## Seminar information archive

Seminar information archive ～05/23｜Today's seminar 05/24 | Future seminars 05/25～

### 2013/06/06

#### Geometry Colloquium

10:00-11:30 Room #122 (Graduate School of Math. Sci. Bldg.)

Equivariant local index and transverse index for circle action (JAPANESE)

**Hajime Fujita**(Japan Women's University)Equivariant local index and transverse index for circle action (JAPANESE)

[ Abstract ]

In our joint work with Furuta and Yoshida we gave a formulation of index theory of Dirac-type operator on open Riemannian manifolds. We used a torus fibration and a perturbation by Dirac-type operator along fibers. In this talk we develop an equivariant version for circle action and apply it for Hamiltonian circle action case. We investigate the relation between our equivariant index and index of transverse elliptic operator/symbol developed by Atiyah, Paradan-Vergne and Braverman. We give a computation for the standard cylinder, which shows the difference between two equivariant indices.

In our joint work with Furuta and Yoshida we gave a formulation of index theory of Dirac-type operator on open Riemannian manifolds. We used a torus fibration and a perturbation by Dirac-type operator along fibers. In this talk we develop an equivariant version for circle action and apply it for Hamiltonian circle action case. We investigate the relation between our equivariant index and index of transverse elliptic operator/symbol developed by Atiyah, Paradan-Vergne and Braverman. We give a computation for the standard cylinder, which shows the difference between two equivariant indices.

#### FMSP Lectures

16:00-17:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Real-valued and circle-valued Morse theory:

an introduction

(ENGLISH)

**Andrei Pajitnov**(Univ. de Nantes)Real-valued and circle-valued Morse theory:

an introduction

(ENGLISH)

[ Abstract ]

Classical Morse theory relates the number of critical points of a Morse

function f on a manifold M to the topology of M. The main technical

ingredient of this theory is a chain complex generated by the critical points

of the function. In 1981 S.P. Novikov generalized this theory to the case of

circle-valued Morse functions. In this talk we describe the construction of

both chain complexes, based on the idea of E. Witten (1982), which allows, in

particular, to compute the boundary operators in the Morse complex from

the count of flow lines of the gradient of f. We discuss geometric applications

of these constructions.

Classical Morse theory relates the number of critical points of a Morse

function f on a manifold M to the topology of M. The main technical

ingredient of this theory is a chain complex generated by the critical points

of the function. In 1981 S.P. Novikov generalized this theory to the case of

circle-valued Morse functions. In this talk we describe the construction of

both chain complexes, based on the idea of E. Witten (1982), which allows, in

particular, to compute the boundary operators in the Morse complex from

the count of flow lines of the gradient of f. We discuss geometric applications

of these constructions.

#### Applied Analysis

16:00-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)

The Geometry of Critical Points of Green functions On Tori (ENGLISH)

**Chang-Shou Lin**(National Taiwan University)The Geometry of Critical Points of Green functions On Tori (ENGLISH)

[ Abstract ]

The Green function of a torus can be expressed by elliptic functions or Jacobic theta functions. It is not surprising the geometry of its critical points would be involved with behaviors of those classical functions. Thus, the non-degeneracy of critical points gives rise to some inequality for elliptic functions. One of consequences of our analysis is to prove any saddle point is non-degenerate, i.e., the Hessian is negative.

We will also show that the number of the critical points of Green function in any torus is either three or five critical points. Furthermore, the moduli space of tori which Green function has five critical points is a simple-connected connected set. The proof of these results use a nonlinear PDE (mean field equation) and the formula for counting zeros of modular form. For a N torsion point,the related modular form is the Eisenstein series of weight one, which was discovered by Hecke (1926). Thus, our PDE method gives a deformation of those Eisenstein series and allows us to find the zeros of those Eisenstein series.

We can generalize our results to a sum of two Green functions.

The Green function of a torus can be expressed by elliptic functions or Jacobic theta functions. It is not surprising the geometry of its critical points would be involved with behaviors of those classical functions. Thus, the non-degeneracy of critical points gives rise to some inequality for elliptic functions. One of consequences of our analysis is to prove any saddle point is non-degenerate, i.e., the Hessian is negative.

