## Seminar information archive

Seminar information archive ～02/06｜Today's seminar 02/07 | Future seminars 02/08～

#### Lectures

10:40-12:10 Room #123 (Graduate School of Math. Sci. Bldg.)

Market, Liquidity and Counterparty Risk (ENGLISH)

**Jean Meyer, Yasuko HISAMATSU**(Risk Capital Market Tokyo, BNP Paribas)Market, Liquidity and Counterparty Risk (ENGLISH)

[ Abstract ]

1. Introduction to the market risk

- Introduction to the Risk Management

in the Financial institutions

- Overview of the main market risks

2. Market & Liquidity Risks –Basics

-Presentation of the main Greeks

-Focus on volatility risk

-Focus on correlation risk

-Conclusion (common features of the market risks)

3. Risk measure

- Stress test

- Value at risk

- Risks measure for counterparty risk

1. Introduction to the market risk

- Introduction to the Risk Management

in the Financial institutions

- Overview of the main market risks

2. Market & Liquidity Risks –Basics

-Presentation of the main Greeks

-Focus on volatility risk

-Focus on correlation risk

-Conclusion (common features of the market risks)

3. Risk measure

- Stress test

- Value at risk

- Risks measure for counterparty risk

### 2010/11/02

#### Lectures

13:00-16:10 Room #122 (Graduate School of Math. Sci. Bldg.)

The Malliavin calculus on configuration spaces and applications (ENGLISH)

**Vladimir Bogachev**(Moscow)The Malliavin calculus on configuration spaces and applications (ENGLISH)

[ Abstract ]

It is planned to discuss first a general scheme of the Malliavin

calculus on an abstract measurable

manifold with minimal assumptions about the manifold.

Then a practical realization of this scheme will be discussed in

several concrete examples with emphasis

on configuration spaces, i.e., spaces of locally finite configurations

in a given manifold (for example, just

a finite-dimensional Euclidean space), which can be alternatively

described as the spaces of integer-valued

discrete measures equipped with suitable differential structures.

No acquaintance with the Malliavin calculus and differential geometry

is assumed.

It is planned to discuss first a general scheme of the Malliavin

calculus on an abstract measurable

manifold with minimal assumptions about the manifold.

Then a practical realization of this scheme will be discussed in

several concrete examples with emphasis

on configuration spaces, i.e., spaces of locally finite configurations

in a given manifold (for example, just

a finite-dimensional Euclidean space), which can be alternatively

described as the spaces of integer-valued

discrete measures equipped with suitable differential structures.

No acquaintance with the Malliavin calculus and differential geometry

is assumed.

#### Tuesday Seminar on Topology

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

Periodic-end manifolds and SW theory (ENGLISH)

**Daniel Ruberman**(Brandeis University)Periodic-end manifolds and SW theory (ENGLISH)

[ Abstract ]

We study an extension of Seiberg-Witten invariants to

4-manifolds with the homology of S^1 \\times S^3. This extension has

many potential applications in low-dimensional topology, including the

study of the homology cobordism group. Because b_2^+ =0, the usual

strategy for defining invariants does not work--one cannot disregard

reducible solutions. In fact, the count of solutions can jump in a

family of metrics or perturbations. To remedy this, we define an

index-theoretic counter-term that jumps by the same amount. The

counterterm is the index of the Dirac operator on a manifold with a

periodic end modeled at infinity by the infinite cyclic cover of the

manifold. This is joint work with Tomasz Mrowka and Nikolai Saveliev.

We study an extension of Seiberg-Witten invariants to

4-manifolds with the homology of S^1 \\times S^3. This extension has

many potential applications in low-dimensional topology, including the

study of the homology cobordism group. Because b_2^+ =0, the usual

strategy for defining invariants does not work--one cannot disregard

reducible solutions. In fact, the count of solutions can jump in a

family of metrics or perturbations. To remedy this, we define an

index-theoretic counter-term that jumps by the same amount. The

counterterm is the index of the Dirac operator on a manifold with a

periodic end modeled at infinity by the infinite cyclic cover of the

manifold. This is joint work with Tomasz Mrowka and Nikolai Saveliev.

