## Seminar information archive

Seminar information archive ～05/20｜Today's seminar 05/21 | Future seminars 05/22～

### 2014/05/27

#### Tuesday Seminar of Analysis

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

The regularity theorem for elliptic equations and the smoothness of domains (JAPANESE)

**Yoichi Miyazaki**(NIHON UNIVERSITY, SCHOOL OF DENTISTRY)The regularity theorem for elliptic equations and the smoothness of domains (JAPANESE)

[ Abstract ]

We consider the Dirichlet boundary problem for a strongly elliptic operator of order $2m$ with non-smooth coefficients, and prove the regularity theorem for $L_p$-based Sobolev spaces when the domain has a boundary of limited smoothness. Compared to the known results, we can weaken the smoothness assumption on the boundary by $m-1$.

We consider the Dirichlet boundary problem for a strongly elliptic operator of order $2m$ with non-smooth coefficients, and prove the regularity theorem for $L_p$-based Sobolev spaces when the domain has a boundary of limited smoothness. Compared to the known results, we can weaken the smoothness assumption on the boundary by $m-1$.

#### Tuesday Seminar on Topology

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

The Teichmuller space and the stable quasiconformal mapping class group for a Riemann surface of infinite type (JAPANESE)

**Ege Fujikawa**(Chiba University)The Teichmuller space and the stable quasiconformal mapping class group for a Riemann surface of infinite type (JAPANESE)

[ Abstract ]

We explain recent developments of the theory of infinite dimensional Teichmuller space. In particular, we observe the dynamics of the orbits by the action of the stable quasiconformal mapping class group on the Teichmuller space and consider the relationship with the asymptotic Teichmuller space. We also introduce the generalized fixed point theorem and the Nielsen realization theorem. Furthermore, we investigate the moduli space of Riemann surface of infinite type.

We explain recent developments of the theory of infinite dimensional Teichmuller space. In particular, we observe the dynamics of the orbits by the action of the stable quasiconformal mapping class group on the Teichmuller space and consider the relationship with the asymptotic Teichmuller space. We also introduce the generalized fixed point theorem and the Nielsen realization theorem. Furthermore, we investigate the moduli space of Riemann surface of infinite type.

#### Lie Groups and Representation Theory

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

On the structure of Schubert modules and filtration by Schubert modules

(JAPANESE)

**Masaki Watanabe**(the University of Tokyo, Graduate School of Mathematical Sciences)On the structure of Schubert modules and filtration by Schubert modules

(JAPANESE)

[ Abstract ]

One of the methods for studying Schubert polynomials is using

Schubert modules introduced by Kraskiewicz and Pragacz.

In this seminar I will talk about a new result on the structure of

Schubert modules, and give a criterion for a module to have a filtration by Schubert modules.

I will also talk about a problem concerning Schubert polynomials

which motivated this research.

One of the methods for studying Schubert polynomials is using

Schubert modules introduced by Kraskiewicz and Pragacz.

In this seminar I will talk about a new result on the structure of

Schubert modules, and give a criterion for a module to have a filtration by Schubert modules.

I will also talk about a problem concerning Schubert polynomials

which motivated this research.

### 2014/05/22

#### Geometry Colloquium

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

Godbillon-Vey invariants for maximal isotropic foliations (ENGLISH)

**Boris Hasselblatt**(Tufts Univ)Godbillon-Vey invariants for maximal isotropic foliations (ENGLISH)

[ Abstract ]

The combination of a contact structure and an orientable maximal isotropic foliation gives rise to m+1 Godbillon-Vey invariants for an m+1-dimensional maximal isotropic foliation that are of interest with respect to geometric rigidity: by studying these jointly, we give new proofs of famous "rigidity'' results from the 1980s that require only a very few simple lines of reasoning rather than the elaborate original proofs.

The combination of a contact structure and an orientable maximal isotropic foliation gives rise to m+1 Godbillon-Vey invariants for an m+1-dimensional maximal isotropic foliation that are of interest with respect to geometric rigidity: by studying these jointly, we give new proofs of famous "rigidity'' results from the 1980s that require only a very few simple lines of reasoning rather than the elaborate original proofs.