We will also show that the number of the critical points of Green function in any torus is either three or five critical points. Furthermore, the moduli space of tori which Green function has five critical points is a simple-connected connected set. The proof of these results use a nonlinear PDE (mean field equation) and the formula for counting zeros of modular form. For a N torsion point,the related modular form is the Eisenstein series of weight one, which was discovered by Hecke (1926). Thus, our PDE method gives a deformation of those Eisenstein series and allows us to find the zeros of those Eisenstein series.

We can generalize our results to a sum of two Green functions.

### 2013/06/04

#### Tuesday Seminar on Topology

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

Low-dimensional linear representations of mapping class groups. (ENGLISH)

**Mustafa Korkmaz**(Middle East Technical University)Low-dimensional linear representations of mapping class groups. (ENGLISH)

[ Abstract ]

For a compact connected orientable surface, the mapping class group

of it is defined as the group of isotopy classes of orientation-preserving

self-diffeomorphisms of S which are identity on the boundary. The action

of the mapping class group on the first homology of the surface

gives rise to the classical 2g-dimensional symplectic representation.

The existence of a faithful linear representation of the mapping class

group is still unknown. In my talk, I will show the following three results;

there is no lower dimensional (complex) linear representation,

up to conjugation the symplectic representation is the unique nontrivial representation in dimension 2g, and there is no faithful linear representation

of the mapping class group in dimensions up to 3g-3. I will also discuss a few applications of these theorems, including some algebraic consequences.

For a compact connected orientable surface, the mapping class group

of it is defined as the group of isotopy classes of orientation-preserving

self-diffeomorphisms of S which are identity on the boundary. The action

of the mapping class group on the first homology of the surface

gives rise to the classical 2g-dimensional symplectic representation.

The existence of a faithful linear representation of the mapping class

group is still unknown. In my talk, I will show the following three results;

there is no lower dimensional (complex) linear representation,

up to conjugation the symplectic representation is the unique nontrivial representation in dimension 2g, and there is no faithful linear representation

of the mapping class group in dimensions up to 3g-3. I will also discuss a few applications of these theorems, including some algebraic consequences.

#### Numerical Analysis Seminar

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

On the theories of discrete differential forms and their applications to structure-preserving numerical methods (JAPANESE)

[ Reference URL ]

http://www.infsup.jp/utnas/

**Takaharu Yaguchi**(Kobe University )On the theories of discrete differential forms and their applications to structure-preserving numerical methods (JAPANESE)

[ Reference URL ]

http://www.infsup.jp/utnas/

### 2013/06/03

#### Seminar on Geometric Complex Analysis

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

Generalized deformation theory of CR structures (JAPANESE)

**Takao Akahori**(University of Hyogo)Generalized deformation theory of CR structures (JAPANESE)

[ Abstract ]

Let $(M, {}^0 T^{''})$ be a compact strongly pseudo convex CR manifold with dimension $2n-1 \geq 5$, embedded in a complex manifold $N$ as a real hypersurface. In our former papers (T. Akahori, Invent. Math. 63 (1981); T. Akahori, P. M. Garfield, and J. M. Lee, Michigan Math. J. 50 (2002)), we constructed the versal family of CR structures. The purpose of this talk is to show that in more wide scope, our family is versal.

Let $(M, {}^0 T^{''})$ be a compact strongly pseudo convex CR manifold with dimension $2n-1 \geq 5$, embedded in a complex manifold $N$ as a real hypersurface. In our former papers (T. Akahori, Invent. Math. 63 (1981); T. Akahori, P. M. Garfield, and J. M. Lee, Michigan Math. J. 50 (2002)), we constructed the versal family of CR structures. The purpose of this talk is to show that in more wide scope, our family is versal.

### 2013/05/31

#### Colloquium

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

Stable patterns and the nonlinear ``hot spots'' conjecture (JAPANESE)

**Yasuhito MIYAMOTO**(Graduate School of Mathematical Sciences, The University of Tokyo)Stable patterns and the nonlinear ``hot spots'' conjecture (JAPANESE)

### 2013/05/30

#### FMSP Lectures

13:00-14:30 Room #122 (Graduate School of Math. Sci. Bldg.)