#### Lie Groups and Representation Theory

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

Twistor theory and the harmonic hull (ENGLISH)

**Michael Eastwood**(University of Adelaide)Twistor theory and the harmonic hull (ENGLISH)

[ Abstract ]

Harmonic functions are real-analytic and so automatically extend from being functions of real variables to being functions of complex variables. But how far do they extend? This question may be answered by twistor theory, the Penrose transform, and associated geometry. I shall base the constructions on a formula of Bateman from 1904. This is joint work with Feng Xu.

Harmonic functions are real-analytic and so automatically extend from being functions of real variables to being functions of complex variables. But how far do they extend? This question may be answered by twistor theory, the Penrose transform, and associated geometry. I shall base the constructions on a formula of Bateman from 1904. This is joint work with Feng Xu.

### 2010/11/01

#### Algebraic Geometry Seminar

16:40-18:10 Room #126 (Graduate School of Math. Sci. Bldg.)

How to estimate Seshadri constants (JAPANESE)

**Atsushi Ito**(Univ. of Tokyo)How to estimate Seshadri constants (JAPANESE)

[ Abstract ]

Seshadri constant is an invariant which measures the positivities of ample line bundles. This relates with adjoint bundles, Nagata conjecture, slope stabilities, Gromov width (an invariant of symplectic manifolds) and so on. But it is very diffiult to compute or estimate Seshadri constants in general, especially in higher dimension.

In this talk, we first study Seshadri constants of toric varieties, and next consider about non-toric cases using toric degenerations. For example, good estimations are obtained for complete intersections in projective spaces.

Seshadri constant is an invariant which measures the positivities of ample line bundles. This relates with adjoint bundles, Nagata conjecture, slope stabilities, Gromov width (an invariant of symplectic manifolds) and so on. But it is very diffiult to compute or estimate Seshadri constants in general, especially in higher dimension.

In this talk, we first study Seshadri constants of toric varieties, and next consider about non-toric cases using toric degenerations. For example, good estimations are obtained for complete intersections in projective spaces.

#### Lectures

16:00-18:15 Room #270 (Graduate School of Math. Sci. Bldg.)

Inverse problems in non linear parabolic equations : Two differents approaches (ENGLISH)

[ Reference URL ]

https://www.ms.u-tokyo.ac.jp/~kengok/abstractTokyo.pdf

Inverse Problems for parabolic System

(ENGLISH)

**Michel Cristofol**(マルセイユ大学) 16:00-17:00Inverse problems in non linear parabolic equations : Two differents approaches (ENGLISH)

[ Reference URL ]

https://www.ms.u-tokyo.ac.jp/~kengok/abstractTokyo.pdf

**Patricia Gaitan**(マルセイユ大学) 17:15-18:15Inverse Problems for parabolic System

(ENGLISH)

[ Abstract ]

I will present a review of stability and controllability results for linear parabolic coupled systems with coupling of first and zeroth-order terms by data of reduced number of components. The key ingredients are global Carleman estimates.

I will present a review of stability and controllability results for linear parabolic coupled systems with coupling of first and zeroth-order terms by data of reduced number of components. The key ingredients are global Carleman estimates.

### 2010/10/29

#### Colloquium

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

Ambient metrics and exceptional holonomy (ENGLISH)

**Robin Graham**(University of Washington)Ambient metrics and exceptional holonomy (ENGLISH)

[ Abstract ]

The holonomy of a pseudo-Riemannian metric is a subgroup of the orthogonal group which measures the structure preserved by parallel translation. Construction of pseudo-Riemannian metrics whose holonomy is an exceptional Lie group has been of great interest in recent years. This talk will outline a construction of metrics in dimension 7 whose holonomy is contained in the split real form of the exceptional group $G_2$. The datum for the construction is a generic real-analytic 2-plane field on a manifold of dimension 5; the metric in dimension 7 arises as the ambient metric of a conformal structure on the 5-manifold defined by Nurowski in terms of the 2-plane field.

The holonomy of a pseudo-Riemannian metric is a subgroup of the orthogonal group which measures the structure preserved by parallel translation. Construction of pseudo-Riemannian metrics whose holonomy is an exceptional Lie group has been of great interest in recent years. This talk will outline a construction of metrics in dimension 7 whose holonomy is contained in the split real form of the exceptional group $G_2$. The datum for the construction is a generic real-analytic 2-plane field on a manifold of dimension 5; the metric in dimension 7 arises as the ambient metric of a conformal structure on the 5-manifold defined by Nurowski in terms of the 2-plane field.