### 2014/05/21

#### Number Theory Seminar

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

Parity of Betti numbers in étale cohomology (ENGLISH)

**Shenghao Sun**(Mathematical Sciences Center of Tsinghua University)Parity of Betti numbers in étale cohomology (ENGLISH)

[ Abstract ]

By Hodge symmetry, the Betti numbers of a complex projective smooth variety in odd degrees are even. When the base field has characteristic p, Deligne proved the hard Lefschetz theorem in etale cohomology, and the parity result follows from this. Suh has generalized this to proper smooth varieties in characteristic p, using crystalline cohomology.

The purity of intersection cohomology group of proper varieties suggests that the same parity property should hold for these groups in characteristic p. We proved this by investigating the symmetry in the categorical level.

In particular, we reproved Suh's result, using merely etale cohomology. Some related results will be discussed. This is joint work with Weizhe Zheng.

By Hodge symmetry, the Betti numbers of a complex projective smooth variety in odd degrees are even. When the base field has characteristic p, Deligne proved the hard Lefschetz theorem in etale cohomology, and the parity result follows from this. Suh has generalized this to proper smooth varieties in characteristic p, using crystalline cohomology.

The purity of intersection cohomology group of proper varieties suggests that the same parity property should hold for these groups in characteristic p. We proved this by investigating the symmetry in the categorical level.

In particular, we reproved Suh's result, using merely etale cohomology. Some related results will be discussed. This is joint work with Weizhe Zheng.

#### Operator Algebra Seminars

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

Group approximation in Cayley topology and coarse geometry

part I: coarse embeddings of amenable groups (ENGLISH)

**Masato Mimura**(Tohoku Univ.)Group approximation in Cayley topology and coarse geometry

part I: coarse embeddings of amenable groups (ENGLISH)

### 2014/05/20

#### Tuesday Seminar on Topology

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

An application of torus graphs to characterize torus manifolds

with extended actions (JAPANESE)

**Shintaro Kuroki**(The Univeristy of Tokyo)An application of torus graphs to characterize torus manifolds

with extended actions (JAPANESE)

[ Abstract ]

A torus manifold is a compact, oriented 2n-dimensional T^n-

manifolds with fixed points. This notion is introduced by Hattori and

Masuda as a topological generalization of toric manifolds. For a given

torus manifold, we can define a labelled graph called a torus graph (

this may be regarded as a generalization of some class of GKM graphs).

It is known that the equivariant cohomology ring of some nice class of

torus manifolds can be computed by using a combinatorial data of torus

graphs. In this talk, we study which torus action of torus manifolds can

be extended to a non-abelian compact connected Lie group. To do this, we

introduce root systems of (abstract) torus graphs and characterize

extended actions of torus manifolds. This is a joint work with Mikiya

Masuda.

A torus manifold is a compact, oriented 2n-dimensional T^n-

manifolds with fixed points. This notion is introduced by Hattori and

Masuda as a topological generalization of toric manifolds. For a given

torus manifold, we can define a labelled graph called a torus graph (

this may be regarded as a generalization of some class of GKM graphs).

It is known that the equivariant cohomology ring of some nice class of

torus manifolds can be computed by using a combinatorial data of torus

graphs. In this talk, we study which torus action of torus manifolds can

be extended to a non-abelian compact connected Lie group. To do this, we

introduce root systems of (abstract) torus graphs and characterize

extended actions of torus manifolds. This is a joint work with Mikiya

Masuda.

#### Seminar on Probability and Statistics

13:00-14:10 Room #052 (Graduate School of Math. Sci. Bldg.)