Low-dimensional linear representations of mapping class groups (III) (ENGLISH)

[ Reference URL ]

http://faculty.ms.u-tokyo.ac.jp/~topology/Korkmaz.pdf

**Mustafa Korkmaz**(Middle East Technical University)Low-dimensional linear representations of mapping class groups (III) (ENGLISH)

[ Reference URL ]

http://faculty.ms.u-tokyo.ac.jp/~topology/Korkmaz.pdf

### 2013/05/29

#### FMSP Lectures

13:00-14:30 Room #122 (Graduate School of Math. Sci. Bldg.)

Low-dimensional linear representations of mapping class groups (II) (ENGLISH)

[ Reference URL ]

http://faculty.ms.u-tokyo.ac.jp/~topology/Korkmaz.pdf

**Mustafa Korkmaz**(Middle East Technical University)Low-dimensional linear representations of mapping class groups (II) (ENGLISH)

[ Reference URL ]

http://faculty.ms.u-tokyo.ac.jp/~topology/Korkmaz.pdf

#### Number Theory Seminar

16:40-17:40 Room #002 (Graduate School of Math. Sci. Bldg.)

On logarithmic nonabelian Hodge theory of higher level in characteristic p (JAPANESE)

**Sachio Ohkawa**(University of Tokyo)On logarithmic nonabelian Hodge theory of higher level in characteristic p (JAPANESE)

[ Abstract ]

Ogus and Vologodsky studied a positive characteristic analogue of Simpson’s nonanelian Hodge theory over the complex number field. Now most part of their theory has been generalized to the case of log schemes by Schepler. In this talk, we generalize the global Cartier transform, which is one of the main theorem in nonabelian Hodge theory in positive characteristic, to the case of log schemes and of higher level. This can be regarded as a higher level version of a result of Schepler.

Ogus and Vologodsky studied a positive characteristic analogue of Simpson’s nonanelian Hodge theory over the complex number field. Now most part of their theory has been generalized to the case of log schemes by Schepler. In this talk, we generalize the global Cartier transform, which is one of the main theorem in nonabelian Hodge theory in positive characteristic, to the case of log schemes and of higher level. This can be regarded as a higher level version of a result of Schepler.

#### Operator Algebra Seminars

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

On full group C*-algebras of discrete quantum groups (ENGLISH)

**Yuki Arano**(Univ. Tokyo)On full group C*-algebras of discrete quantum groups (ENGLISH)

### 2013/05/28

#### FMSP Lectures

17:10-18:40 Room #117 (Graduate School of Math. Sci. Bldg.)

Low-dimensional linear representations of mapping class groups (I) (ENGLISH)

[ Reference URL ]

http://faculty.ms.u-tokyo.ac.jp/~topology/Korkmaz.pdf

**Mustafa Korkmaz**(Middle East Technical University)Low-dimensional linear representations of mapping class groups (I) (ENGLISH)

[ Reference URL ]

http://faculty.ms.u-tokyo.ac.jp/~topology/Korkmaz.pdf

### 2013/05/27

#### Seminar on Geometric Complex Analysis

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

2次元擬斉次特異点の接層のコホモロジーについて (JAPANESE)

**Tomohiro Okuma**(Yamagata University)2次元擬斉次特異点の接層のコホモロジーについて (JAPANESE)

[ Abstract ]

複素2次元特異点の特異点解消上の接層のコホモロジーの次元は解析的不変量である. セミナーでは, リンクが有理ホモロジー球面であるような2次元擬斉次特異点の場合にはそれが位相的不変量であり, グラフから計算できることを紹介する.

複素2次元特異点の特異点解消上の接層のコホモロジーの次元は解析的不変量である. セミナーでは, リンクが有理ホモロジー球面であるような2次元擬斉次特異点の場合にはそれが位相的不変量であり, グラフから計算できることを紹介する.

