomorphism

of the two dimensional disc. (ENGLISH)

[ Abstract ]

Let D be the open unit disc in the Euclidean plane and let

G:=Diff(D, area) be the group of smooth compactly supported

area-preserving diffeomorphisms of D. A diffeomorphism is called

autonomous if it is the time one map of the flow of a time independent

vector field. Every diffeomorphism in G is a composition of a number

of autonomous diffeomorphisms. The least amount of such

diffeomorphisms defines a norm on G. In the talk I will investigate

geometric properties of such a norm.

In particular I will construct a bi-Lipschitz embedding of the free

abelian group of arbitrary rank to G. I will also show that the space

of homogeneous quasi-morphisms vanishing on all autonomous

diffeomorphisms in G is infinite dimensional.

This is a joint work with Michael Brandenbursky.

Let D be the open unit disc in the Euclidean plane and let

G:=Diff(D, area) be the group of smooth compactly supported

area-preserving diffeomorphisms of D. A diffeomorphism is called

autonomous if it is the time one map of the flow of a time independent

vector field. Every diffeomorphism in G is a composition of a number

of autonomous diffeomorphisms. The least amount of such

diffeomorphisms defines a norm on G. In the talk I will investigate

geometric properties of such a norm.

In particular I will construct a bi-Lipschitz embedding of the free

abelian group of arbitrary rank to G. I will also show that the space

of homogeneous quasi-morphisms vanishing on all autonomous

diffeomorphisms in G is infinite dimensional.

This is a joint work with Michael Brandenbursky.

### 2013/01/21

16:30-17:30 Room #002 (Graduate School of Math. Sci. Bldg.)

)

Lie foliations transversely modeled on nilpotent Lie

algebras

(JAPANESE)

**Naoki Kato**(The University of Tokyo)

Lie foliations transversely modeled on nilpotent Lie

algebras

(JAPANESE)

[ Abstract ]

To each Lie $\\mathfrak{g}$-foliation, there is an associated subalgebra

$\\mathfrak{h}$ of $\\mathfrak{g}$ with the foliation, which is called the

structure Lie algabra. In this talk, we will explain the inverse problem,

that is, which pair $(\\mathfrak{g},\\mathfrak{h})$ can be realized as a

Lie $\\mathfrak{g}$-foliation with the structure Lie algabra $\\mathfrak{h}

$, under the assumption that $\\mathfrak{g}$ is nilpotent.

To each Lie $\\mathfrak{g}$-foliation, there is an associated subalgebra

$\\mathfrak{h}$ of $\\mathfrak{g}$ with the foliation, which is called the

structure Lie algabra. In this talk, we will explain the inverse problem,

that is, which pair $(\\mathfrak{g},\\mathfrak{h})$ can be realized as a

Lie $\\mathfrak{g}$-foliation with the structure Lie algabra $\\mathfrak{h}

$, under the assumption that $\\mathfrak{g}$ is nilpotent.

### 2013/01/21

17:30-18:30 Room #002 (Graduate School of Math. Sci. Bldg.)

Quasi-morphisms on the group of area-preserving diffeomorphisms of

the 2-disk

(JAPANESE)

**Tomohiko Ishida**(The University of Tokyo)Quasi-morphisms on the group of area-preserving diffeomorphisms of

the 2-disk

(JAPANESE)

[ Abstract ]

Gambaudo and Ghys constructed linearly independent countably many quasi-

morphisms on the group of area-preserving diffeomorphisms of the 2-disk

from quasi-morphisms on braid groups.

In this talk, we will explain that their construction is injective as a

homomorphism between vector spaces of quasi-morphisms.

If time permits, we introduce an application by Brandenbursky and K\\c{e}

dra.

Gambaudo and Ghys constructed linearly independent countably many quasi-

morphisms on the group of area-preserving diffeomorphisms of the 2-disk

from quasi-morphisms on braid groups.

In this talk, we will explain that their construction is injective as a

homomorphism between vector spaces of quasi-morphisms.

If time permits, we introduce an application by Brandenbursky and K\\c{e}

dra.

### 2012/12/11

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

Homotopy-theoretic methods in the study of spaces of knots and links (ENGLISH)

**Ismar Volic**(Wellesley College)Homotopy-theoretic methods in the study of spaces of knots and links (ENGLISH)

[ Abstract ]

I will survey the ways in which some homotopy-theoretic

methods, manifold calculus of functors main among them, have in recent

years been used for extracting information about the topology of

spaces of knots and links. Cosimplicial spaces and operads will also

be featured. I will end with some recent results about spaces of

homotopy string links and in particular about how one can use functor

calculus in combination with configuration space integrals to extract

information about Milnor invariants.

I will survey the ways in which some homotopy-theoretic

methods, manifold calculus of functors main among them, have in recent

years been used for extracting information about the topology of

spaces of knots and links. Cosimplicial spaces and operads will also

be featured. I will end with some recent results about spaces of

homotopy string links and in particular about how one can use functor

calculus in combination with configuration space integrals to extract

information about Milnor invariants.

### 2012/12/04

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

Conformal field theory for C2-cofinite vertex algebras (JAPANESE)

**Yoshitake Hashimoto**(Tokyo City University)Conformal field theory for C2-cofinite vertex algebras (JAPANESE)

[ Abstract ]

This is a jount work with Akihiro Tsuchiya (Kavli IPMU).

