## Tuesday Seminar on Topology

Seminar information archive ～09/23｜Next seminar｜Future seminars 09/24～

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**

### 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.

### 2011/01/25

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

On unknotting of surface-knots with small sheet numbers

(JAPANESE)

**Chikara Haruta**(Graduate School of Mathematical Sciences, the University of Tokyo )On unknotting of surface-knots with small sheet numbers

(JAPANESE)

[ Abstract ]

A connected surface smoothly embedded in ${\\mathbb R}^4$ is called a surface-knot. In particular, if a surface-knot $F$ is homeomorphic to the $2$-sphere or the torus, then it is called an $S^2$-knot or a $T^2$-knot, respectively. The sheet number of a surface-knot is an invariant analogous to the crossing number of a $1$-knot. M. Saito and S. Satoh proved some results concerning the sheet number of an $S^2$-knot. In particular, it is known that an $S^2$-knot is trivial if and only if its sheet number is $1$, and there is no $S^2$-knot whose sheet number is $2$. In this talk, we show that there is no $S^2$-knot whose sheet number is $3$, and a $T^2$-knot is trivial if and only if its sheet number is $1$.

A connected surface smoothly embedded in ${\\mathbb R}^4$ is called a surface-knot. In particular, if a surface-knot $F$ is homeomorphic to the $2$-sphere or the torus, then it is called an $S^2$-knot or a $T^2$-knot, respectively. The sheet number of a surface-knot is an invariant analogous to the crossing number of a $1$-knot. M. Saito and S. Satoh proved some results concerning the sheet number of an $S^2$-knot. In particular, it is known that an $S^2$-knot is trivial if and only if its sheet number is $1$, and there is no $S^2$-knot whose sheet number is $2$. In this talk, we show that there is no $S^2$-knot whose sheet number is $3$, and a $T^2$-knot is trivial if and only if its sheet number is $1$.

### 2011/01/11

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

The Chas-Sullivan conjecture for a surface of infinite genus (JAPANESE)

**Nariya Kawazumi**(The University of Tokyo)The Chas-Sullivan conjecture for a surface of infinite genus (JAPANESE)

[ Abstract ]

Let \\Sigma_{\\infty,1} be the inductive limit of compact

oriented surfaces with one boundary component. We prove the

center of the Goldman Lie algebra of the surface \\Sigma_{\\infty,1}

is spanned by the constant loop.

A similar statement for a closed oriented surface was conjectured

by Chas and Sullivan, and proved by Etingof. Our result is deduced

from a computation of the center of the Lie algebra of oriented chord

diagrams.

If time permits, the Lie bracket on the space of linear chord diagrams

will be discussed. This talk is based on a joint work with Yusuke Kuno

(Hiroshima U./JSPS).

Let \\Sigma_{\\infty,1} be the inductive limit of compact

oriented surfaces with one boundary component. We prove the

center of the Goldman Lie algebra of the surface \\Sigma_{\\infty,1}

is spanned by the constant loop.

A similar statement for a closed oriented surface was conjectured

by Chas and Sullivan, and proved by Etingof. Our result is deduced

from a computation of the center of the Lie algebra of oriented chord

diagrams.

If time permits, the Lie bracket on the space of linear chord diagrams

will be discussed. This talk is based on a joint work with Yusuke Kuno

(Hiroshima U./JSPS).

### 2010/12/14

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

On the coarse geometry of Teichmueller space (ENGLISH)

**Kenneth Schackleton**(IPMU)On the coarse geometry of Teichmueller space (ENGLISH)

[ Abstract ]

We discuss the synthetic geometry of the pants graph in

comparison with the Weil-Petersson metric, whose geometry the

pants graph coarsely models following work of Brock’s. We also

restrict our attention to the 5-holed sphere, studying the Gromov

bordification of the pants graph and the dynamics of pseudo-Anosov

mapping classes.

We discuss the synthetic geometry of the pants graph in

comparison with the Weil-Petersson metric, whose geometry the

pants graph coarsely models following work of Brock’s. We also

restrict our attention to the 5-holed sphere, studying the Gromov

bordification of the pants graph and the dynamics of pseudo-Anosov

mapping classes.