Maximum likelihood type estimation of diffusion processes with non synchronous observations contaminated by market microstructure noise (JAPANESE)

[ Reference URL ]

http://www.sigmath.es.osaka-u.ac.jp/~kamatani/statseminar/2014/02.html

**OGIHARA, Teppei**(Center for the Study of Finance and Insurance, Osaka University)Maximum likelihood type estimation of diffusion processes with non synchronous observations contaminated by market microstructure noise (JAPANESE)

[ Reference URL ]

http://www.sigmath.es.osaka-u.ac.jp/~kamatani/statseminar/2014/02.html

### 2014/05/19

#### Seminar on Geometric Complex Analysis

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

On degenerations of Ricci-flat Kähler manifolds (JAPANESE)

**Shigeharu Takayama**(University of Tokyo)On degenerations of Ricci-flat Kähler manifolds (JAPANESE)

### 2014/05/17

#### Harmonic Analysis Komaba Seminar

13:30-17:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Bounded small solutions to a chemotaxis system with non-diffusive chemical (JAPANESE)

Heat kernel and Schroedinger kernel on the Heisenberg group (JAPANESE)

**Yohei Tsutsui**(The University of Tokyo) 13:30-15:00Bounded small solutions to a chemotaxis system with non-diffusive chemical (JAPANESE)

[ Abstract ]

We consider a chemotaxis system with a logarithmic sensitivity and a non-diffusive chemical substance. For some chemotactic sensitivity constants, Ahn and Kang proved the existence of bounded global solutions to the system. An entropy functional was used in their argument to control the cell density by the density of the chemical substance. Our purpose is to show the existence of bounded global solutions for all the chemotactic sensitivity constants. Assuming the smallness on the initial data in some sense, we can get uniform estimates for time. These estimates are used to extend local solutions.

This talk is partially based on joint work with Yoshie Sugiyama (Kyusyu Univ.) and Juan J.L. Vel\\'azquez (Univ. of Bonn).

We consider a chemotaxis system with a logarithmic sensitivity and a non-diffusive chemical substance. For some chemotactic sensitivity constants, Ahn and Kang proved the existence of bounded global solutions to the system. An entropy functional was used in their argument to control the cell density by the density of the chemical substance. Our purpose is to show the existence of bounded global solutions for all the chemotactic sensitivity constants. Assuming the smallness on the initial data in some sense, we can get uniform estimates for time. These estimates are used to extend local solutions.

This talk is partially based on joint work with Yoshie Sugiyama (Kyusyu Univ.) and Juan J.L. Vel\\'azquez (Univ. of Bonn).

**Toshinao Kagawa**(Tokyo City University) 15:30-17:00Heat kernel and Schroedinger kernel on the Heisenberg group (JAPANESE)

### 2014/05/15

#### Geometry Colloquium

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

Gap theorems for compact gradient Sasaki Ricci solitons (JAPANESE)

**Homare TADANO**(Osaka University)Gap theorems for compact gradient Sasaki Ricci solitons (JAPANESE)

[ Abstract ]

In this talk we give some necessary and sufficient conditions for compact gradient Sasaki-Ricci solitons to be Sasaki-Einstein. Our result may be considered as a Sasaki geometry version of recent works by H. Li, and M. Fern¥'andez-L¥'opez-E. Garc¥'ia-Rio.

In this talk we give some necessary and sufficient conditions for compact gradient Sasaki-Ricci solitons to be Sasaki-Einstein. Our result may be considered as a Sasaki geometry version of recent works by H. Li, and M. Fern¥'andez-L¥'opez-E. Garc¥'ia-Rio.

#### Lectures

16:30-17:30 Room #050 (Graduate School of Math. Sci. Bldg.)

Synthetic theory of Ricci curvature

― When Monge, Riemann and Boltzmann meet ― (ENGLISH)

http://faculty.ms.u-tokyo.ac.jp/Villani.html

**Cédric Villani**(Université de Lyon, Institut Henri Poincaré)Synthetic theory of Ricci curvature

― When Monge, Riemann and Boltzmann meet ― (ENGLISH)

[ Abstract ]

Optimal transport theory, non-Euclidean geometry and statistical physics met fifteen years ago with the discovery that Ricci curvature can be studied quantitatively thanks to entropy and

Monge-Kantorovich transport.

