## Tuesday Seminar on Topology

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

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

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

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

**Seminar information archive**

### 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.

### 2015/05/19

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

Quiver mutation loops and partition q-series (JAPANESE)

**Akishi Kato**(The University of Tokyo)Quiver mutation loops and partition q-series (JAPANESE)

[ Abstract ]

Quivers and their mutations are ubiquitous in mathematics and

mathematical physics; they play a key role in cluster algebras,

wall-crossing phenomena, gluing of ideal tetrahedra, etc.

Recently, we introduced a partition q-series for a quiver mutation loop

(a loop in a quiver exchange graph) using the idea of state sum of statistical

mechanics. The partition q-series enjoy some nice properties such

as pentagon move invariance. We also discuss their relation with combinatorial

Donaldson-Thomas invariants, as well as fermionic character formulas of

certain conformal field theories.

This is a joint work with Yuji Terashima.

Quivers and their mutations are ubiquitous in mathematics and

mathematical physics; they play a key role in cluster algebras,

wall-crossing phenomena, gluing of ideal tetrahedra, etc.

Recently, we introduced a partition q-series for a quiver mutation loop

(a loop in a quiver exchange graph) using the idea of state sum of statistical

mechanics. The partition q-series enjoy some nice properties such

as pentagon move invariance. We also discuss their relation with combinatorial

Donaldson-Thomas invariants, as well as fermionic character formulas of

certain conformal field theories.

This is a joint work with Yuji Terashima.

### 2015/05/12

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

Growth rate of the number of periodic points for generic dynamical systems (JAPANESE)

**Masayuki Asaoka**(Kyoto University)Growth rate of the number of periodic points for generic dynamical systems (JAPANESE)

[ Abstract ]

For any hyperbolic dynamical system, the number of periodic

points grows at most exponentially and the growth rate

reflects statistic property of the system. For dynamics far

from hyperbolicity, the situation is different. In 1999,

Kaloshin proved genericity of super-exponential growth in the

region where dense set of dynamical systems exhibits homoclinic

tangency (so called the Newhouse region).

How does the number of periodic points grow for generic

partially hyperbolic dynamical systems? Such systems are known

to be far from homoclinic tangency. Is the growth at most

exponential like hyperbolic system, or super-exponential by

a mechanism different from homoclinic tangency?

The speaker, Katsutoshi Shinohara, and Dimitry Turaev proved

super-exponential growth of the number of periodic points for

generic one-dimensional iterated function systems under some

reasonable conditions. Such systems are models of dynamics

of partially hyperbolic systems in neutral direction. So, we

expect genericity of super-exponential growth in a region of

partially hyperbolic systems.

In this talk, we start with a brief history of the problem on

growth rate of the number of periodic point and discuss two

mechanisms which lead to genericity of super-exponential growth,

Kaloshin's one and ours.

For any hyperbolic dynamical system, the number of periodic

points grows at most exponentially and the growth rate

reflects statistic property of the system. For dynamics far

from hyperbolicity, the situation is different. In 1999,

Kaloshin proved genericity of super-exponential growth in the

region where dense set of dynamical systems exhibits homoclinic

tangency (so called the Newhouse region).

How does the number of periodic points grow for generic

partially hyperbolic dynamical systems? Such systems are known

to be far from homoclinic tangency. Is the growth at most

exponential like hyperbolic system, or super-exponential by

a mechanism different from homoclinic tangency?

The speaker, Katsutoshi Shinohara, and Dimitry Turaev proved

super-exponential growth of the number of periodic points for

generic one-dimensional iterated function systems under some

reasonable conditions. Such systems are models of dynamics

of partially hyperbolic systems in neutral direction. So, we

expect genericity of super-exponential growth in a region of

partially hyperbolic systems.

In this talk, we start with a brief history of the problem on

growth rate of the number of periodic point and discuss two

mechanisms which lead to genericity of super-exponential growth,

Kaloshin's one and ours.

### 2015/05/07

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

The group of parenthesized braids (ENGLISH)

**Patrick Dehornoy**(Univ. de Caen)The group of parenthesized braids (ENGLISH)

[ Abstract ]

We describe a group B obtained by gluing in a natural way two well-known

groups, namely Artin's braid group B_infty and Thompson's group F. The

elements of B correspond to braid diagrams in which the distances

between the strands are non uniform and some rescaling operators may

change these distances. The group B shares many properties with B_infty:

as the latter, it can be realized as a subgroup of a mapping class

group, namely that of a sphere with a Cantor set removed, and as a group

of automorphisms of a free group. Technically, the key point is the

existence of a self-distributive operation on B.

