## Tuesday Seminar on Topology

Seminar information archive ～09/26｜Next seminar｜Future seminars 09/27～

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**

### 2016/04/26

17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Arithmetic topology on branched covers of 3-manifolds (JAPANESE)

**Jun Ueki**(The University of Tokyo)Arithmetic topology on branched covers of 3-manifolds (JAPANESE)

[ Abstract ]

The analogy between 3-dimensional topology and number theory was first pointed out by Mazur in the 1960s, and it has been studied systematically by Kapranov, Reznikov, Morishita, and others. In their analogies, for example, knots and 3-manifolds correspond to primes and number rings respectively. The study of these analogies is called arithmetic topology now.

In my talk, based on their dictionary of analogies, we study analogues of idelic class field theory, Iwasawa theory, and Galois deformation theory in the context of 3-dimensional topology, and establish various foundational analogies in arithmetic topology.

The analogy between 3-dimensional topology and number theory was first pointed out by Mazur in the 1960s, and it has been studied systematically by Kapranov, Reznikov, Morishita, and others. In their analogies, for example, knots and 3-manifolds correspond to primes and number rings respectively. The study of these analogies is called arithmetic topology now.

In my talk, based on their dictionary of analogies, we study analogues of idelic class field theory, Iwasawa theory, and Galois deformation theory in the context of 3-dimensional topology, and establish various foundational analogies in arithmetic topology.

### 2016/04/19

17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Topological rigidity of finite cyclic group actions on compact surfaces (ENGLISH)

**Błażej Szepietowski**(Gdansk University)Topological rigidity of finite cyclic group actions on compact surfaces (ENGLISH)

[ Abstract ]

Two actions of a group on a surface are called topologically equivalent if they are conjugate by a homeomorphism of the surface. I will describe a method of enumeration (and classification) of topological equivalence classes of actions of a finite group on a compact surface, based on the combinatorial theory of noneuclidean crystallographic groups (NEC groups in short) and a relationship between the outer automorphism group of an NEC group and certain mapping class group. By this method we study topological equivalence of actions of a finite cyclic group on a compact surface, in the situation where the order of the group is large relative to the genus of the surface.

Two actions of a group on a surface are called topologically equivalent if they are conjugate by a homeomorphism of the surface. I will describe a method of enumeration (and classification) of topological equivalence classes of actions of a finite group on a compact surface, based on the combinatorial theory of noneuclidean crystallographic groups (NEC groups in short) and a relationship between the outer automorphism group of an NEC group and certain mapping class group. By this method we study topological equivalence of actions of a finite cyclic group on a compact surface, in the situation where the order of the group is large relative to the genus of the surface.

### 2016/04/12

17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Homotopy theory of differential graded Lie algebras (ENGLISH)

**Aniceto Murillo**(Universidad de Malaga)Homotopy theory of differential graded Lie algebras (ENGLISH)

[ Abstract ]

Having as motivation the Deligne's principle by which every deformation functor is governed by a differential graded Lie algebra, we build a homotopy theory for these algebras which extend the classical Quillen approach and let us model any (non necessarily 1-connected nor path connected) complex. This is joint work with Urtzi Buijs, Yves Félix and Daniel Tanré.

Having as motivation the Deligne's principle by which every deformation functor is governed by a differential graded Lie algebra, we build a homotopy theory for these algebras which extend the classical Quillen approach and let us model any (non necessarily 1-connected nor path connected) complex. This is joint work with Urtzi Buijs, Yves Félix and Daniel Tanré.

### 2016/04/05

17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Torsion invariants and representation varieties for non-positively curved cube complexes (JAPANESE)

**Takahiro Kitayama**(The University of Tokyo)Torsion invariants and representation varieties for non-positively curved cube complexes (JAPANESE)

[ Abstract ]

Applications of torsion invariants and representation varieties have been extensively studied for 3-manifolds. Twisted Alexander polynomials are known to detect the Thurston norm and fiberedness of a 3-manifold. Ideal points of character varieties are known to detect essential surfaces in a 3-manifold in a certain extension of Culler-Shalen theory. In view of cubulation of 3-manifolds one can expect that these results naturally extend to a wider framework and, in particular, the case of virtually special cube complexes. We formulate and discuss such analogous questions for non-positively curved cube complexes.