This unexpected encounter was very fruitful, leading to progress in each of these fields.

[ Reference URL ]Optimal transport theory, non-Euclidean geometry and statistical physics met fifteen years ago with the discovery that Ricci curvature can be studied quantitatively thanks to entropy and

Monge-Kantorovich transport.

This unexpected encounter was very fruitful, leading to progress in each of these fields.

http://faculty.ms.u-tokyo.ac.jp/Villani.html

#### FMSP Lectures

16:30-17:30 Room #050 (Graduate School of Math. Sci. Bldg.)

Synthetic theory of Ricci curvature ― When Monge, Riemann and Boltzmann meet ― (ENGLISH)

[ Reference URL ]

http://faculty.ms.u-tokyo.ac.jp/Villani.html

**Cédric Villani**(Université de Lyon, Institut Henri Poincaré)Synthetic theory of Ricci curvature ― When Monge, Riemann and Boltzmann meet ― (ENGLISH)

[ Reference URL ]

http://faculty.ms.u-tokyo.ac.jp/Villani.html

### 2014/05/14

#### Operator Algebra Seminars

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

A Supersymmetric model in AQFT (after Buchholz and Grundling) (ENGLISH)

**Yul Otani**(Univ. Tokyo)A Supersymmetric model in AQFT (after Buchholz and Grundling) (ENGLISH)

### 2014/05/13

#### Tuesday Seminar of Analysis

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

Ultra-differentiable classes and intersection theorems (JAPANESE)

**Yasunori Okada**(Graduate School of Science and Technology, Chiba University)Ultra-differentiable classes and intersection theorems (JAPANESE)

[ Abstract ]

There are two ways to define notions of

ultra-differentiability: one in terms of estimates on derivatives, and

the other in terms of growth properties of Fourier transforms of

suitably localized functions.

In this talk, we study the relation between BMT-classes and

inhomogeneous Gevrey classes, which are examples of such two kinds of

notions of ultra-differentiability.

We also mention intersection theorems on these classes.

This talk is based on a joint work with Otto Liess (Bologna University).

There are two ways to define notions of

ultra-differentiability: one in terms of estimates on derivatives, and

the other in terms of growth properties of Fourier transforms of

suitably localized functions.

In this talk, we study the relation between BMT-classes and

inhomogeneous Gevrey classes, which are examples of such two kinds of

notions of ultra-differentiability.

We also mention intersection theorems on these classes.

This talk is based on a joint work with Otto Liess (Bologna University).

#### Seminar on Probability and Statistics

13:00-14:10 Room #052 (Graduate School of Math. Sci. Bldg.)

On High Frequency Estimation of the Frictionless Price: The Use of Observed Liquidity Variables (ENGLISH)

http://www.sigmath.es.osaka-u.ac.jp/~kamatani/statseminar/2014/01.html

**Selma Chaker**(Bank of Canada)On High Frequency Estimation of the Frictionless Price: The Use of Observed Liquidity Variables (ENGLISH)

[ Abstract ]

Observed high-frequency prices are always contaminated with liquidity costs or market microstructure noise. Inspired by the market microstructure literature, I explicitly model this noise and remove it from observed prices to obtain an estimate of the frictionless price. I then formally test whether the prices adjusted for the estimated liquidity costs are either totally or partially free from noise. If the liquidity costs are only partially removed, the residual noise is smaller and closer to an exogenous white noise than the original noise is. To illustrate my approach, I use the adjusted prices to improve volatility estimation in the presence of noise. If the noise is totally absorbed, I show that the sum of squared returns - which would be inconsistent for return variance when based on observed returns - becomes consistent when based on adjusted returns.