### 2010/10/28

#### Operator Algebra Seminars

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

Type III representations of the infinite symmetric group (ENGLISH)

**Makoto Yamashita**(Univ. Tokyo)Type III representations of the infinite symmetric group (ENGLISH)

[ Abstract ]

Based on earlier results about the structure of the II$_1$ representations of the infinite symmetric group, we investigate its type III representations and the related inclusion of von Neumann algebras of type III.

Based on earlier results about the structure of the II$_1$ representations of the infinite symmetric group, we investigate its type III representations and the related inclusion of von Neumann algebras of type III.

#### Lectures

10:40-12:10 Room #123 (Graduate School of Math. Sci. Bldg.)

Market, Liquidity and Counterparty Risk (ENGLISH)

**Jean Meyer, Yasuko HISAMATSU**(Risk Capital Market Tokyo, BNP Paribas)Market, Liquidity and Counterparty Risk (ENGLISH)

[ Abstract ]

1. Introduction to the market risk

- Introduction to the Risk Management

in the Financial institutions

- Overview of the main market risks

2. Market & Liquidity Risks –Basics

-Presentation of the main Greeks

-Focus on volatility risk

-Focus on correlation risk

-Conclusion (common features of the market risks)

3. Risk measure

- Stress test

- Value at risk

- Risks measure for counterparty risk

1. Introduction to the market risk

- Introduction to the Risk Management

in the Financial institutions

- Overview of the main market risks

2. Market & Liquidity Risks –Basics

-Presentation of the main Greeks

-Focus on volatility risk

-Focus on correlation risk

-Conclusion (common features of the market risks)

3. Risk measure

- Stress test

- Value at risk

- Risks measure for counterparty risk

### 2010/10/26

#### Seminar on Geometric Complex Analysis

13:00-14:30 Room #123 (Graduate School of Math. Sci. Bldg.)

Limits of Moishezon Manifolds under Holomorphic Deformations (ENGLISH)

**Dan Popovici**(Toulouse)Limits of Moishezon Manifolds under Holomorphic Deformations (ENGLISH)

[ Abstract ]

We prove that if all the fibres, except one, of a holomorphic family of compact complex manifolds are supposed to be Moishezon (i.e. bimeromorphic to projective manifolds), then the remaining (limit) fibre is again Moishezon. The two ingredients of the proof are the relative Barlet space of divisors contained in the fibres for which we show properness over the base of the family and the "strongly Gauduchon" (sG) metrics that we have introduced for the purpose of controlling volumes of cycles. These new metrics enjoy stability properties under both deformations and modifications and play a crucial role in obtaining a uniform control on volumes of relative divisors that prove the above-mentioned properness.

We prove that if all the fibres, except one, of a holomorphic family of compact complex manifolds are supposed to be Moishezon (i.e. bimeromorphic to projective manifolds), then the remaining (limit) fibre is again Moishezon. The two ingredients of the proof are the relative Barlet space of divisors contained in the fibres for which we show properness over the base of the family and the "strongly Gauduchon" (sG) metrics that we have introduced for the purpose of controlling volumes of cycles. These new metrics enjoy stability properties under both deformations and modifications and play a crucial role in obtaining a uniform control on volumes of relative divisors that prove the above-mentioned properness.

#### Tuesday Seminar on Topology

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

Quantum fundamental groups and quantum representation varieties for 3-manifolds (JAPANESE)

**Kazuo Habiro**(RIMS, Kyoto University)Quantum fundamental groups and quantum representation varieties for 3-manifolds (JAPANESE)

[ Abstract ]

We define a refinement of the fundamental groups of 3-manifolds and

a generalization of representation variety of the fundamental group

of 3-manifolds. We consider the category $H$ whose morphisms are

nonnegative integers, where $n$ corresponds to a genus $n$ handlebody

equipped with an embedding of a disc into the boundary, and whose

morphisms are the isotopy classes of embeddings of handlebodies

compatible with the embeddings of the disc into the boundaries. For

each 3-manifold $M$ with an embedding of a disc into the boundary, we

can construct a contravariant functor from $H$ to the category of

sets, where the object $n$ of $H$ is mapped to the set of isotopy

classes of embedding of the genus $n$ handlebody into $M$, compatible

with the embeddings of the disc into the boundaries. This functor can

be regarded as a refinement of the fundamental group of $M$, and we

call it the quantum fundamental group of $M$. Using this invariant, we

can construct for each co-ribbon Hopf algebra $A$ an invariant of

3-manifolds which may be regarded as (the space of regular functions

on) the representation variety of $M$ with respect to $A$.