Applications of torsion invariants and representation varieties have been extensively studied for 3-manifolds. Twisted Alexander polynomials are known to detect the Thurston norm and fiberedness of a 3-manifold. Ideal points of character varieties are known to detect essential surfaces in a 3-manifold in a certain extension of Culler-Shalen theory. In view of cubulation of 3-manifolds one can expect that these results naturally extend to a wider framework and, in particular, the case of virtually special cube complexes. We formulate and discuss such analogous questions for non-positively curved cube complexes.

### 2016/02/16

17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

String Topology, Euler Class and TNCZ free loop fibrations (ENGLISH)

**Luc Menichi**(University of Angers)String Topology, Euler Class and TNCZ free loop fibrations (ENGLISH)

[ Abstract ]

Let $M$ be a connected, closed oriented manifold.

Chas and Sullivan have defined a loop product and a loop coproduct on

$H_*(LM;¥mathbb{F})$, the homology of the

free loops on $M$ with coefficients in the field $¥mathbb{F}$.

By studying this loop coproduct, I will show that if the free loop

fibration

$¥Omega M¥buildrel{i}¥over¥hookrightarrow

LM¥buildrel{ev}¥over¥twoheadrightarrow M$

is homologically trivial, i.e. $i^*:H^*(LM;¥mathbb{F})¥twoheadrightarrow

H^*(¥Omega M;¥mathbb{F})$ is onto,

then the Euler characteristic of $M$ is divisible by the characteristic

of the field $¥mathbb{F}$

(or $M$ is a point).

Let $M$ be a connected, closed oriented manifold.

Chas and Sullivan have defined a loop product and a loop coproduct on

$H_*(LM;¥mathbb{F})$, the homology of the

free loops on $M$ with coefficients in the field $¥mathbb{F}$.

By studying this loop coproduct, I will show that if the free loop

fibration

$¥Omega M¥buildrel{i}¥over¥hookrightarrow

LM¥buildrel{ev}¥over¥twoheadrightarrow M$

is homologically trivial, i.e. $i^*:H^*(LM;¥mathbb{F})¥twoheadrightarrow

H^*(¥Omega M;¥mathbb{F})$ is onto,

then the Euler characteristic of $M$ is divisible by the characteristic

of the field $¥mathbb{F}$

(or $M$ is a point).

### 2016/01/19

15:00-16:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Ricci-mean curvature flows in gradient shrinking Ricci solitons (JAPANESE)

**Hikaru Yamamoto**(The University of Tokyo)Ricci-mean curvature flows in gradient shrinking Ricci solitons (JAPANESE)

[ Abstract ]

A Ricci-mean curvature flow is a coupled parabolic PDE system of a mean

curvature flow and a Ricci flow.

In this talk, we consider a Ricci-mean curvature flow in a gradient

shrinking Ricci soliton, and give a generalization of a well-known result

of Huisken which states that if a mean curvature flow in a Euclidean space

develops a singularity of type I, then its parabolic rescaling near the singular

point converges to a self-shrinker.

A Ricci-mean curvature flow is a coupled parabolic PDE system of a mean

curvature flow and a Ricci flow.

In this talk, we consider a Ricci-mean curvature flow in a gradient

shrinking Ricci soliton, and give a generalization of a well-known result

of Huisken which states that if a mean curvature flow in a Euclidean space

develops a singularity of type I, then its parabolic rescaling near the singular

point converges to a self-shrinker.

### 2016/01/12

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

Heavy subsets and non-contractible trajectories (JAPANESE)

On codimension two contact embeddings in the standard spheres (JAPANESE)

**Morimichi Kawasaki**(The University of Tokyo) 16:30-17:30Heavy subsets and non-contractible trajectories (JAPANESE)

[ Abstract ]

For a compact set Y of an open symplectic manifold $(N,¥omega)$ and a free

homotopy class $¥alpha¥in [S^1,N]$, Biran, Polterovich and Salamon

defined the relative symplectic capacity $C_{BPS}(N,Y;¥alpha)$ which

measures the existence of non-contractible 1-periodic trajectories of

Hamiltonian isotopies.

On the hand, Entov and Polterovich defined heaviness for closed subsets

of a symplectic manifold by using spectral invarinats of the Hamiltonian

Floer theory on contractible trajectories.

Heavy subsets are known to be non-displaceable.

In this talk, we prove the finiteness of $C(M,X,¥alpha)$ (i.e. the

existence of non-contractible 1-periodic trajectories under some setting)

by using heaviness.

For a compact set Y of an open symplectic manifold $(N,¥omega)$ and a free

homotopy class $¥alpha¥in [S^1,N]$, Biran, Polterovich and Salamon

defined the relative symplectic capacity $C_{BPS}(N,Y;¥alpha)$ which

measures the existence of non-contractible 1-periodic trajectories of

Hamiltonian isotopies.