### 2010/12/07

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

Diffeomorphism-invariant geometries and noncommutative geometry (ENGLISH)

**Raphael Ponge**(The University of Tokyo)Diffeomorphism-invariant geometries and noncommutative geometry (ENGLISH)

[ Abstract ]

In many geometric situations we may encounter the action of

a group $G$ on a manifold $M$, e.g., in the context of foliations. If

the action is free and proper, then the quotient $M/G$ is a smooth

manifold. However, in general the quotient $M/G$ need not even be

Hausdorff. Furthermore, it is well-known that a manifold has structure

invariant under the full group of diffeomorphisms except the

differentiable structure itself. Under these conditions how can one do

diffeomorphism-invariant geometry?

Noncommutative geometry provides us with the solution of trading the

ill-behaved space $M/G$ for a non-commutative algebra which

essentially plays the role of the algebra of smooth functions on that

space. The local index formula of Atiyah-Singer ultimately holds in

the setting of noncommutative geometry. Using this framework Connes

and Moscovici then obtained in the 90s a striking reformulation of the

local index formula in diffeomorphism-invariant geometry.

An important part the talk will be devoted to reviewing noncommutative

geometry and Connes-Moscovici's index formula. We will then hint to on-

going projects about reformulating the local index formula in two new

geometric settings: biholomorphism-invariant geometry of strictly

pseudo-convex domains and contactomorphism-invariant geometry.

In many geometric situations we may encounter the action of

a group $G$ on a manifold $M$, e.g., in the context of foliations. If

the action is free and proper, then the quotient $M/G$ is a smooth

manifold. However, in general the quotient $M/G$ need not even be

Hausdorff. Furthermore, it is well-known that a manifold has structure

invariant under the full group of diffeomorphisms except the

differentiable structure itself. Under these conditions how can one do

diffeomorphism-invariant geometry?

Noncommutative geometry provides us with the solution of trading the

ill-behaved space $M/G$ for a non-commutative algebra which

essentially plays the role of the algebra of smooth functions on that

space. The local index formula of Atiyah-Singer ultimately holds in

the setting of noncommutative geometry. Using this framework Connes

and Moscovici then obtained in the 90s a striking reformulation of the

local index formula in diffeomorphism-invariant geometry.

An important part the talk will be devoted to reviewing noncommutative

geometry and Connes-Moscovici's index formula. We will then hint to on-

going projects about reformulating the local index formula in two new

geometric settings: biholomorphism-invariant geometry of strictly

pseudo-convex domains and contactomorphism-invariant geometry.

### 2010/11/30

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

Pin^-(2)-monopole equations and intersection forms with local coefficients of 4-manifolds (JAPANESE)

**Nobuhiro Nakamura**(The University of Tokyo)Pin^-(2)-monopole equations and intersection forms with local coefficients of 4-manifolds (JAPANESE)

[ Abstract ]

We introduce a variant of the Seiberg-Witten equations, Pin^-(2)-monopole equations, and explain its applications to intersection forms with local coefficients of 4-manifolds.

The first application is an analogue of Froyshov's results on 4-manifolds which have definite forms with local coefficients.

The second one is a local coefficient version of Furuta's 10/8-inequality.

As a corollary, we construct nonsmoothable spin 4-manifolds satisfying Rohlin's theorem and the 10/8-inequality.

We introduce a variant of the Seiberg-Witten equations, Pin^-(2)-monopole equations, and explain its applications to intersection forms with local coefficients of 4-manifolds.

The first application is an analogue of Froyshov's results on 4-manifolds which have definite forms with local coefficients.

The second one is a local coefficient version of Furuta's 10/8-inequality.

As a corollary, we construct nonsmoothable spin 4-manifolds satisfying Rohlin's theorem and the 10/8-inequality.

### 2010/11/16

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

On a colored Khovanov bicomplex (JAPANESE)

**Noboru Ito**(Waseda University)On a colored Khovanov bicomplex (JAPANESE)

[ Abstract ]

We discuss the existence of a bicomplex which is a Khovanov-type

complex associated with categorification of a colored Jones polynomial.

This is an answer to the question proposed by A. Beliakova and S. Wehrli.