We define a refinement of the fundamental groups of 3-manifolds and

a generalization of representation variety of the fundamental group

of 3-manifolds. We consider the category $H$ whose morphisms are

nonnegative integers, where $n$ corresponds to a genus $n$ handlebody

equipped with an embedding of a disc into the boundary, and whose

morphisms are the isotopy classes of embeddings of handlebodies

compatible with the embeddings of the disc into the boundaries. For

each 3-manifold $M$ with an embedding of a disc into the boundary, we

can construct a contravariant functor from $H$ to the category of

sets, where the object $n$ of $H$ is mapped to the set of isotopy

classes of embedding of the genus $n$ handlebody into $M$, compatible

with the embeddings of the disc into the boundaries. This functor can

be regarded as a refinement of the fundamental group of $M$, and we

call it the quantum fundamental group of $M$. Using this invariant, we

can construct for each co-ribbon Hopf algebra $A$ an invariant of

3-manifolds which may be regarded as (the space of regular functions

on) the representation variety of $M$ with respect to $A$.

#### Numerical Analysis Seminar

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

Theoretical analysis of Sinc schemes for integral equations of the second kind (JAPANESE)

[ Reference URL ]

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

**Tomoaki Okayama**(Hitotsubashi University)Theoretical analysis of Sinc schemes for integral equations of the second kind (JAPANESE)

[ Reference URL ]

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

#### Lie Groups and Representation Theory

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

Some instances of the reasonable effectiveness (and limitations) of symmetries and deformations in fundamental physics (ENGLISH)

https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar2010.html

**Daniel Sternheimer**(Keio University and Institut de Mathematiques de Bourgogne)Some instances of the reasonable effectiveness (and limitations) of symmetries and deformations in fundamental physics (ENGLISH)

[ Abstract ]

In this talk we survey some applications of group theory and deformation theory (including quantization) in mathematical physics. We start with sketching applications of rotation and discrete groups representations in molecular physics (``dynamical" symmetry breaking in crystals, Racah-Flato-Kibler; chains of groups and symmetry breaking). These methods led to the use of ``classification Lie groups" (``internal symmetries") in particle physics. Their relation with space-time symmetries will be discussed. Symmetries are naturally deformed, which eventually brought to Flato's deformation philosophy and the realization that quantization can be viewed as a deformation, including the many avatars of deformation quantization (such as quantum groups and quantized spaces). Nonlinear representations of Lie groups can be viewed as deformations (of their linear part), with applications to covariant nonlinear evolution equations. Combining all these suggests an Ansatz based on Anti de Sitter space-time and group, a deformation of the Poincare group of Minkowski space-time, which could eventually be quantized, with possible implications in particle physics and cosmology. Prospects for future developments between mathematics and physics will be indicated.

[ Reference URL ]In this talk we survey some applications of group theory and deformation theory (including quantization) in mathematical physics. We start with sketching applications of rotation and discrete groups representations in molecular physics (``dynamical" symmetry breaking in crystals, Racah-Flato-Kibler; chains of groups and symmetry breaking). These methods led to the use of ``classification Lie groups" (``internal symmetries") in particle physics. Their relation with space-time symmetries will be discussed. Symmetries are naturally deformed, which eventually brought to Flato's deformation philosophy and the realization that quantization can be viewed as a deformation, including the many avatars of deformation quantization (such as quantum groups and quantized spaces). Nonlinear representations of Lie groups can be viewed as deformations (of their linear part), with applications to covariant nonlinear evolution equations. Combining all these suggests an Ansatz based on Anti de Sitter space-time and group, a deformation of the Poincare group of Minkowski space-time, which could eventually be quantized, with possible implications in particle physics and cosmology. Prospects for future developments between mathematics and physics will be indicated.

https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar2010.html

### 2010/10/21

#### Operator Algebra Seminars

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

Free probability and entropy additivity problems for Quantum information theory (ENGLISH)

**Benoit Collins**(Univ. Ottawa)Free probability and entropy additivity problems for Quantum information theory (ENGLISH)

### 2010/10/20

#### Seminar on Mathematics for various disciplines

10:30-11:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Curvature Dependent Diffusion Flow on Surface with Thickness (JAPANESE)

**Naohisa Ogawa**(Hokkaido Institute of Technology)Curvature Dependent Diffusion Flow on Surface with Thickness (JAPANESE)

[ Abstract ]

Particle diffusion in a two dimensional curved surface with thickness

embedded in $R_3$ is considered.