On the hand, Entov and Polterovich defined heaviness for closed subsets

of a symplectic manifold by using spectral invarinats of the Hamiltonian

Floer theory on contractible trajectories.

Heavy subsets are known to be non-displaceable.

In this talk, we prove the finiteness of $C(M,X,¥alpha)$ (i.e. the

existence of non-contractible 1-periodic trajectories under some setting)

by using heaviness.

**Ryo Furukawa**(The University of Tokyo) 17:30-18:30On codimension two contact embeddings in the standard spheres (JAPANESE)

[ Abstract ]

In this talk we consider codimension two contact

embedding problem by using higher dimensional braids.

First, we focus on embeddings of contact $3$-manifolds to the standard $

S^5$ and give some results, for example, any contact structure on $S^3$

can embed so that it is smoothly isotopic to the standard embedding.

These are joint work with John Etnyre. Second, we consider the relative

Euler number of codimension two contact submanifolds and its Seifert

hypersurfaces which is a generalization of the self-linking number of

transverse knots in contact $3$-manifolds. We give a way to calculate

the relative Euler number of certain contact submanifolds obtained by

braids and as an application we give examples of embeddings of one

contact manifold which are isotopic as smooth embeddings but not

isotopic as contact embeddings in higher dimension.

In this talk we consider codimension two contact

embedding problem by using higher dimensional braids.

First, we focus on embeddings of contact $3$-manifolds to the standard $

S^5$ and give some results, for example, any contact structure on $S^3$

can embed so that it is smoothly isotopic to the standard embedding.

These are joint work with John Etnyre. Second, we consider the relative

Euler number of codimension two contact submanifolds and its Seifert

hypersurfaces which is a generalization of the self-linking number of

transverse knots in contact $3$-manifolds. We give a way to calculate

the relative Euler number of certain contact submanifolds obtained by

braids and as an application we give examples of embeddings of one

contact manifold which are isotopic as smooth embeddings but not

isotopic as contact embeddings in higher dimension.

### 2015/12/15

17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

The Curved Cartan Complex (ENGLISH)

**Constantin Teleman**(University of California, Berkeley)The Curved Cartan Complex (ENGLISH)

[ Abstract ]

The Cartan model computes the equivariant cohomology of a smooth manifold X with

differentiable action of a compact Lie group G, from the invariant polynomial

functions on the Lie algebra with values in differential forms and a deformation

of the de Rham differential. Before extracting invariants, the Cartan differential

does not square to zero and is apparently meaningless. Unrecognised was the fact

that the full complex is a curved algebra, computing the quotient by G of the

algebra of differential forms on X. This generates, for example, a gauged version of

string topology. Another instance of the construction, applied to deformation

quantisation of symplectic manifolds, gives the BRST construction of the symplectic

quotient. Finally, the theory for a X point with an additional quadratic curving

computes the representation category of the compact group G, and this generalises

to the loop group of G and even to real semi-simple groups.

The Cartan model computes the equivariant cohomology of a smooth manifold X with

differentiable action of a compact Lie group G, from the invariant polynomial

functions on the Lie algebra with values in differential forms and a deformation

of the de Rham differential. Before extracting invariants, the Cartan differential

does not square to zero and is apparently meaningless. Unrecognised was the fact

that the full complex is a curved algebra, computing the quotient by G of the

algebra of differential forms on X. This generates, for example, a gauged version of

string topology. Another instance of the construction, applied to deformation

quantisation of symplectic manifolds, gives the BRST construction of the symplectic

quotient. Finally, the theory for a X point with an additional quadratic curving

computes the representation category of the compact group G, and this generalises

to the loop group of G and even to real semi-simple groups.

### 2015/12/08

17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Lens space surgery and Kirby calculus of 4-manifolds (JAPANESE)

**Yuichi Yamada**(The Univ. of Electro-Comm.)Lens space surgery and Kirby calculus of 4-manifolds (JAPANESE)

[ Abstract ]

The problem asking "Which knot yields a lens space by Dehn surgery" is

called "lens space surgery". Berge's list ('90) is believed to be the

complete list, but it is still unproved, even after some progress by

Heegaard Floer Homology.

This problem seems to enter a new aspect: study using 4-manifolds, lens

space surgery from lens spaces, checking hyperbolicity by computer.

In the talk, we review the structure of Berge's list and talk on our

study on pairs of distinct knots but yield same lens spaces, and

4-maniolds constructed from such pairs. This is joint work with Motoo

Tange (Tsukuba University).

