## Seminar information archive

Seminar information archive ～11/15｜Today's seminar 11/16 | Future seminars 11/17～

#### Algebraic Geometry Seminar

15:00-16:30 Room #122 (Graduate School of Math. Sci. Bldg.)

Divisorial Extractions from Singular Curves in Smooth 3-Folds (ENGLISH)

**Thomas Ducat**(University of Warwick)Divisorial Extractions from Singular Curves in Smooth 3-Folds (ENGLISH)

[ Abstract ]

Consider a singular curve C contained in a smooth 3-fold X.

Assuming the existence of a Du Val general elephant S containing C,

I give a normal form for the equations of C in X and an outline of how to

construct a divisorial extraction from this curve. If the general S is

Du Val of type D_{2k}, E_6 or E_7 then I can give some explicit

conditions for the existence of a terminal extraction. A treatment of

the D_{2k+1} case should be possible by similar means.

Consider a singular curve C contained in a smooth 3-fold X.

Assuming the existence of a Du Val general elephant S containing C,

I give a normal form for the equations of C in X and an outline of how to

construct a divisorial extraction from this curve. If the general S is

Du Val of type D_{2k}, E_6 or E_7 then I can give some explicit

conditions for the existence of a terminal extraction. A treatment of

the D_{2k+1} case should be possible by similar means.

### 2014/01/21

#### Tuesday Seminar of Analysis

16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

An improved level set method based on comparison with a signed distance function (JAPANESE)

**Nao Hamamuki**(Graduate shool of Mathematical Sciences, the University of Tokyo)An improved level set method based on comparison with a signed distance function (JAPANESE)

[ Abstract ]

In the classical level set method, a slope of a solution to level set

equations can be close to zero as time develops even if the initial

slope is large, and this prevents one from computing interfaces given as

the level set of the solution. To overcome this issue we introduce an

improved equation by adding an extra term to the original equation.

Then, by applying a comparison principle to the signed distance function

to the interface, we prove that, globally in time, the slope of a

solution of the initial value problem is preserved near the zero level set.

In the classical level set method, a slope of a solution to level set

equations can be close to zero as time develops even if the initial

slope is large, and this prevents one from computing interfaces given as

the level set of the solution. To overcome this issue we introduce an

improved equation by adding an extra term to the original equation.

Then, by applying a comparison principle to the signed distance function

to the interface, we prove that, globally in time, the slope of a

solution of the initial value problem is preserved near the zero level set.

#### Tuesday Seminar on Topology

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

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

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

[ Abstract ]

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

obstruction for codimension two contact embeddings in the odd

dimensional Euclidean space.

By Gromov's h-principle,

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

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

embedding.

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

obstruction for codimension two contact embeddings in the odd

dimensional Euclidean space.

By Gromov's h-principle,

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

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

embedding.

#### Tuesday Seminar on Topology

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

Weak eigenvalues in homoclinic classes: perturbations from saddles

with small angles (ENGLISH)

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

with small angles (ENGLISH)

[ Abstract ]

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

new sufficient condition for creating weak contracting eigenvalues is

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

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

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

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

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

heterodimensional cycles. In particular, this sufficient condition is

satisfied by a substantial class of saddles with homoclinic tangencies.

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

new sufficient condition for creating weak contracting eigenvalues is

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

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

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

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

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

heterodimensional cycles. In particular, this sufficient condition is

satisfied by a substantial class of saddles with homoclinic tangencies.

### 2014/01/20

#### Seminar on Geometric Complex Analysis

10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)

タイヒミュラー距離の幾何学とその応用 (JAPANESE)

**Hideki Miyachi**(Osaka University)タイヒミュラー距離の幾何学とその応用 (JAPANESE)

#### GCOE Seminars

16:00-17:00 Room #118 (Graduate School of Math. Sci. Bldg.)

Increasing stability in the inverse problems for the Helmholtz type prposed equations (ENGLISH)

**Victor Isakov**(The Wichita State University)Increasing stability in the inverse problems for the Helmholtz type prposed equations (ENGLISH)

[ Abstract ]

We report on new stability estimates for recovery of the near field from the prposed scattering amplitude prposed and for Schroedinger potential from the Dirichlet-to Neumann map. In these prposed esrtimates prposed unstable (logarithmic part) goes to zero as the wave number grows. Proofs prposed are using prposed new bounds for Hankel functions and complex and real geometrical optics prposed solutions.