We describe a group B obtained by gluing in a natural way two well-known

groups, namely Artin's braid group B_infty and Thompson's group F. The

elements of B correspond to braid diagrams in which the distances

between the strands are non uniform and some rescaling operators may

change these distances. The group B shares many properties with B_infty:

as the latter, it can be realized as a subgroup of a mapping class

group, namely that of a sphere with a Cantor set removed, and as a group

of automorphisms of a free group. Technically, the key point is the

existence of a self-distributive operation on B.

### 2015/04/28

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

Verify hyperbolicity of 3-manifolds by computer and its applications. (JAPANESE)

**Hidetoshi Masai**(The University of Tokyo, JSPS)Verify hyperbolicity of 3-manifolds by computer and its applications. (JAPANESE)

[ Abstract ]

In this talk I will talk about the program called HIKMOT which

rigorously proves hyperbolicity of a given triangulated 3-manifold. To

prove hyperbolicity of a given triangulated 3-manifold, it suffices to

get a solution of Thurston's gluing equation. We use the notion called

interval arithmetic to overcome two types errors; round-off errors,

and truncated errors. I will also talk about its application to

exceptional surgeries along alternating knots. This talk is based on

joint work with N. Hoffman, K. Ichihara, M. Kashiwagi, S. Oishi, and

A. Takayasu.

In this talk I will talk about the program called HIKMOT which

rigorously proves hyperbolicity of a given triangulated 3-manifold. To

prove hyperbolicity of a given triangulated 3-manifold, it suffices to

get a solution of Thurston's gluing equation. We use the notion called

interval arithmetic to overcome two types errors; round-off errors,

and truncated errors. I will also talk about its application to

exceptional surgeries along alternating knots. This talk is based on

joint work with N. Hoffman, K. Ichihara, M. Kashiwagi, S. Oishi, and

A. Takayasu.

### 2015/04/21

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

Orbit equivalence relations arising from Baumslag-Solitar groups (JAPANESE)

**Yoshikata Kida**(The University of Tokyo)Orbit equivalence relations arising from Baumslag-Solitar groups (JAPANESE)

[ Abstract ]

This talk is about measure-preserving actions of countable groups on probability

measure spaces and their orbit structure. Two such actions are called orbit equivalent

if there exists an isomorphism between the spaces preserving orbits. In this talk, I focus

on actions of Baumslag-Solitar groups that have two generators, a and t, with the relation

ta^p=a^qt, where p and q are given integers. This group is well studied in combinatorial

and geometric group theory. Whether Baumslag-Solitar groups with different p and q can

have orbit-equivalent actions is still a big open problem. I will discuss invariants under

orbit equivalence, motivating background and some results toward this problem.

This talk is about measure-preserving actions of countable groups on probability

measure spaces and their orbit structure. Two such actions are called orbit equivalent

if there exists an isomorphism between the spaces preserving orbits. In this talk, I focus

on actions of Baumslag-Solitar groups that have two generators, a and t, with the relation

ta^p=a^qt, where p and q are given integers. This group is well studied in combinatorial

and geometric group theory. Whether Baumslag-Solitar groups with different p and q can

have orbit-equivalent actions is still a big open problem. I will discuss invariants under

orbit equivalence, motivating background and some results toward this problem.

### 2015/04/14

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

Pin(2)-monopole invariants for 4-manifolds (JAPANESE)

**Nobuhiro Nakamura**(Gakushuin University)Pin(2)-monopole invariants for 4-manifolds (JAPANESE)

[ Abstract ]

The Pin(2)-monopole equations are a variant of the Seiberg-Witten equations

which can be considered as a real version of the SW equations. A Pin(2)-mono

pole version of the Seiberg-Witten invariants is defined, and a special feature of

this is that the Pin(2)-monopole invariant can be nontrivial even when all of

the Donaldson and Seiberg-Witten invariants vanish. As an application, we

construct a new series of exotic 4-manifolds.

The Pin(2)-monopole equations are a variant of the Seiberg-Witten equations

which can be considered as a real version of the SW equations. A Pin(2)-mono

pole version of the Seiberg-Witten invariants is defined, and a special feature of

this is that the Pin(2)-monopole invariant can be nontrivial even when all of

the Donaldson and Seiberg-Witten invariants vanish. As an application, we

construct a new series of exotic 4-manifolds.