The problem asking "Which knot yields a lens space by Dehn surgery" is

called "lens space surgery". Berge's list ('90) is believed to be the

complete list, but it is still unproved, even after some progress by

Heegaard Floer Homology.

This problem seems to enter a new aspect: study using 4-manifolds, lens

space surgery from lens spaces, checking hyperbolicity by computer.

In the talk, we review the structure of Berge's list and talk on our

study on pairs of distinct knots but yield same lens spaces, and

4-maniolds constructed from such pairs. This is joint work with Motoo

Tange (Tsukuba University).

### 2015/12/01

17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Monodromies of splitting families for singular fibers (JAPANESE)

**Takayuki Okuda**(The University of Tokyo)Monodromies of splitting families for singular fibers (JAPANESE)

[ Abstract ]

A degeneration of Riemann surfaces is a family of complex curves

over a disk allowed to have a singular fiber.

A singular fiber may split into several simpler singular fibers

under a deformation family of such families,

which is called a splitting family for the singular fiber.

We are interested in the topology of splitting families.

For the topological types of degenerations of Riemann surfaces,

it is known that there is a good relationship with

the surface mapping classes, via topological monodromy.

In this talk,

we introduce the "topological monodromies of splitting families",

and give a description of those of certain splitting families.

A degeneration of Riemann surfaces is a family of complex curves

over a disk allowed to have a singular fiber.

A singular fiber may split into several simpler singular fibers

under a deformation family of such families,

which is called a splitting family for the singular fiber.

We are interested in the topology of splitting families.

For the topological types of degenerations of Riemann surfaces,

it is known that there is a good relationship with

the surface mapping classes, via topological monodromy.

In this talk,

we introduce the "topological monodromies of splitting families",

and give a description of those of certain splitting families.

### 2015/11/24

17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

On the cohomology ring of the handlebody mapping class group of genus two (JAPANESE)

**Masatoshi Sato**(Tokyo Denki University)On the cohomology ring of the handlebody mapping class group of genus two (JAPANESE)

[ Abstract ]

The genus two handlebody mapping class group acts on a tree

constructed by Kramer from the disk complex,

and decomposes into an amalgamated product of two subgroups.

We determine the integral cohomology ring of the genus two handlebody

mapping class group by examining these two subgroups

and the Mayer-Vietoris exact sequence.

Using this result, we estimate the ranks of low dimensional homology

groups of the genus three handlebody mapping class group.

The genus two handlebody mapping class group acts on a tree

constructed by Kramer from the disk complex,

and decomposes into an amalgamated product of two subgroups.

We determine the integral cohomology ring of the genus two handlebody

mapping class group by examining these two subgroups

and the Mayer-Vietoris exact sequence.

Using this result, we estimate the ranks of low dimensional homology

groups of the genus three handlebody mapping class group.

### 2015/11/17

17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Topology of some three-dimensional singularities related to algebraic geometry (ENGLISH)

**Atsuko Katanaga**(Shinshu University)Topology of some three-dimensional singularities related to algebraic geometry (ENGLISH)

[ Abstract ]

In this talk, we deal with hypersurface isolated singularities. First, we will recall

some topological results of singularities. Next, we will sketch the classification of

singularities in algebraic geometry. Finally, we will focus on the three-dimensional

case and discuss some results obtained so far.

In this talk, we deal with hypersurface isolated singularities. First, we will recall

some topological results of singularities. Next, we will sketch the classification of

singularities in algebraic geometry. Finally, we will focus on the three-dimensional

case and discuss some results obtained so far.

### 2015/11/10

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

Topological T-duality for "Real" circle bundle (JAPANESE)

**Kiyonori Gomi**(Shinshu University)Topological T-duality for "Real" circle bundle (JAPANESE)

[ Abstract ]

Topological T-duality originates from T-duality in superstring theory,

and is first studied by Bouwkneght, Evslin and Mathai. The duality

basically consists of two parts: The first part is that, for any pair

of a principal circle bundle with `H-flux', there is another `T-dual'

pair on the same base space. The second part states that the twisted

K-groups of the total spaces of principal circle bundles in duality

are isomorphic under degree shift. This is the most simple topological

T-duality following Bunke and Schick, and there are a number of

generalizations. The generalization I will talk about is a topological

T-duality for "Real" circle bundles, motivated by T-duality in type II

orbifold string theory. In this duality, a variant of Z_2-equivariant

K-theory appears.