Then the second term of the spectral sequence of the bicomplex corresponds

to the Khovanov-type homology group. In this talk, we explain how to define

the bicomplex. If time permits, we also define a colored Rasmussen invariant

by using another spectral sequence of the colored Jones polynomial.

We discuss the existence of a bicomplex which is a Khovanov-type

complex associated with categorification of a colored Jones polynomial.

This is an answer to the question proposed by A. Beliakova and S. Wehrli.

Then the second term of the spectral sequence of the bicomplex corresponds

to the Khovanov-type homology group. In this talk, we explain how to define

the bicomplex. If time permits, we also define a colored Rasmussen invariant

by using another spectral sequence of the colored Jones polynomial.

### 2010/11/09

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

Characterising bumping points on deformation spaces of Kleinian groups (JAPANESE)

**Ken'ichi Ohshika**(Osaka University)Characterising bumping points on deformation spaces of Kleinian groups (JAPANESE)

[ Abstract ]

It is known that components of the interior of a deformation space of a Kleinian group can bump, and a component of the interior can bump itself, on the boundary of the deformation space.

Anderson-Canary-McCullogh gave a necessary and sufficient condition for two components to bump.

In this talk, I shall give a criterion for points on the boundary to be bumping points.

It is known that components of the interior of a deformation space of a Kleinian group can bump, and a component of the interior can bump itself, on the boundary of the deformation space.

Anderson-Canary-McCullogh gave a necessary and sufficient condition for two components to bump.

In this talk, I shall give a criterion for points on the boundary to be bumping points.

### 2010/11/02

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

Periodic-end manifolds and SW theory (ENGLISH)

**Daniel Ruberman**(Brandeis University)Periodic-end manifolds and SW theory (ENGLISH)

[ Abstract ]

We study an extension of Seiberg-Witten invariants to

4-manifolds with the homology of S^1 \\times S^3. This extension has

many potential applications in low-dimensional topology, including the

study of the homology cobordism group. Because b_2^+ =0, the usual

strategy for defining invariants does not work--one cannot disregard

reducible solutions. In fact, the count of solutions can jump in a

family of metrics or perturbations. To remedy this, we define an

index-theoretic counter-term that jumps by the same amount. The

counterterm is the index of the Dirac operator on a manifold with a

periodic end modeled at infinity by the infinite cyclic cover of the

manifold. This is joint work with Tomasz Mrowka and Nikolai Saveliev.

We study an extension of Seiberg-Witten invariants to

4-manifolds with the homology of S^1 \\times S^3. This extension has

many potential applications in low-dimensional topology, including the

study of the homology cobordism group. Because b_2^+ =0, the usual

strategy for defining invariants does not work--one cannot disregard

reducible solutions. In fact, the count of solutions can jump in a

family of metrics or perturbations. To remedy this, we define an

index-theoretic counter-term that jumps by the same amount. The

counterterm is the index of the Dirac operator on a manifold with a

periodic end modeled at infinity by the infinite cyclic cover of the

manifold. This is joint work with Tomasz Mrowka and Nikolai Saveliev.

### 2010/10/26

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

Quantum fundamental groups and quantum representation varieties for 3-manifolds (JAPANESE)

**Kazuo Habiro**(RIMS, Kyoto University)Quantum fundamental groups and quantum representation varieties for 3-manifolds (JAPANESE)

[ Abstract ]

We define a refinement of the fundamental groups of 3-manifolds and

a generalization of representation variety of the fundamental group

of 3-manifolds. We consider the category $H$ whose morphisms are

nonnegative integers, where $n$ corresponds to a genus $n$ handlebody

equipped with an embedding of a disc into the boundary, and whose

morphisms are the isotopy classes of embeddings of handlebodies

compatible with the embeddings of the disc into the boundaries. For

each 3-manifold $M$ with an embedding of a disc into the boundary, we

can construct a contravariant functor from $H$ to the category of

sets, where the object $n$ of $H$ is mapped to the set of isotopy

classes of embedding of the genus $n$ handlebody into $M$, compatible

with the embeddings of the disc into the boundaries. This functor can

be regarded as a refinement of the fundamental group of $M$, and we

call it the quantum fundamental group of $M$. Using this invariant, we

can construct for each co-ribbon Hopf algebra $A$ an invariant of

3-manifolds which may be regarded as (the space of regular functions

on) the representation variety of $M$ with respect to $A$.