### 2013/05/25

#### Harmonic Analysis Komaba Seminar

13:00-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Multilinear fractional integral operators on weighted Morrey spaces (JAPANESE)

On factorization of divergence form elliptic operators

and its application

(JAPANESE)

**Takeshi Iida**(Fukushima National College of Technology) 13:30-15:00Multilinear fractional integral operators on weighted Morrey spaces (JAPANESE)

**Yasunori Maekawa**(Tohoku University) 15:30-17:00On factorization of divergence form elliptic operators

and its application

(JAPANESE)

### 2013/05/24

#### Lectures

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

Kernels of Langlands' automorphic transfer and non-linear Poisson formulas (JAPANESE)

**Laurent Lafforgue**(IHES)Kernels of Langlands' automorphic transfer and non-linear Poisson formulas (JAPANESE)

### 2013/05/21

#### Tuesday Seminar of Analysis

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

Homogenization in a Thin Layer with an Oscillating Interface and Highly Contrast Coefficients (JAPANESE)

**Masaaki Uesaka**(Graduate School of Mathematical Sciences, The University of Tokyo)Homogenization in a Thin Layer with an Oscillating Interface and Highly Contrast Coefficients (JAPANESE)

[ Abstract ]

We consider the homogenization problem of the elliptic boundary value problem in a thin domain which has a high and low conductivity zones. In our model, two media are separated by a highly oscillating interface. The asymptotic behavior is governed by the order of the thickness of the domain, oscillation period of the interface and contrast between two media. In this talk, we show that the limit problem is changed by these parameters. We also introduce the two-scale convergence result in a thin domain which is the key ingredient of the proof.

We consider the homogenization problem of the elliptic boundary value problem in a thin domain which has a high and low conductivity zones. In our model, two media are separated by a highly oscillating interface. The asymptotic behavior is governed by the order of the thickness of the domain, oscillation period of the interface and contrast between two media. In this talk, we show that the limit problem is changed by these parameters. We also introduce the two-scale convergence result in a thin domain which is the key ingredient of the proof.

#### Lectures

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

Kernels of Langlands' automorphic transfer and non-linear Poisson formulas (ENGLISH)

**Laurent Lafforgue**(IHES)Kernels of Langlands' automorphic transfer and non-linear Poisson formulas (ENGLISH)

#### Tuesday Seminar on Topology

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

A Heegaard Floer homology for bipartite spatial graphs and its

properties (ENGLISH)

**Yuanyuan Bao**(The University of Tokyo)A Heegaard Floer homology for bipartite spatial graphs and its

properties (ENGLISH)

[ Abstract ]

A spatial graph is a smooth embedding of a graph into a given

3-manifold. We can regard a link as a particular spatial graph.

So it is natural to ask whether it is possible to extend the idea

of link Floer homology to define a Heegaard Floer homology for

spatial graphs. In this talk, we discuss some ideas towards this

question. In particular, we define a Heegaard Floer homology for

bipartite spatial graphs and discuss some further observations

about this construction. We remark that Harvey and O’Donnol

have announced a combinatorial Floer homology for spatial graphs by

considering grid diagrams.

A spatial graph is a smooth embedding of a graph into a given

3-manifold. We can regard a link as a particular spatial graph.

So it is natural to ask whether it is possible to extend the idea

of link Floer homology to define a Heegaard Floer homology for

spatial graphs. In this talk, we discuss some ideas towards this

question. In particular, we define a Heegaard Floer homology for

bipartite spatial graphs and discuss some further observations

about this construction. We remark that Harvey and O’Donnol

have announced a combinatorial Floer homology for spatial graphs by

considering grid diagrams.

### 2013/05/20

#### Seminar on Geometric Complex Analysis

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

レヴィ平坦面の分類に関する最近の進展 (JAPANESE)

**Takeo Ohsawa**(Nagoya University)レヴィ平坦面の分類に関する最近の進展 (JAPANESE)

[ Abstract ]

レヴィ平坦面の分類がCP^2の場合にできていないことから、種々の興味深い問題が生じているように思われる。ここではトーラスの場合に観察されたことをホップ曲面に拡げたとき、ホップ曲面においてならレヴィ平坦面の分類が(実解析的な場合に限るが)完全にできることを報告する。

レヴィ平坦面の分類がCP^2の場合にできていないことから、種々の興味深い問題が生じているように思われる。ここではトーラスの場合に観察されたことをホップ曲面に拡げたとき、ホップ曲面においてならレヴィ平坦面の分類が(実解析的な場合に限るが)完全にできることを報告する。

### 2013/05/17

#### Seminar on Probability and Statistics

13:30-14:40 Room #052 (Graduate School of Math. Sci. Bldg.)