[ Reference URL ]Observed high-frequency prices are always contaminated with liquidity costs or market microstructure noise. Inspired by the market microstructure literature, I explicitly model this noise and remove it from observed prices to obtain an estimate of the frictionless price. I then formally test whether the prices adjusted for the estimated liquidity costs are either totally or partially free from noise. If the liquidity costs are only partially removed, the residual noise is smaller and closer to an exogenous white noise than the original noise is. To illustrate my approach, I use the adjusted prices to improve volatility estimation in the presence of noise. If the noise is totally absorbed, I show that the sum of squared returns - which would be inconsistent for return variance when based on observed returns - becomes consistent when based on adjusted returns.

http://www.sigmath.es.osaka-u.ac.jp/~kamatani/statseminar/2014/01.html

#### Tuesday Seminar on Topology

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

Transverse projective structures of foliations and deformations of the Godbillon-Vey class (JAPANESE)

**Taro Asuke**(The University of Tokyo)Transverse projective structures of foliations and deformations of the Godbillon-Vey class (JAPANESE)

[ Abstract ]

Given a smooth family of foliations, we can define the derivative of the Godbillon-Vey class

with respect to the family. The derivative is known to be represented in terms of the projective

Schwarzians of holonomy maps. In this talk, we will study transverse projective structures

and connections, and show that the derivative is in fact determined by the projective structure

and the family.

Given a smooth family of foliations, we can define the derivative of the Godbillon-Vey class

with respect to the family. The derivative is known to be represented in terms of the projective

Schwarzians of holonomy maps. In this talk, we will study transverse projective structures

and connections, and show that the derivative is in fact determined by the projective structure

and the family.

#### Lie Groups and Representation Theory

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

)

Global q,t-hypergeometric and q-Whittaker functions (ENGLISH)

**Ivan Cherednik**(The University of North Carolina at Chapel Hill, RIMS)

Global q,t-hypergeometric and q-Whittaker functions (ENGLISH)

[ Abstract ]

The lectures will be devoted to the new theory of global

difference hypergeometric and Whittaker functions, one of

the major applications of the double affine Hecke algebras

and a breakthrough in the classical harmonic analysis. They

integrate the Ruijsenaars-Macdonald difference QMBP and

"Q-Toda" (any root systems), and are analytic everywhere

("global") with superb asymptotic behavior.

The definition of the global functions was suggested about

14 years ago; it is conceptually different from the definition

Heine gave in 1846, which remained unchanged and unchallenged

since then. Algebraically, the new functions are closer to

Bessel functions than to the classical hypergeometric and

Whittaker functions. The analytic theory of these functions was

completed only recently (the speaker and Jasper Stokman).

The construction is based on DAHA. The global functions are defined

as reproducing kernels of Fourier-DAHA transforms. Their

specializations are Macdonald polynomials, which is a powerful

generalization of the Shintani and Casselman-Shalika p-adic formulas.

If time permits, the connection of the Harish-Chandra theory of global

q-Whittaker functions will be discussed with the Givental-Lee formula

(Gromov-Witten invariants of flag varieties) and its generalizations due

to Braverman and Finkelberg (algebraic theory of affine flag varieties).

The lectures will be devoted to the new theory of global

difference hypergeometric and Whittaker functions, one of

the major applications of the double affine Hecke algebras

and a breakthrough in the classical harmonic analysis. They

integrate the Ruijsenaars-Macdonald difference QMBP and

"Q-Toda" (any root systems), and are analytic everywhere

("global") with superb asymptotic behavior.

The definition of the global functions was suggested about

14 years ago; it is conceptually different from the definition

Heine gave in 1846, which remained unchanged and unchallenged

since then. Algebraically, the new functions are closer to

Bessel functions than to the classical hypergeometric and

Whittaker functions. The analytic theory of these functions was

completed only recently (the speaker and Jasper Stokman).

The construction is based on DAHA. The global functions are defined

as reproducing kernels of Fourier-DAHA transforms. Their

specializations are Macdonald polynomials, which is a powerful

generalization of the Shintani and Casselman-Shalika p-adic formulas.

If time permits, the connection of the Harish-Chandra theory of global

q-Whittaker functions will be discussed with the Givental-Lee formula

(Gromov-Witten invariants of flag varieties) and its generalizations due

to Braverman and Finkelberg (algebraic theory of affine flag varieties).