We report on new stability estimates for recovery of the near field from the prposed scattering amplitude prposed and for Schroedinger potential from the Dirichlet-to Neumann map. In these prposed esrtimates prposed unstable (logarithmic part) goes to zero as the wave number grows. Proofs prposed are using prposed new bounds for Hankel functions and complex and real geometrical optics prposed solutions.

#### Algebraic Geometry Seminar

15:30-17:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Deforming elephants of Q-Fano 3-folds (ENGLISH)

**Taro Sano**(University of Warwick)Deforming elephants of Q-Fano 3-folds (ENGLISH)

[ Abstract ]

Shokurov and Reid proved that a Fano 3-fold with canonical

Gorenstein singularities has a Du Val elephant, that is,

a member of the anticanonical linear system with only Du Val singularities.

The classification of Fano 3-folds is based on this fact.

However, for a Fano 3-fold with non-Gorenstein terminal singularities,

the anticanonical system does not contain such a member in general.

Alt{\\i}nok--Brown--Reid conjectured that, if the anticanonical system is non-empty,

a Q-Fano 3-fold can be deformed to that with a Du Val elephant.

In this talk, I will explain how to deform an elephant with isolated

singularities to a Du Val elephant.

Shokurov and Reid proved that a Fano 3-fold with canonical

Gorenstein singularities has a Du Val elephant, that is,

a member of the anticanonical linear system with only Du Val singularities.

The classification of Fano 3-folds is based on this fact.

However, for a Fano 3-fold with non-Gorenstein terminal singularities,

the anticanonical system does not contain such a member in general.

Alt{\\i}nok--Brown--Reid conjectured that, if the anticanonical system is non-empty,

a Q-Fano 3-fold can be deformed to that with a Du Val elephant.

In this talk, I will explain how to deform an elephant with isolated

singularities to a Du Val elephant.

### 2014/01/15

#### FMSP Lectures

14:50-16:20 Room #056 (Graduate School of Math. Sci. Bldg.)

Lectures on quantum Teichmüller theory IV (ENGLISH)

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Kashaev.pdf

**Rinat Kashaev**(University of Geneva)Lectures on quantum Teichmüller theory IV (ENGLISH)

[ Abstract ]

Quantum Teichmüller theory leads to a family of unitary infinite dimensional projective representations of the mapping class groups of punctured surfaces. One of the recent applications of this theory is the construction of state integral three-manifold invariants related with hyperbolic geometry.

In these lectures it is planned to address the following subjects:

1) Penner’s coordinates in the decorated Teichmüller space.

2) Ratio coordinates.

3) Quantization.

4) The length spectrum of simple closed curves.

[ Reference URL ]Quantum Teichmüller theory leads to a family of unitary infinite dimensional projective representations of the mapping class groups of punctured surfaces. One of the recent applications of this theory is the construction of state integral three-manifold invariants related with hyperbolic geometry.

In these lectures it is planned to address the following subjects:

1) Penner’s coordinates in the decorated Teichmüller space.

2) Ratio coordinates.

3) Quantization.

4) The length spectrum of simple closed curves.

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Kashaev.pdf

#### Operator Algebra Seminars

16:30-18:00 Room #118 (Graduate School of Math. Sci. Bldg.)

Wedge-local fields in integrable models with bound states (JAPANESE)

**Yoh Tanimoto**(Univ. Tokyo)Wedge-local fields in integrable models with bound states (JAPANESE)

#### GCOE Seminars

16:00-17:00 Room #370 (Graduate School of Math. Sci. Bldg.)

Increasing stability of the continuation for the Helmholtz type equations (ENGLISH)

**Victor Isakov**(The Wichita State University)Increasing stability of the continuation for the Helmholtz type equations (ENGLISH)

[ Abstract ]

We derive conditional stability estimates for the Helmholtz type equations which are becoming of Lipschitz type for large frequencies/wave numbers. Proofs use splitting solutions into low and high frequencies parts where we use energy (in particular) Carleman estimates. We discuss numerical confirmation and open problems.

We derive conditional stability estimates for the Helmholtz type equations which are becoming of Lipschitz type for large frequencies/wave numbers. Proofs use splitting solutions into low and high frequencies parts where we use energy (in particular) Carleman estimates. We discuss numerical confirmation and open problems.

#### GCOE Seminars

17:00-18:00 Room #370 (Graduate School of Math. Sci. Bldg.)