Computing the normalizing constant of the Bingham family by the holonomic gradient method (JAPANESE)

[ Reference URL ]

http://www.sigmath.es.osaka-u.ac.jp/~kamatani/statseminar/2013/00.html

**SEI, Tomonari**(Department of Mathematics, Keio University)Computing the normalizing constant of the Bingham family by the holonomic gradient method (JAPANESE)

[ Reference URL ]

http://www.sigmath.es.osaka-u.ac.jp/~kamatani/statseminar/2013/00.html

### 2013/05/16

#### Geometry Colloquium

10:00-11:30 Room #122 (Graduate School of Math. Sci. Bldg.)

A generalization of Taub-NUT deformations (JAPANESE)

**Kota Hattori**(University of Tokyo)A generalization of Taub-NUT deformations (JAPANESE)

[ Abstract ]

Taub-NUT metric on C^2 is a complete Ricci-flat Kaehler metric which is not flat. It is obtained by the Taub-NUT deformations of the Euclidean metric on C^2 using an S^1 action. Taub-NUT deformations are known to be defined for toric hyperKaehler manifolds, and deform ALE metrics to non-ALE metrics. In this talk, I explain a generalization of Taub-NUT deformations by using noncommutative Lie groups.

Taub-NUT metric on C^2 is a complete Ricci-flat Kaehler metric which is not flat. It is obtained by the Taub-NUT deformations of the Euclidean metric on C^2 using an S^1 action. Taub-NUT deformations are known to be defined for toric hyperKaehler manifolds, and deform ALE metrics to non-ALE metrics. In this talk, I explain a generalization of Taub-NUT deformations by using noncommutative Lie groups.

### 2013/05/15

#### Operator Algebra Seminars

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

Rohlin Flows on Amalgamated Free Product Factors (ENGLISH)

**Koichi Shimada**(Univ. Tokyo)Rohlin Flows on Amalgamated Free Product Factors (ENGLISH)

#### Number Theory Seminar

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

Special values of zeta functions of singular varieties over finite fields via higher chow groups (JAPANESE)

**Hiroyasu Miyazaki**(University of Tokyo)Special values of zeta functions of singular varieties over finite fields via higher chow groups (JAPANESE)

### 2013/05/14

#### Tuesday Seminar on Topology

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

Vanishing cycles and homotopies of wrinkled fibrations (JAPANESE)

**Kenta Hayano**(Osaka University)Vanishing cycles and homotopies of wrinkled fibrations (JAPANESE)

[ Abstract ]

Wrinkled fibrations on closed 4-manifolds are stable

maps to closed surfaces with only indefinite singularities. Such

fibrations and variants of them have been studied for the past few years

to obtain new descriptions of 4-manifolds using mapping class groups.

Vanishing cycles of wrinkled fibrations play a key role in these studies.

In this talk, we will explain how homotopies of wrinkled fibrtions affect

their vanishing cycles. Part of the results in this talk is a joint work

with Stefan Behrens (Max Planck Institute for Mathematics).

Wrinkled fibrations on closed 4-manifolds are stable

maps to closed surfaces with only indefinite singularities. Such

fibrations and variants of them have been studied for the past few years

to obtain new descriptions of 4-manifolds using mapping class groups.

Vanishing cycles of wrinkled fibrations play a key role in these studies.

In this talk, we will explain how homotopies of wrinkled fibrtions affect

their vanishing cycles. Part of the results in this talk is a joint work

with Stefan Behrens (Max Planck Institute for Mathematics).

#### Lectures

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

Kernels of Langlands' automorphic transfer and non-linear Poisson formulas (ENGLISH)

**Laurent Lafforgue**(IHES)Kernels of Langlands' automorphic transfer and non-linear Poisson formulas (ENGLISH)

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