### 2015/04/07

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

Potential functions for Grassmannians (JAPANESE)

**Kazushi Ueda**(The University of Tokyo)Potential functions for Grassmannians (JAPANESE)

[ Abstract ]

Potential functions are Floer-theoretic invariants

obtained by counting Maslov index 2 disks

with Lagrangian boundary conditions.

In the talk, we will discuss our joint work

with Yanki Lekili and Yuichi Nohara

on Lagrangian torus fibrations on the Grassmannian

of 2-planes in an n-space,

the potential functions of their Lagrangian torus fibers,

and their relation with mirror symmetry for Grassmannians.

Potential functions are Floer-theoretic invariants

obtained by counting Maslov index 2 disks

with Lagrangian boundary conditions.

In the talk, we will discuss our joint work

with Yanki Lekili and Yuichi Nohara

on Lagrangian torus fibrations on the Grassmannian

of 2-planes in an n-space,

the potential functions of their Lagrangian torus fibers,

and their relation with mirror symmetry for Grassmannians.

### 2015/03/24

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

Knots and Mirror Symmetry (ENGLISH)

**Mina Aganagic**(University of California, Berkeley)Knots and Mirror Symmetry (ENGLISH)

[ Abstract ]

I will describe two conjectures relating knot theory and mirror symmetry. One can associate, to every knot K, one a Calabi-Yau manifold Y(K), which depends on the homotopy type of the knot only. The first conjecture is that Y(K) arises by a generalization of SYZ mirror symmetry, as mirror to the conifold, O(-1)+O(-1)->P^1. The second conjecture is that topological string provides a quantization of Y(K) which leads to quantum HOMFLY invariants of the knot. The conjectures are based on joint work with C. Vafa and also with T.Ekholm, L. Ng.

I will describe two conjectures relating knot theory and mirror symmetry. One can associate, to every knot K, one a Calabi-Yau manifold Y(K), which depends on the homotopy type of the knot only. The first conjecture is that Y(K) arises by a generalization of SYZ mirror symmetry, as mirror to the conifold, O(-1)+O(-1)->P^1. The second conjecture is that topological string provides a quantization of Y(K) which leads to quantum HOMFLY invariants of the knot. The conjectures are based on joint work with C. Vafa and also with T.Ekholm, L. Ng.

### 2015/03/10

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

Arnold conjecture, Floer homology,

and augmentation ideals of finite groups.

(ENGLISH)

**Andrei Pajitnov**(Univ. de Nantes)Arnold conjecture, Floer homology,

and augmentation ideals of finite groups.

(ENGLISH)

[ Abstract ]

Let H be a generic time-dependent 1-periodic

Hamiltonian on a closed weakly monotone

symplectic manifold M. We construct a refined version

of the Floer chain complex associated to (M,H),

and use it to obtain new lower bounds for the number P(H)

of the 1-periodic orbits of the corresponding hamiltonian

vector field. We prove in particular that

if the fundamental group of M is finite

and solvable or simple, then P(H)

is not less than the minimal number

of generators of the fundamental group.

This is joint work with Kaoru Ono.

Let H be a generic time-dependent 1-periodic

Hamiltonian on a closed weakly monotone

symplectic manifold M. We construct a refined version

of the Floer chain complex associated to (M,H),

and use it to obtain new lower bounds for the number P(H)

of the 1-periodic orbits of the corresponding hamiltonian

vector field. We prove in particular that

if the fundamental group of M is finite

and solvable or simple, then P(H)

is not less than the minimal number

of generators of the fundamental group.

This is joint work with Kaoru Ono.

### 2015/01/20

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

On Lagrangian caps and their applications (JAPANESE)

**Toru Yoshiyasu**(The University of Tokyo)On Lagrangian caps and their applications (JAPANESE)

[ Abstract ]

In 2013, Y. Eliashberg and E. Murphy established the $h$-principle for

exact Lagrangian embeddings with a concave Legendrian boundary. In this

talk, I will explain a modification of their $h$-principle and show

applications to Lagrangian submanifolds in the complex projective spaces.

In 2013, Y. Eliashberg and E. Murphy established the $h$-principle for

exact Lagrangian embeddings with a concave Legendrian boundary. In this

talk, I will explain a modification of their $h$-principle and show

applications to Lagrangian submanifolds in the complex projective spaces.