Topological T-duality originates from T-duality in superstring theory,

and is first studied by Bouwkneght, Evslin and Mathai. The duality

basically consists of two parts: The first part is that, for any pair

of a principal circle bundle with `H-flux', there is another `T-dual'

pair on the same base space. The second part states that the twisted

K-groups of the total spaces of principal circle bundles in duality

are isomorphic under degree shift. This is the most simple topological

T-duality following Bunke and Schick, and there are a number of

generalizations. The generalization I will talk about is a topological

T-duality for "Real" circle bundles, motivated by T-duality in type II

orbifold string theory. In this duality, a variant of Z_2-equivariant

K-theory appears.

### 2015/10/27

17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Heegaard Floer homology for graphs (JAPANESE)

**Yuanyuan Bao**(The University of Tokyo)Heegaard Floer homology for graphs (JAPANESE)

[ Abstract ]

Ozsváth and Szabó defined the Heegaard Floer homology (HF) for a closed oriented 3-manifold. The definition was then generalized to links embedded in a 3-manifold and the manifolds with boundary (sutured and bordered manifolds). In the case of links, there is a beautiful combinatorial way to rewrite the original definition of HF, which was defined on a Heegaard diagram of the given link, by using grid diagram. For a balanced bipartite graph, we defined its Heegaard diagram and the HF for it. Around the same time, Harvey and O’Donnol defined the combinatorial HF for transverse graphs (see the definition in [arXiv:1506.04785v1]). In this talk, we compare these two methods.

Ozsváth and Szabó defined the Heegaard Floer homology (HF) for a closed oriented 3-manifold. The definition was then generalized to links embedded in a 3-manifold and the manifolds with boundary (sutured and bordered manifolds). In the case of links, there is a beautiful combinatorial way to rewrite the original definition of HF, which was defined on a Heegaard diagram of the given link, by using grid diagram. For a balanced bipartite graph, we defined its Heegaard diagram and the HF for it. Around the same time, Harvey and O’Donnol defined the combinatorial HF for transverse graphs (see the definition in [arXiv:1506.04785v1]). In this talk, we compare these two methods.

### 2015/10/27

15:00-16:30 Room #056 (Graduate School of Math. Sci. Bldg.)

The unfolded Seiberg-Witten-Floer spectrum and its applications

(ENGLISH)

**Jianfeng Lin**(UCLA)The unfolded Seiberg-Witten-Floer spectrum and its applications

(ENGLISH)

[ Abstract ]

Following Furuta's idea of finite dimensional approximation in

the Seiberg-Witten theory, Manolescu defined the Seiberg-Witten-Floer

stable homotopy type for rational homology three-spheres in 2003. In

this talk, I will explain how to construct similar invariants for a

general three-manifold and discuss some applications of these new

invariants. This is a joint work with Tirasan Khandhawit and Hirofumi

Sasahira.

Following Furuta's idea of finite dimensional approximation in

the Seiberg-Witten theory, Manolescu defined the Seiberg-Witten-Floer

stable homotopy type for rational homology three-spheres in 2003. In

this talk, I will explain how to construct similar invariants for a

general three-manifold and discuss some applications of these new

invariants. This is a joint work with Tirasan Khandhawit and Hirofumi

Sasahira.

### 2015/10/20

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

On the existence of stable compact leaves for

transversely holomorphic foliations (ENGLISH)

**Bruno Scardua**(Universidade Federal do Rio de Janeiro)On the existence of stable compact leaves for

transversely holomorphic foliations (ENGLISH)

[ Abstract ]

One of the most important results in the theory of foliations is

the celebrated Local stability theorem of Reeb :

A compact leaf of a foliation having finite holonomy group is

stable, indeed, it admits a fundamental system of invariant

neighborhoods where each leaf is compact with finite holonomy

group. This result, together with the Global stability theorem of Reeb

(for codimension one real foliations), has many important consequences

and motivates several questions in the theory of foliations. In this talk

we show how to prove:

A transversely holomorphic foliation on a compact complex manifold, exhibits a compact stable

leaf if and only if the set of compact leaves is not a zero measure subset of the manifold.

This is a joint work with Cesar Camacho.

One of the most important results in the theory of foliations is

the celebrated Local stability theorem of Reeb :

A compact leaf of a foliation having finite holonomy group is

stable, indeed, it admits a fundamental system of invariant

neighborhoods where each leaf is compact with finite holonomy

group. This result, together with the Global stability theorem of Reeb

(for codimension one real foliations), has many important consequences

and motivates several questions in the theory of foliations. In this talk

we show how to prove:

A transversely holomorphic foliation on a compact complex manifold, exhibits a compact stable

leaf if and only if the set of compact leaves is not a zero measure subset of the manifold.