We define a refinement of the fundamental groups of 3-manifolds and

a generalization of representation variety of the fundamental group

of 3-manifolds. We consider the category $H$ whose morphisms are

nonnegative integers, where $n$ corresponds to a genus $n$ handlebody

equipped with an embedding of a disc into the boundary, and whose

morphisms are the isotopy classes of embeddings of handlebodies

compatible with the embeddings of the disc into the boundaries. For

each 3-manifold $M$ with an embedding of a disc into the boundary, we

can construct a contravariant functor from $H$ to the category of

sets, where the object $n$ of $H$ is mapped to the set of isotopy

classes of embedding of the genus $n$ handlebody into $M$, compatible

with the embeddings of the disc into the boundaries. This functor can

be regarded as a refinement of the fundamental group of $M$, and we

call it the quantum fundamental group of $M$. Using this invariant, we

can construct for each co-ribbon Hopf algebra $A$ an invariant of

3-manifolds which may be regarded as (the space of regular functions

on) the representation variety of $M$ with respect to $A$.

### 2010/10/19

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

Optimistic limits of colored Jones invariants (ENGLISH)

**Jinseok Cho**(Waseda University)Optimistic limits of colored Jones invariants (ENGLISH)

[ Abstract ]

Yokota made a wonderful theory on the optimistic limit of Kashaev

invariant of a hyperbolic knot

that the limit determines the hyperbolic volume and the Chern-Simons

invariant of the knot.

Especially, his theory enables us to calculate the volume of a knot

combinatorially from its diagram for many cases.

We will briefly discuss Yokota theory, and then move to the optimistic

limit of colored Jones invariant.

We will explain a parallel version of Yokota theory based on the

optimistic limit of colored Jones invariant.

Especially, we will show the optimistic limit of colored Jones

invariant coincides with that of Kashaev invariant modulo 2\\pi^2.

This implies the optimistic limit of colored Jones invariant also

determines the volume and Chern-Simons invariant of the knot, and

probably more information.

This is a joint-work with Jun Murakami of Waseda University.

Yokota made a wonderful theory on the optimistic limit of Kashaev

invariant of a hyperbolic knot

that the limit determines the hyperbolic volume and the Chern-Simons

invariant of the knot.

Especially, his theory enables us to calculate the volume of a knot

combinatorially from its diagram for many cases.

We will briefly discuss Yokota theory, and then move to the optimistic

limit of colored Jones invariant.

We will explain a parallel version of Yokota theory based on the

optimistic limit of colored Jones invariant.

Especially, we will show the optimistic limit of colored Jones

invariant coincides with that of Kashaev invariant modulo 2\\pi^2.

This implies the optimistic limit of colored Jones invariant also

determines the volume and Chern-Simons invariant of the knot, and

probably more information.

This is a joint-work with Jun Murakami of Waseda University.

### 2010/10/12

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

Asymptotics of Morse numbers of finite coverings of manifolds (ENGLISH)

**Andrei Pajitnov**(Univ. de Nantes, The University of Tokyo)Asymptotics of Morse numbers of finite coverings of manifolds (ENGLISH)

[ Abstract ]

Let X be a closed manifold;

denote by m(X) the Morse number of X

(that is, the minimal number of critical

points of a Morse function on X).

Let Y be a finite covering of X of degree d.

In this survey talk we will address the following question

posed by M. Gromov: What are the asymptotic properties

of m(N) as d goes to infinity?

It turns out that for high-dimensional manifolds with

free abelian fundamental group the asymptotics of

the number m(N)/d is directly related to the Novikov homology

of N. We prove this theorem and discuss related results.

Let X be a closed manifold;

denote by m(X) the Morse number of X

(that is, the minimal number of critical

points of a Morse function on X).

Let Y be a finite covering of X of degree d.

In this survey talk we will address the following question

posed by M. Gromov: What are the asymptotic properties

of m(N) as d goes to infinity?

It turns out that for high-dimensional manifolds with

free abelian fundamental group the asymptotics of

the number m(N)/d is directly related to the Novikov homology

of N. We prove this theorem and discuss related results.