We consider sheaves of covacua and conformal blocks over parameter spaces of n-pointed Riemann surfaces

for a vertex algebra of which the category of modules is not necessarily semi-simple.

We assume the C2-cofiniteness condition for vertex algebras.

We define "tensor product" of two modules over a C2-cofinite vertex algebra.

This is a jount work with Akihiro Tsuchiya (Kavli IPMU).

We consider sheaves of covacua and conformal blocks over parameter spaces of n-pointed Riemann surfaces

for a vertex algebra of which the category of modules is not necessarily semi-simple.

We assume the C2-cofiniteness condition for vertex algebras.

We define "tensor product" of two modules over a C2-cofinite vertex algebra.

### 2012/11/27

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

On a finite aspect of characteristic classes of foliations (JAPANESE)

**Hiraku Nozawa**(JSPS-IHES fellow)On a finite aspect of characteristic classes of foliations (JAPANESE)

[ Abstract ]

Characteristic classes of foliations are not bounded due to Thurston.

In this talk, we will explain finiteness of characteristic classes for

foliations with certain transverse structures (e.g. transverse

conformally flat structure) and its relation to unboundedness and

rigidity of foliations.

(This talk is based on a joint work with Jesús Antonio

Álvarez López at University of Santiago de Compostela,

which is available as arXiv:1205.3375.)

Characteristic classes of foliations are not bounded due to Thurston.

In this talk, we will explain finiteness of characteristic classes for

foliations with certain transverse structures (e.g. transverse

conformally flat structure) and its relation to unboundedness and

rigidity of foliations.

(This talk is based on a joint work with Jesús Antonio

Álvarez López at University of Santiago de Compostela,

which is available as arXiv:1205.3375.)

### 2012/11/20

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

3 dimensional hyperbolic geometry and cluster algebras (JAPANESE)

**Kentaro Nagao**(Nagoya University)3 dimensional hyperbolic geometry and cluster algebras (JAPANESE)

[ Abstract ]

The cluster algebra was discovered by Fomin-Zelevinsky in 2000.

Recently, the structures of cluster algebras are recovered in

many areas including the theory of quantum groups, low

dimensional topology, discrete integrable systems, Donaldson-Thomas

theory, and string theory and there is dynamic development in the

research on these subjects. In this talk I introduce a relation between

3 dimensional hyperbolic geometry and cluster algebras motivated

by some duality in string theory.

The cluster algebra was discovered by Fomin-Zelevinsky in 2000.

Recently, the structures of cluster algebras are recovered in

many areas including the theory of quantum groups, low

dimensional topology, discrete integrable systems, Donaldson-Thomas

theory, and string theory and there is dynamic development in the

research on these subjects. In this talk I introduce a relation between

3 dimensional hyperbolic geometry and cluster algebras motivated

by some duality in string theory.

### 2012/11/13

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

The virtual fibering theorem and sutured manifold hierarchies (JAPANESE)

**Takahiro Kitayama**(RIMS, Kyoto University,JSPS PD)The virtual fibering theorem and sutured manifold hierarchies (JAPANESE)

[ Abstract ]

In 2007 Agol showed that every irreducible 3-manifold whose fundamental

group is nontrivial and virtually residually finite rationally solvable

(RFRS) is virtually fibered. In the proof he used the theory of

least-weight taut normal surfaces introduced and developed by Oertel and

Tollefson-Wang. We give another proof using complexities of sutured

manifolds. This is a joint work with Stefan Friedl (University of

Cologne).

In 2007 Agol showed that every irreducible 3-manifold whose fundamental

group is nontrivial and virtually residually finite rationally solvable

(RFRS) is virtually fibered. In the proof he used the theory of

least-weight taut normal surfaces introduced and developed by Oertel and

Tollefson-Wang. We give another proof using complexities of sutured

manifolds. This is a joint work with Stefan Friedl (University of

Cologne).

### 2012/11/06

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

Galois action on knots (JAPANESE)

**Furusho Hidekazu**(Nagoya University)Galois action on knots (JAPANESE)

[ Abstract ]

I will explain a motivic structure on knots.

Then I will explain that the absolute Galois group of

the rational number field acts non-trivially

on 'the space of knots' in a non-trivial way.

I will explain a motivic structure on knots.

Then I will explain that the absolute Galois group of

the rational number field acts non-trivially

on 'the space of knots' in a non-trivial way.

### 2012/10/30

16:30-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Applications of knot theory to molecular biology (JAPANESE)

**Koya Shimokawa**(Saitama University)Applications of knot theory to molecular biology (JAPANESE)

[ Abstract ]

In this talk we discuss applications of knot theory to studies of DNA

and proteins.

Especially we will consider (1)topological characterization of

mechanisms of site-specific recombination systems,

(2)modeling knotted DNA and proteins in confined regions using lattice

knots, and

(3)mechanism of topoisomerases and signed crossing changes.

In this talk we discuss applications of knot theory to studies of DNA

and proteins.

Especially we will consider (1)topological characterization of

mechanisms of site-specific recombination systems,

(2)modeling knotted DNA and proteins in confined regions using lattice

knots, and

(3)mechanism of topoisomerases and signed crossing changes.

### 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.