This is a joint work with Cesar Camacho.

### 2015/10/06

17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Delooping theorem in K-theory (JAPANESE)

**Sho Saito**(Kavli IPMU)Delooping theorem in K-theory (JAPANESE)

[ Abstract ]

There is an important special class of infinite dimensional vector spaces, formed by those called Tate vector spaces. Since their first appearance in Tate’s work on residues of differentials on curves, they have been playing important roles in several different contexts including the study of formal loop spaces and semi-infinite Hodge theory. They have more sophisticated linear algebraic invariant than finite dimensional vector spaces, for instance the dimension of a Tate vector spaces is not a single integer, but a torsor acted upon by the all integers, and the determinant of an automorphism is not a single invertible scalar, but a torsor acted upon by the all invertible scalars. In this talk I will show how a delooping theorem in K-theory provides a clarified perspective on this phenomenon, using the recently developed higher categorical framework of infinity-topoi.

There is an important special class of infinite dimensional vector spaces, formed by those called Tate vector spaces. Since their first appearance in Tate’s work on residues of differentials on curves, they have been playing important roles in several different contexts including the study of formal loop spaces and semi-infinite Hodge theory. They have more sophisticated linear algebraic invariant than finite dimensional vector spaces, for instance the dimension of a Tate vector spaces is not a single integer, but a torsor acted upon by the all integers, and the determinant of an automorphism is not a single invertible scalar, but a torsor acted upon by the all invertible scalars. In this talk I will show how a delooping theorem in K-theory provides a clarified perspective on this phenomenon, using the recently developed higher categorical framework of infinity-topoi.

### 2015/07/21

17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Ribbon concordance and 0-surgeries along knots (JAPANESE)

**Keiji Tagami**(Tokyo Institute of Technology)Ribbon concordance and 0-surgeries along knots (JAPANESE)

[ Abstract ]

Akbulut and Kirby conjectured that two knots with

the same 0-surgery are concordant. Recently, Yasui

gave a counterexample of this conjecture.

In this talk, we introduce a technique to construct

non-ribbon concordant knots with the same 0-surgery.

Moreover, we give a potential counterexample of the

slice-ribbon conjecture. This is a joint work with

Tetsuya Abe (Osaka City University, OCAMI).

Akbulut and Kirby conjectured that two knots with

the same 0-surgery are concordant. Recently, Yasui

gave a counterexample of this conjecture.

In this talk, we introduce a technique to construct

non-ribbon concordant knots with the same 0-surgery.

Moreover, we give a potential counterexample of the

slice-ribbon conjecture. This is a joint work with

Tetsuya Abe (Osaka City University, OCAMI).

### 2015/07/14

17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

How homoclinic orbits explain some algebraic relations holding in Novikov rings. (ENGLISH)

**Carlos Moraga Ferrandiz**(The University of Tokyo, JSPS)How homoclinic orbits explain some algebraic relations holding in Novikov rings. (ENGLISH)

[ Abstract ]

Given u, a de-Rham cohomology class of degree 1 of a closed manifold M, we consider the space F_u of (closed) Morse 1-forms in this class. In Morse theory, it is important to equip each α in F_u with a descending pseudo-gradient X. The case u=0 yields usual Morse theory, while u ≠ 0 yields Morse-Novikov theory, which is devoted to the understanding of the space of equipped 1-forms (α,X) with α in F_u.

Here, X is a descending pseudo-gradient, which is said to be adapted to α.

The morphism π1(M) → R induced by u (given by the integral of any α in F_u over a loop of M) determines a set of u-negative loops.

We show that for every u-negative g in π1(M), there exists a co-dimension 1 C∞-stratum Sg of F_u which is naturally co-oriented. The stratum Sg is made of elements (α, X) such that X has exactly one homoclinic orbit L whose homotopy class is g.

The goal of this talk is to show that there exists a co-dimension 1 C∞-stratum Sg (0) of Sg which lies in the closure of Sg^2. This result explains geometrically an easy algebraic relation holding in the Novikov ring associated with u.

We will mention how this study generalizes to produce some non-evident symmetric formulas holding in the Novikov ring.

Given u, a de-Rham cohomology class of degree 1 of a closed manifold M, we consider the space F_u of (closed) Morse 1-forms in this class. In Morse theory, it is important to equip each α in F_u with a descending pseudo-gradient X. The case u=0 yields usual Morse theory, while u ≠ 0 yields Morse-Novikov theory, which is devoted to the understanding of the space of equipped 1-forms (α,X) with α in F_u.