### 2014/05/12

#### Seminar on Geometric Complex Analysis

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

Resolution of singularities via Newton polyhedra and its application to analysis (JAPANESE)

**Joe Kamimoto**(Kyushu university)Resolution of singularities via Newton polyhedra and its application to analysis (JAPANESE)

[ Abstract ]

In the 1970s, A. N. Varchenko precisely investigated the leading term of the asymptotic expansion of an oscillatory integral with real analytic phase by using the geometry of the Newton polyhedron of the phase. Since his study, the importance of the resolution of singularities by means of Newton polyhedra has been strongly recognized. The purpose of this talk is to consider studies around this theme and to explain their relationship with some problems in several complex variables.

In the 1970s, A. N. Varchenko precisely investigated the leading term of the asymptotic expansion of an oscillatory integral with real analytic phase by using the geometry of the Newton polyhedron of the phase. Since his study, the importance of the resolution of singularities by means of Newton polyhedra has been strongly recognized. The purpose of this talk is to consider studies around this theme and to explain their relationship with some problems in several complex variables.

#### Algebraic Geometry Seminar

15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Higher Nash blowup on normal toric varieties and a higher order version of Nobile's theorem (ENGLISH)

**Andrés Daniel Duarte**(Institut de Mathématiques de Toulouse)Higher Nash blowup on normal toric varieties and a higher order version of Nobile's theorem (ENGLISH)

[ Abstract ]

The higher Nash blowup of an algebraic variety replaces singular points with limits of certain vector spaces carrying first or higher order data associated to the variety at non-singular points. In the case of normal toric varieties, the higher Nash blowup has a combinatorial description in terms of the Gröbner fan. This description will allows us to prove a higher version of Nobile's theorem in this context: for a normal toric variety, the higher Nash blowup is an isomorphism if and only if the variety is non-singular. We will also present some further observations coming from computational experiments.

The higher Nash blowup of an algebraic variety replaces singular points with limits of certain vector spaces carrying first or higher order data associated to the variety at non-singular points. In the case of normal toric varieties, the higher Nash blowup has a combinatorial description in terms of the Gröbner fan. This description will allows us to prove a higher version of Nobile's theorem in this context: for a normal toric variety, the higher Nash blowup is an isomorphism if and only if the variety is non-singular. We will also present some further observations coming from computational experiments.

#### Numerical Analysis Seminar

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

On the finite difference approximation for blow-up solutions of the nonlinear wave equation (JAPANESE)

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

**Chien-Hong Cho**(National Chung Cheng University)On the finite difference approximation for blow-up solutions of the nonlinear wave equation (JAPANESE)

[ Abstract ]

We consider in this paper the 1-dim nonlinear wave equation $u_{tt}=u_{xx}+u^{1+\\alpha}$ $(\\alpha > 0)$ and its finite difference analogue. It is known that the solutions of the current equation becomes unbounded in finite time, a phenomenon which is often called blow-up. Numerical approaches on such kind of problems are widely investigated in the last decade. However, those results are mainly about parabolic blow-up problems. Compared with the parabolic ones, there is a remarkable property for the solution of the nonlinear wave equation -- the existence of the blow-up curve. That is, even though the solution has become unbounded at certain points, the solution continues to exist at other points and blows up at later times. We are concerned in this paper as to how a finite difference scheme can reproduce such a phenomenon.

[ Reference URL ]We consider in this paper the 1-dim nonlinear wave equation $u_{tt}=u_{xx}+u^{1+\\alpha}$ $(\\alpha > 0)$ and its finite difference analogue. It is known that the solutions of the current equation becomes unbounded in finite time, a phenomenon which is often called blow-up. Numerical approaches on such kind of problems are widely investigated in the last decade. However, those results are mainly about parabolic blow-up problems. Compared with the parabolic ones, there is a remarkable property for the solution of the nonlinear wave equation -- the existence of the blow-up curve. That is, even though the solution has become unbounded at certain points, the solution continues to exist at other points and blows up at later times. We are concerned in this paper as to how a finite difference scheme can reproduce such a phenomenon.