A numerical method for solving the inverse heat conduction problem without initial value (ENGLISH)

**Jin Cheng**(Fudan University)A numerical method for solving the inverse heat conduction problem without initial value (ENGLISH)

[ Abstract ]

In this talk, we will present some results for the inverse heat conduction problem for the heat equation of determining a boundary value at in an unreachable part of the boundary. The main difficulty for this problem is that the initial value is unknown by the practical reason. A new method is prposed to solve this problem and the nuemrical tests show the effective of this method. Some theoretic analysis will be presented. This is a joint work with J Nakagawa, YB Wang, M Yamamoto.

In this talk, we will present some results for the inverse heat conduction problem for the heat equation of determining a boundary value at in an unreachable part of the boundary. The main difficulty for this problem is that the initial value is unknown by the practical reason. A new method is prposed to solve this problem and the nuemrical tests show the effective of this method. Some theoretic analysis will be presented. This is a joint work with J Nakagawa, YB Wang, M Yamamoto.

#### Number Theory Seminar

16:40-17:40 Room #056 (Graduate School of Math. Sci. Bldg.)

Special values of zeta-functions of schemes (ENGLISH)

**Stephen Lichtenbaum**(Brown University)Special values of zeta-functions of schemes (ENGLISH)

[ Abstract ]

We will give conjectured formulas giving the behavior of the

seta-function of regular schemes projective and flat over Spec Z at

non-positive integers in terms of Weil-etale cohomology. We will also

explain the conjectured relationship of Weil-etale cohomology to etale

cohomology, which makes it possible to express these formulas also in terms

of etale cohomology.

We will give conjectured formulas giving the behavior of the

seta-function of regular schemes projective and flat over Spec Z at

non-positive integers in terms of Weil-etale cohomology. We will also

explain the conjectured relationship of Weil-etale cohomology to etale

cohomology, which makes it possible to express these formulas also in terms

of etale cohomology.

### 2014/01/14

#### FMSP Lectures

14:50-16:20 Room #056 (Graduate School of Math. Sci. Bldg.)

Lectures on quantum Teichmüller theory III (ENGLISH)

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Kashaev.pdf

**Rinat Kashaev**(University of Geneva)Lectures on quantum Teichmüller theory III (ENGLISH)

[ Abstract ]

Quantum Teichmüller theory leads to a family of unitary infinite dimensional projective representations of the mapping class groups of punctured surfaces. One of the recent applications of this theory is the construction of state integral three-manifold invariants related with hyperbolic geometry.

In these lectures it is planned to address the following subjects:

1) Penner’s coordinates in the decorated Teichmüller space.

2) Ratio coordinates.

3) Quantization.

4) The length spectrum of simple closed curves.

[ Reference URL ]Quantum Teichmüller theory leads to a family of unitary infinite dimensional projective representations of the mapping class groups of punctured surfaces. One of the recent applications of this theory is the construction of state integral three-manifold invariants related with hyperbolic geometry.

In these lectures it is planned to address the following subjects:

1) Penner’s coordinates in the decorated Teichmüller space.

2) Ratio coordinates.

3) Quantization.

4) The length spectrum of simple closed curves.

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Kashaev.pdf

#### Tuesday Seminar on Topology

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

State-integral partition functions on shaped triangulations (ENGLISH)

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

[ Abstract ]

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

generalized TQFT within the combinatorial framework of shaped

triangulations with the tetrahedral weight functions given in

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

quantum dilogarithm. By using simple examples, I will

illustrate the connection of this theory with the hyperbolic

geometry in three dimensions.

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

generalized TQFT within the combinatorial framework of shaped

triangulations with the tetrahedral weight functions given in

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

quantum dilogarithm. By using simple examples, I will

illustrate the connection of this theory with the hyperbolic

geometry in three dimensions.

#### Tuesday Seminar of Analysis

16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Threshold properties for one-dimensional discrete Schr\\"odinger operators (JAPANESE)

**Kenichi Ito**(University of Tsukuba)Threshold properties for one-dimensional discrete Schr\\"odinger operators (JAPANESE)

[ Abstract ]

We study the relation between the generalized eigenspace and the asymptotic expansion of the resolvent around the threshold $0$ for the one-dimensional discrete Schr\\"odinger operator on $\\mathbb Z$. We decompose the generalized eigenspace into the subspaces corresponding to the eigenstates and the resonance states only by their asymptotics at infinity, and classify the coefficient operators of the singlar part of resolvent expansion completely in terms of these eigenspaces. Here the generalized eigenspace we consider is largest possible. For an explicit computation of the resolvent expansion we apply the expansion scheme of Jensen-Nenciu (2001). This talk is based on the recent joint work with Arne Jensen (Aalborg University).