### 2015/01/13

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

Stable presentation length of 3-manifold groups (JAPANESE)

**Ken'ichi Yoshida**(The University of Tokyo)Stable presentation length of 3-manifold groups (JAPANESE)

[ Abstract ]

We will introduce the stable presentation length

of a finitely presented group, which is defined

by stabilizing the presentation length for the

finite index subgroups. The stable presentation

length of the fundamental group of a 3-manifold

is an analogue of the simplicial volume and the

stable complexity introduced by Francaviglia,

Frigerio and Martelli. We will explain some

similarities of stable presentation length with

simplicial volume and stable complexity.

We will introduce the stable presentation length

of a finitely presented group, which is defined

by stabilizing the presentation length for the

finite index subgroups. The stable presentation

length of the fundamental group of a 3-manifold

is an analogue of the simplicial volume and the

stable complexity introduced by Francaviglia,

Frigerio and Martelli. We will explain some

similarities of stable presentation length with

simplicial volume and stable complexity.

### 2014/12/16

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

Differential forms in diffeological spaces (JAPANESE)

**Norio Iwase**(Kyushu University)Differential forms in diffeological spaces (JAPANESE)

[ Abstract ]

The idea of a space with smooth structure is first introduced by K. T. Chen in his study of a loop space to employ the idea of iterated path integrals.

Following the pattern established by Chen, J. M. Souriau introduced his version of a space with smooth structure which is now called diffeology and become one of the most exciting topics in Algebraic Topology. Following Souriau, P. I.-Zenmour presented de Rham theory associated to a diffeology of a space. However, if one tries to show a version of de Rham theorem for a general diffeological space, he must encounter a difficulty to show the existence of a partition of unity and thus the exactness of the Mayer-Vietoris sequence. To resolve such difficulties, we introduce a new definition of differential forms.

The idea of a space with smooth structure is first introduced by K. T. Chen in his study of a loop space to employ the idea of iterated path integrals.

Following the pattern established by Chen, J. M. Souriau introduced his version of a space with smooth structure which is now called diffeology and become one of the most exciting topics in Algebraic Topology. Following Souriau, P. I.-Zenmour presented de Rham theory associated to a diffeology of a space. However, if one tries to show a version of de Rham theorem for a general diffeological space, he must encounter a difficulty to show the existence of a partition of unity and thus the exactness of the Mayer-Vietoris sequence. To resolve such difficulties, we introduce a new definition of differential forms.

### 2014/12/09

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

Stable commutator length on mapping class groups (JAPANESE)

**Koji Fujiwara**(Kyoto University)Stable commutator length on mapping class groups (JAPANESE)

[ Abstract ]

Let MCG(S) be the mapping class group of a closed orientable surface S.

We give a precise condition (in terms of the Nielsen-Thurston

decomposition) when an element

in MCG(S) has positive stable commutator length.

Stable commutator length tends to be positive if there is "negative

curvature".

The proofs use our earlier construction in the paper "Constructing group

actions on quasi-trees and applications to mapping class groups" of

group actions on quasi-trees.

This is a joint work with Bestvina and Bromberg.

Let MCG(S) be the mapping class group of a closed orientable surface S.

We give a precise condition (in terms of the Nielsen-Thurston

decomposition) when an element

in MCG(S) has positive stable commutator length.

Stable commutator length tends to be positive if there is "negative

curvature".

The proofs use our earlier construction in the paper "Constructing group

actions on quasi-trees and applications to mapping class groups" of

group actions on quasi-trees.

This is a joint work with Bestvina and Bromberg.

### 2014/12/02

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

The Atiyah-Segal completion theorem in noncommutative topology (JAPANESE)

**Yosuke Kubota**(The University of Tokyo)The Atiyah-Segal completion theorem in noncommutative topology (JAPANESE)

[ Abstract ]

We introduce a new perspevtive on the Atiyah-Segal completion

theorem applying the "noncommutative" topology, which deals with

topological properties of C*-algebras. The homological algebra of the

Kasparov category as a triangulated category, which is developed by R.

Meyer and R. Nest, plays a central role. It contains the Atiyah-Segal

type completion theorems for equivariant K-homology and twisted K-theory.

This is a joint work with Yuki Arano.

We introduce a new perspevtive on the Atiyah-Segal completion

theorem applying the "noncommutative" topology, which deals with

topological properties of C*-algebras. The homological algebra of the

Kasparov category as a triangulated category, which is developed by R.

Meyer and R. Nest, plays a central role. It contains the Atiyah-Segal

type completion theorems for equivariant K-homology and twisted K-theory.

This is a joint work with Yuki Arano.