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

### 2014/05/08

#### Geometry Colloquium

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

On non Hamiltonian volume minimizing H-stable Lagrangian tori (JAPANESE)

**Hajime Ono**(Saitama University)On non Hamiltonian volume minimizing H-stable Lagrangian tori (JAPANESE)

[ Abstract ]

Y. –G. Oh investigated the volume of Lagrangian submanifolds in a Kaehler manifold and introduced the notion of Hamiltonian minimality, Hamiltonian stability and Hamiltonian volume minimizing property. For example, it is known that standard tori in complex Euclidean spaces and torus orbits in complex projective spaces are H-minimal and H-stable. In this talk I show that

1. Almost all of standard tori in the complex Euclidean space of dimension greater than two are not Hamiltonian volume minimizing.

2. There are non Hamiltonian volume minimizing torus orbits in any compact toric Kaehler manifold of dimension greater than two.

Y. –G. Oh investigated the volume of Lagrangian submanifolds in a Kaehler manifold and introduced the notion of Hamiltonian minimality, Hamiltonian stability and Hamiltonian volume minimizing property. For example, it is known that standard tori in complex Euclidean spaces and torus orbits in complex projective spaces are H-minimal and H-stable. In this talk I show that

1. Almost all of standard tori in the complex Euclidean space of dimension greater than two are not Hamiltonian volume minimizing.

2. There are non Hamiltonian volume minimizing torus orbits in any compact toric Kaehler manifold of dimension greater than two.

### 2014/05/07

#### Mathematical Biology Seminar

14:50-16:20 Room #128 (Graduate School of Math. Sci. Bldg.)

Asymptotic behavior of differential equation systems for age-structured epidemic models (JAPANESE)

**Yoichi Enatsu**(Graduate School of Mathematical Sciences, University fo Tokyo)Asymptotic behavior of differential equation systems for age-structured epidemic models (JAPANESE)

### 2014/05/02

#### Colloquium

16:30-17:30 Room #123 (Graduate School of Math. Sci. Bldg.)

From hyperplane arrangements to Deligne-Mumford moduli spaces: Kohno-Drinfeld way (ENGLISH)

**A.P. Veselov**(Loughborough, UK and Tokyo, Japan)From hyperplane arrangements to Deligne-Mumford moduli spaces: Kohno-Drinfeld way (ENGLISH)

[ Abstract ]

Gaudin subalgebras are abelian Lie subalgebras of maximal

dimension spanned by generators of the Kohno-Drinfeld Lie algebra t_n,

associated to A-type hyperplane arrangement.

It turns out that Gaudin subalgebras form a smooth algebraic variety

isomorphic to the Deligne-Mumford moduli space \\bar M_{0,n+1} of

stable genus zero curves with n+1 marked points.

A real version of this result allows to describe the

moduli space of integrable n-dimensional tops and

separation coordinates on the unit sphere

in terms of the geometry of Stasheff polytope.

The talk is based on joint works with L. Aguirre and G. Felder and with K.

Schoebel.

Gaudin subalgebras are abelian Lie subalgebras of maximal

dimension spanned by generators of the Kohno-Drinfeld Lie algebra t_n,

associated to A-type hyperplane arrangement.

It turns out that Gaudin subalgebras form a smooth algebraic variety

isomorphic to the Deligne-Mumford moduli space \\bar M_{0,n+1} of

stable genus zero curves with n+1 marked points.

A real version of this result allows to describe the

moduli space of integrable n-dimensional tops and

separation coordinates on the unit sphere

in terms of the geometry of Stasheff polytope.

The talk is based on joint works with L. Aguirre and G. Felder and with K.

Schoebel.

### 2014/04/30

#### Number Theory Seminar

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

An effective upper bound for the number of principally polarized Abelian schemes (JAPANESE)

**Takuya Maruyama**(University of Tokyo)An effective upper bound for the number of principally polarized Abelian schemes (JAPANESE)

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