We study the relation between the generalized eigenspace and the asymptotic expansion of the resolvent around the threshold $0$ for the one-dimensional discrete Schr\\"odinger operator on $\\mathbb Z$. We decompose the generalized eigenspace into the subspaces corresponding to the eigenstates and the resonance states only by their asymptotics at infinity, and classify the coefficient operators of the singlar part of resolvent expansion completely in terms of these eigenspaces. Here the generalized eigenspace we consider is largest possible. For an explicit computation of the resolvent expansion we apply the expansion scheme of Jensen-Nenciu (2001). This talk is based on the recent joint work with Arne Jensen (Aalborg University).

#### Lie Groups and Representation Theory

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

A cellular classification of simple modules of the Hecke-Clifford

superalgebra (JAPANESE)

**Masaki Mori**(the University of Tokyo)A cellular classification of simple modules of the Hecke-Clifford

superalgebra (JAPANESE)

[ Abstract ]

The Hecke--Clifford superalgebra is a super version of

the Iwahori--Hecke algebra of type A. Its simple modules

are classified by Brundan, Kleshchev and Tsuchioka using

a method of categorification of affine Lie algebras.

However their constructions are too abstract to study in practice.

In this talk, we introduce a more concrete way to produce its

simple modules with a generalized theory of cellular algebras

which is originally developed by Graham and Lehrer.

In our construction the key is that there is a right action of

the Clifford superalgebra on the super-analogue of the Specht module.

With the help of the notion of the Morita context, a simple module

of the Hecke--Clifford superalgebra is made from that of

the Clifford superalgebra.

The Hecke--Clifford superalgebra is a super version of

the Iwahori--Hecke algebra of type A. Its simple modules

are classified by Brundan, Kleshchev and Tsuchioka using

a method of categorification of affine Lie algebras.

However their constructions are too abstract to study in practice.

In this talk, we introduce a more concrete way to produce its

simple modules with a generalized theory of cellular algebras

which is originally developed by Graham and Lehrer.

In our construction the key is that there is a right action of

the Clifford superalgebra on the super-analogue of the Specht module.

With the help of the notion of the Morita context, a simple module

of the Hecke--Clifford superalgebra is made from that of

the Clifford superalgebra.

#### GCOE Seminars

15:00-16:00 Room #118 (Graduate School of Math. Sci. Bldg.)

Calderon problem for Maxwell's equations in cylindrical domain (ENGLISH)

**Oleg Emanouilov**(Colorado State University)Calderon problem for Maxwell's equations in cylindrical domain (ENGLISH)

[ Abstract ]

We prove some uniqueness results in determination of the conductivity, the permeability and the permittivity of Maxwell's equations from partial Dirichlet-to-Neumann map.

We prove some uniqueness results in determination of the conductivity, the permeability and the permittivity of Maxwell's equations from partial Dirichlet-to-Neumann map.

### 2014/01/11

#### Monthly Seminar on Arithmetic of Automorphic Forms

14:00-16:00 Room #123 (Graduate School of Math. Sci. Bldg.)

A product of tensor product L-functions of quasi-split classical groups of Hermitian type. (jointly with Lei Zhang) (ENGLISH)

A product of tensor product L-functions of quasi-split classical groups of Hermitian type, Part II. (jointly with Lei Zhang) (ENGLISH)

**Dihua Jiang**(School of Mathematics, University of Minnesota) 14:00-14:45A product of tensor product L-functions of quasi-split classical groups of Hermitian type. (jointly with Lei Zhang) (ENGLISH)

**Dihua Jiang**(School of Mathematics, University of Minnesota) 15:00-15:45A product of tensor product L-functions of quasi-split classical groups of Hermitian type, Part II. (jointly with Lei Zhang) (ENGLISH)

### 2014/01/10

#### FMSP Lectures

14:50-16:20 Room #056 (Graduate School of Math. Sci. Bldg.)

Lectures on quantum Teichmüller theory II (ENGLISH)

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Kashaev.pdf

**Rinat Kashaev**(University of Geneva)Lectures on quantum Teichmüller theory II (ENGLISH)

[ Abstract ]

Quantum Teichmüller theory leads to a family of unitary infinite dimensional projective representations of the mapping class groups of punctured surfaces. One of the recent applications of this theory is the construction of state integral three-manifold invariants related with hyperbolic geometry.

In these lectures it is planned to address the following subjects:

1) Penner’s coordinates in the decorated Teichmüller space.

2) Ratio coordinates.

3) Quantization.

4) The length spectrum of simple closed curves.