In addition to the usual diffusion flow, we find a new flow with an explicit

curvature dependence in $\\epsilon$ (thickness of surface) expansion.

As an example, the surface of elliptic cylinder is considered, and curvature

dependent diffusion coefficient is calculated. In addition, we consider the

1 dimensional object in $R_3$ (Tube),

and check the physical meaning of curvature effect.

Particle diffusion in a two dimensional curved surface with thickness

embedded in $R_3$ is considered.

In addition to the usual diffusion flow, we find a new flow with an explicit

curvature dependence in $\\epsilon$ (thickness of surface) expansion.

As an example, the surface of elliptic cylinder is considered, and curvature

dependent diffusion coefficient is calculated. In addition, we consider the

1 dimensional object in $R_3$ (Tube),

and check the physical meaning of curvature effect.

### 2010/10/19

#### Tuesday Seminar on Topology

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

Optimistic limits of colored Jones invariants (ENGLISH)

**Jinseok Cho**(Waseda University)Optimistic limits of colored Jones invariants (ENGLISH)

[ Abstract ]

Yokota made a wonderful theory on the optimistic limit of Kashaev

invariant of a hyperbolic knot

that the limit determines the hyperbolic volume and the Chern-Simons

invariant of the knot.

Especially, his theory enables us to calculate the volume of a knot

combinatorially from its diagram for many cases.

We will briefly discuss Yokota theory, and then move to the optimistic

limit of colored Jones invariant.

We will explain a parallel version of Yokota theory based on the

optimistic limit of colored Jones invariant.

Especially, we will show the optimistic limit of colored Jones

invariant coincides with that of Kashaev invariant modulo 2\\pi^2.

This implies the optimistic limit of colored Jones invariant also

determines the volume and Chern-Simons invariant of the knot, and

probably more information.

This is a joint-work with Jun Murakami of Waseda University.

Yokota made a wonderful theory on the optimistic limit of Kashaev

invariant of a hyperbolic knot

that the limit determines the hyperbolic volume and the Chern-Simons

invariant of the knot.

Especially, his theory enables us to calculate the volume of a knot

combinatorially from its diagram for many cases.

We will briefly discuss Yokota theory, and then move to the optimistic

limit of colored Jones invariant.

We will explain a parallel version of Yokota theory based on the

optimistic limit of colored Jones invariant.

Especially, we will show the optimistic limit of colored Jones

invariant coincides with that of Kashaev invariant modulo 2\\pi^2.

This implies the optimistic limit of colored Jones invariant also

determines the volume and Chern-Simons invariant of the knot, and

probably more information.

This is a joint-work with Jun Murakami of Waseda University.

### 2010/10/18

#### Seminar on Geometric Complex Analysis

10:30-11:30 Room #128 (Graduate School of Math. Sci. Bldg.)

Limiting behavior of minimal trajectories of parabolic vector fields on the complex projective plane. (ENGLISH)

**Sergey Ivashkovitch**(Univ. de Lille)Limiting behavior of minimal trajectories of parabolic vector fields on the complex projective plane. (ENGLISH)

[ Abstract ]

The classical Poincare-Bendixson theory describes the way a trajectory of a vector field on the real plane behaves when accumulating to the singular locus of the vector field. We shall describe, in the first approximation, the way a minimal trajectory of a parabolic complex polynomial vector field (or, a holomorphic foliation) on the complex projective plane approaches the singular locus. In particular we shall prove that if a holomorphic foliation has an exceptional minimal set then its nef model is necessarily hyperbolic.

The classical Poincare-Bendixson theory describes the way a trajectory of a vector field on the real plane behaves when accumulating to the singular locus of the vector field. We shall describe, in the first approximation, the way a minimal trajectory of a parabolic complex polynomial vector field (or, a holomorphic foliation) on the complex projective plane approaches the singular locus. In particular we shall prove that if a holomorphic foliation has an exceptional minimal set then its nef model is necessarily hyperbolic.