Here, X is a descending pseudo-gradient, which is said to be adapted to α.

The morphism π1(M) → R induced by u (given by the integral of any α in F_u over a loop of M) determines a set of u-negative loops.

We show that for every u-negative g in π1(M), there exists a co-dimension 1 C∞-stratum Sg of F_u which is naturally co-oriented. The stratum Sg is made of elements (α, X) such that X has exactly one homoclinic orbit L whose homotopy class is g.

The goal of this talk is to show that there exists a co-dimension 1 C∞-stratum Sg (0) of Sg which lies in the closure of Sg^2. This result explains geometrically an easy algebraic relation holding in the Novikov ring associated with u.

We will mention how this study generalizes to produce some non-evident symmetric formulas holding in the Novikov ring.

### 2015/07/07

17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Representation varieties detect essential surfaces (JAPANESE)

**Takahiro Kitayama**(Tokyo Institute of Technology)Representation varieties detect essential surfaces (JAPANESE)

[ Abstract ]

Extending Culler-Shalen theory, Hara and I presented a way to construct

certain kinds of branched surfaces (possibly without any branch) in a 3-

manifold from an ideal point of a curve in the SL_n-character variety.

There exists an essential surface in some 3-manifold known to be not

detected in the classical SL_2-theory. We show that every essential

surface in a 3-manifold is given by the ideal point of a line in the SL_

n-character variety for some n. The talk is partially based on joint

works with Stefan Friedl and Matthias Nagel, and also with Takashi Hara.

Extending Culler-Shalen theory, Hara and I presented a way to construct

certain kinds of branched surfaces (possibly without any branch) in a 3-

manifold from an ideal point of a curve in the SL_n-character variety.

There exists an essential surface in some 3-manifold known to be not

detected in the classical SL_2-theory. We show that every essential

surface in a 3-manifold is given by the ideal point of a line in the SL_

n-character variety for some n. The talk is partially based on joint

works with Stefan Friedl and Matthias Nagel, and also with Takashi Hara.

### 2015/06/30

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

The Cannon-Thurston maps and the canonical decompositions of punctured surface bundles over the circle (JAPANESE)

**Makoto Sakuma**(Hiroshima University)The Cannon-Thurston maps and the canonical decompositions of punctured surface bundles over the circle (JAPANESE)

[ Abstract ]

To each once-punctured-torus bundle over the circle with pseudo-Anosov monodromy,

there are associated two tessellations of the complex plane:

one is the triangulation of a horosphere induced by the canonical decomposition into ideal tetrahedra,

and the other is a fractal tessellation given by the Cannon-Thurston map of the fiber group.

In a joint work with Warren Dicks, I had described the relation between these two tessellations.

This result was recently generalized by Francois Gueritaud to punctured surface bundles

with pseudo-Anosov monodromy where all singuraities of the invariant foliations are at punctures.

In this talk, I will explain Gueritaud's work and related work by Naoki Sakata.

To each once-punctured-torus bundle over the circle with pseudo-Anosov monodromy,

there are associated two tessellations of the complex plane:

one is the triangulation of a horosphere induced by the canonical decomposition into ideal tetrahedra,

and the other is a fractal tessellation given by the Cannon-Thurston map of the fiber group.

In a joint work with Warren Dicks, I had described the relation between these two tessellations.

This result was recently generalized by Francois Gueritaud to punctured surface bundles

with pseudo-Anosov monodromy where all singuraities of the invariant foliations are at punctures.

In this talk, I will explain Gueritaud's work and related work by Naoki Sakata.

### 2015/06/23

17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Box complexes and model structures on the category of graphs (JAPANESE)

**Takahiro Matsushita**(The University of Tokyo)Box complexes and model structures on the category of graphs (JAPANESE)

[ Abstract ]

To determine the chromatic numbers of graphs, so-called the graph

coloring problem, is one of the most classical problems in graph theory.

Box complex is a Z_2-space associated to a graph, and it is known that

its equivariant homotopy invariant is related to the chromatic number.

Csorba showed that for each finite Z_2-CW-complex X, there is a graph

whose box complex is Z_2-homotopy equivalent to X. From this result, I

expect that the usual model category of Z_2-topological spaces is

Quillen equivalent to a certain model structure on the category of

graphs, whose weak equivalences are graph homomorphisms inducing Z_2-

homotopy equivalences between their box complexes.