[ Reference URL ]Quantum Teichmüller theory leads to a family of unitary infinite dimensional projective representations of the mapping class groups of punctured surfaces. One of the recent applications of this theory is the construction of state integral three-manifold invariants related with hyperbolic geometry.

In these lectures it is planned to address the following subjects:

1) Penner’s coordinates in the decorated Teichmüller space.

2) Ratio coordinates.

3) Quantization.

4) The length spectrum of simple closed curves.

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Kashaev.pdf

### 2014/01/09

#### FMSP Lectures

14:50-16:20 Room #056 (Graduate School of Math. Sci. Bldg.)

Lectures on quantum Teichmüller theory I (ENGLISH)

[ Abstract ]

Quantum Teichmüller theory leads to a family of unitary infinite dimensional projective representations of the mapping class groups of punctured surfaces. One of the recent applications of this theory is the construction of state integral three-manifold invariants related with hyperbolic geometry.

In these lectures it is planned to address the following subjects:

1) Penner’s coordinates in the decorated Teichmüller space.

2) Ratio coordinates.

3) Quantization.

4) The length spectrum of simple closed curves.
[ Reference URL ]

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Kashaev.pdf

**Rinat Kashaev**(University of Geneva)Lectures on quantum Teichmüller theory I (ENGLISH)

Quantum Teichmüller theory leads to a family of unitary infinite dimensional projective representations of the mapping class groups of punctured surfaces. One of the recent applications of this theory is the construction of state integral three-manifold invariants related with hyperbolic geometry.

In these lectures it is planned to address the following subjects:

1) Penner’s coordinates in the decorated Teichmüller space.

2) Ratio coordinates.

3) Quantization.

4) The length spectrum of simple closed curves.

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Kashaev.pdf

### 2014/01/08

#### Operator Algebra Seminars

16:30-18:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Dow's metrization theorem and beyond (JAPANESE)

**Sakae Fuchino**(Kobe University)Dow's metrization theorem and beyond (JAPANESE)

#### Number Theory Seminar

16:40-17:40 Room #056 (Graduate School of Math. Sci. Bldg.)

Roots of the discriminant of an elliptic curves and its torsion points (JAPANESE)

**Sho Yoshikawa**(University of Tokyo)Roots of the discriminant of an elliptic curves and its torsion points (JAPANESE)

[ Abstract ]

We give an explicit and intrinsic description of (the torsor defined by the 12th roots of) the discriminant of an elliptic curve using the group of its 12-torsion points and the Weil pairing. As an application, we extend a result of Coates (which deals with the characteristic 0 case) to the case where the characteristic of the base field is not 2 or 3. This is a joint work with Kohei Fukuda.

We give an explicit and intrinsic description of (the torsor defined by the 12th roots of) the discriminant of an elliptic curve using the group of its 12-torsion points and the Weil pairing. As an application, we extend a result of Coates (which deals with the characteristic 0 case) to the case where the characteristic of the base field is not 2 or 3. This is a joint work with Kohei Fukuda.

### 2013/12/26

#### Operator Algebra Seminars

16:30-18:00 Room #118 (Graduate School of Math. Sci. Bldg.)

Countable Chain Condition for $C^*$-algebras (ENGLISH)

**Shuhei Masumoto**(Univ. Tokyo)Countable Chain Condition for $C^*$-algebras (ENGLISH)

### 2013/12/25

#### GCOE Seminars

16:00-17:00 Room #270 (Graduate School of Math. Sci. Bldg.)

Nonsmooth Nonconvex Optimization Problems (ENGLISH)

**Kazufumi Ito**(North Carolina State University)Nonsmooth Nonconvex Optimization Problems (ENGLISH)

[ Abstract ]

A general class of nonsmooth nonconvex optimization problems is discussed. A general existence theory of solutions, the Lagrange multiplier theory and sensitivity analysis of the optimal value function are developed. Concrete examples are presented to demonstrate the applicability of our approach.

A general class of nonsmooth nonconvex optimization problems is discussed. A general existence theory of solutions, the Lagrange multiplier theory and sensitivity analysis of the optimal value function are developed. Concrete examples are presented to demonstrate the applicability of our approach.

### 2013/12/24

#### Tuesday Seminar on Topology

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

Stable homotopy type for monopole Floer homology (ENGLISH)

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

[ Abstract ]

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

construction of stable homotopy type for monopole Floer homology. The

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

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

ingredients are finite dimensional approximation technique and Conley

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

3-manifolds such as the 3-torus.

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

construction of stable homotopy type for monopole Floer homology. The

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

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

ingredients are finite dimensional approximation technique and Conley

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

3-manifolds such as the 3-torus.

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