#### Seminar on Geometric Complex Analysis

13:00-14:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Degenerate complex Monge-Ampere equations (ENGLISH)

**Philippe Eyssidieux**(Institut Fourier, Grenoble)Degenerate complex Monge-Ampere equations (ENGLISH)

#### Kavli IPMU Komaba Seminar

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

Quasi-modular forms and Gromov--Witten theory of elliptic orbifold $\\mathbb{P}^1$ (ENGLISH)

**Todor Milanov**(IPMU)Quasi-modular forms and Gromov--Witten theory of elliptic orbifold $\\mathbb{P}^1$ (ENGLISH)

[ Abstract ]

This talk is based on my current work with Y. Ruan. Our project is part of the so called Landau--Ginzburg/Calabi-Yau correspondence. The latter is a conjecture, due to Ruan, that describes the relation between the $W$-spin invariants of a Landau-Ginzburg potential $W$ and the Gromov--Witten invariants of a certain Calabi--Yau orbifold. I am planning first to explain the higher-genus reconstruction formalism of Givental. This formalism together with the work of M. Krawitz and Y. Shen allows us to express the Gromov--Witten invariants of the orbifold $\\mathbb{P}^1$'s with weights $(3,3,3)$, $(2,4,4)$, and $(2,3,6)$ in terms of Saito's Frobenius structure associated with the simple elliptic singularities $P_8$, $X_9$, and $J_{10}$ respectively. After explaining Givental's formalism, my goal would be to discuss the Saito's flat structure, and to explain how its modular behavior fits in the Givental's formalism. This allows us to prove that the Gromov--Witten invariants are quasi-modular forms on an appropriate modular group.

This talk is based on my current work with Y. Ruan. Our project is part of the so called Landau--Ginzburg/Calabi-Yau correspondence. The latter is a conjecture, due to Ruan, that describes the relation between the $W$-spin invariants of a Landau-Ginzburg potential $W$ and the Gromov--Witten invariants of a certain Calabi--Yau orbifold. I am planning first to explain the higher-genus reconstruction formalism of Givental. This formalism together with the work of M. Krawitz and Y. Shen allows us to express the Gromov--Witten invariants of the orbifold $\\mathbb{P}^1$'s with weights $(3,3,3)$, $(2,4,4)$, and $(2,3,6)$ in terms of Saito's Frobenius structure associated with the simple elliptic singularities $P_8$, $X_9$, and $J_{10}$ respectively. After explaining Givental's formalism, my goal would be to discuss the Saito's flat structure, and to explain how its modular behavior fits in the Givental's formalism. This allows us to prove that the Gromov--Witten invariants are quasi-modular forms on an appropriate modular group.

#### Algebraic Geometry Seminar

16:40-18:10 Room #126 (Graduate School of Math. Sci. Bldg.)

Galois extensions and maps on local cohomology (JAPANESE)

**Akiyoshi Sannai**(Univ. of Tokyo)Galois extensions and maps on local cohomology (JAPANESE)

### 2010/10/14

#### Operator Algebra Seminars

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

Nonstandard analysis for operator algebraists (JAPANESE)

**Yasuyuki Kawahigashi**(Univ. Tokyo)Nonstandard analysis for operator algebraists (JAPANESE)

### 2010/10/12

#### Tuesday Seminar on Topology

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

Asymptotics of Morse numbers of finite coverings of manifolds (ENGLISH)

**Andrei Pajitnov**(Univ. de Nantes, The University of Tokyo)Asymptotics of Morse numbers of finite coverings of manifolds (ENGLISH)

[ Abstract ]

Let X be a closed manifold;

denote by m(X) the Morse number of X

(that is, the minimal number of critical

points of a Morse function on X).

Let Y be a finite covering of X of degree d.

In this survey talk we will address the following question

posed by M. Gromov: What are the asymptotic properties

of m(N) as d goes to infinity?

It turns out that for high-dimensional manifolds with

free abelian fundamental group the asymptotics of

the number m(N)/d is directly related to the Novikov homology

of N. We prove this theorem and discuss related results.

Let X be a closed manifold;

denote by m(X) the Morse number of X

(that is, the minimal number of critical

points of a Morse function on X).

Let Y be a finite covering of X of degree d.

In this survey talk we will address the following question

posed by M. Gromov: What are the asymptotic properties

of m(N) as d goes to infinity?