In this talk, we introduce model structures on the category of graphs

whose weak equivalences are described as above. We also compare our

model categories of graphs with the category of Z_2-topological spaces.

To determine the chromatic numbers of graphs, so-called the graph

coloring problem, is one of the most classical problems in graph theory.

Box complex is a Z_2-space associated to a graph, and it is known that

its equivariant homotopy invariant is related to the chromatic number.

Csorba showed that for each finite Z_2-CW-complex X, there is a graph

whose box complex is Z_2-homotopy equivalent to X. From this result, I

expect that the usual model category of Z_2-topological spaces is

Quillen equivalent to a certain model structure on the category of

graphs, whose weak equivalences are graph homomorphisms inducing Z_2-

homotopy equivalences between their box complexes.

In this talk, we introduce model structures on the category of graphs

whose weak equivalences are described as above. We also compare our

model categories of graphs with the category of Z_2-topological spaces.

### 2015/06/16

17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Stable maps and branched shadows of 3-manifolds (JAPANESE)

**Masaharu Ishikawa**(Tohoku University)Stable maps and branched shadows of 3-manifolds (JAPANESE)

[ Abstract ]

We study what kind of stable map to the real plane a 3-manifold has. It

is known by O. Saeki that there exists a stable map without certain

singular fibers if and only if the 3-manifold is a graph manifold. According to

F. Costantino and D. Thurston, we identify the Stein factorization of a

stable map with a shadow of the 3-manifold under some modification,

where the above singular fibers correspond to the vertices of the shadow. We

define the notion of stable map complexity by counting the number of

such singular fibers and prove that this equals the branched shadow

complexity. With this equality, we give an estimation of the Gromov norm of the

3-manifold by the stable map complexity. This is a joint work with Yuya Koda.

We study what kind of stable map to the real plane a 3-manifold has. It

is known by O. Saeki that there exists a stable map without certain

singular fibers if and only if the 3-manifold is a graph manifold. According to

F. Costantino and D. Thurston, we identify the Stein factorization of a

stable map with a shadow of the 3-manifold under some modification,

where the above singular fibers correspond to the vertices of the shadow. We

define the notion of stable map complexity by counting the number of

such singular fibers and prove that this equals the branched shadow

complexity. With this equality, we give an estimation of the Gromov norm of the

3-manifold by the stable map complexity. This is a joint work with Yuya Koda.

### 2015/06/09

17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Symplectic displacement energy for exact Lagrangian immersions (JAPANESE)

**Manabu Akaho**(Tokyo Metropolitan University)Symplectic displacement energy for exact Lagrangian immersions (JAPANESE)

[ Abstract ]

We give an inequality of the displacement energy for exact Lagrangian

immersions and the symplectic area of punctured holomorphic discs. Our

approach is based on Floer homology for Lagrangian immersions and

Chekanov's homotopy technique of continuations. Moreover, we discuss our

inequality and the Hofer--Zehnder capacity.

We give an inequality of the displacement energy for exact Lagrangian

immersions and the symplectic area of punctured holomorphic discs. Our

approach is based on Floer homology for Lagrangian immersions and

Chekanov's homotopy technique of continuations. Moreover, we discuss our

inequality and the Hofer--Zehnder capacity.

### 2015/05/26

17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Introduction to formalization of topology using a proof assistant. (JAPANESE)

**Ken'ichi Kuga**(Chiba University)Introduction to formalization of topology using a proof assistant. (JAPANESE)

[ Abstract ]

Although the program of formalization goes back to David

Hilbert, it is only recently that we can actually formalize

substantial theorems in modern mathematics. It is made possible by the

development of certain type theory and a computer software called a

proof assistant. We begin this talk by showing our formalization of

some basic geometric topology using a proof assistant COQ. Then we

introduce homotopy type theory (HoTT) of Voevodsky et al., which

interprets type theory from abstract homotopy theoretic perspective.

HoTT proposes "univalent" foundation of mathematics which is

particularly suited for computer formalization.

Although the program of formalization goes back to David

Hilbert, it is only recently that we can actually formalize

substantial theorems in modern mathematics. It is made possible by the

development of certain type theory and a computer software called a

proof assistant. We begin this talk by showing our formalization of

some basic geometric topology using a proof assistant COQ. Then we

introduce homotopy type theory (HoTT) of Voevodsky et al., which

interprets type theory from abstract homotopy theoretic perspective.

HoTT proposes "univalent" foundation of mathematics which is

particularly suited for computer formalization.