It turns out that for high-dimensional manifolds with

free abelian fundamental group the asymptotics of

the number m(N)/d is directly related to the Novikov homology

of N. We prove this theorem and discuss related results.

### 2010/10/08

#### Colloquium

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

Exceptional Jacobi polynomials as solutions of a Schroedinger

(Sturm-Liouville) equation with $3 +¥ell$ ($¥ell=1,2,¥ldots) regular

singularities (JAPANESE)

**Ryu Sasaki**(Yukawa Institute for Theoretical Physics, Kyoto University)Exceptional Jacobi polynomials as solutions of a Schroedinger

(Sturm-Liouville) equation with $3 +¥ell$ ($¥ell=1,2,¥ldots) regular

singularities (JAPANESE)

[ Abstract ]

Global solutions of Fuchsian differential equations with more than 3 (hypergeometric) or four (Heun) regular singularities had been virtually unkown. Here I present a complete set of eigenfunctions of a Schroedinger (Sturm-Liouville) equation with $3 + ¥ell$ ($¥ell=1,2,¥ldots$) regular singularities. They are deformations of the Darboux-P¥" oschl-Teller potential with the Hamiltonian (Schroedinger operator) ¥[ ¥mathcal{H}=-¥frac{d^2}{dx^2}+¥frac{g(g-1)}{¥sin^2x}+¥frac{h(h-1)} {¥cos^2x}¥] The eigenfunctions consist of the {¥em exceptional Jacobi polynomials} $¥{P_{¥ell,n}(¥eta)¥}$, $n=0,1,2,¥ldots$, with deg($P_{¥ell,n}$)$=n+¥ell$. Thus the restriction due to Bochner's theorem does not apply. The confluent limit produces two sets of the exceptional Laguerre polynomials for $¥ell=1,2,¥ldots$. Similar deformation method provides the exceptional Wilson and Askey-Wilson polynomials for $¥ell=1,2,¥ldots$.

Global solutions of Fuchsian differential equations with more than 3 (hypergeometric) or four (Heun) regular singularities had been virtually unkown. Here I present a complete set of eigenfunctions of a Schroedinger (Sturm-Liouville) equation with $3 + ¥ell$ ($¥ell=1,2,¥ldots$) regular singularities. They are deformations of the Darboux-P¥" oschl-Teller potential with the Hamiltonian (Schroedinger operator) ¥[ ¥mathcal{H}=-¥frac{d^2}{dx^2}+¥frac{g(g-1)}{¥sin^2x}+¥frac{h(h-1)} {¥cos^2x}¥] The eigenfunctions consist of the {¥em exceptional Jacobi polynomials} $¥{P_{¥ell,n}(¥eta)¥}$, $n=0,1,2,¥ldots$, with deg($P_{¥ell,n}$)$=n+¥ell$. Thus the restriction due to Bochner's theorem does not apply. The confluent limit produces two sets of the exceptional Laguerre polynomials for $¥ell=1,2,¥ldots$. Similar deformation method provides the exceptional Wilson and Askey-Wilson polynomials for $¥ell=1,2,¥ldots$.

### 2010/10/06

#### Geometry Seminar

14:45-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Handle attaching in wrapped Floer homology and brake orbits in classical Hamiltonian systems (JAPANESE)

Mirror Symmetry for Weighted Homogeneous Polynomials (JAPANESE)

**Kei Irie**(Kyoto Univ.) 14:45-16:15Handle attaching in wrapped Floer homology and brake orbits in classical Hamiltonian systems (JAPANESE)

[ Abstract ]

In this talk, the term "classical Hamiltonian systems" means special types of Hamiltonian systems, which describe solutions of classical equations of motion. The study of periodic solutions of Hamiltonian systems is an interesting problem, and for classical Hamiltonian systems, the following result is known : for any compact and regular energy surface $S$, there exists a brake orbit (a particular type of periodic solutions) on $S$. This result is first proved by S.V.Bolotin in 1978, and it is a special case of the Arnold chord conjecture. In this talk, I will explain that calculations of wrapped Floer homology (which is a variant of Lagrangian Floer homology) give a new proof of the above result.

In this talk, the term "classical Hamiltonian systems" means special types of Hamiltonian systems, which describe solutions of classical equations of motion. The study of periodic solutions of Hamiltonian systems is an interesting problem, and for classical Hamiltonian systems, the following result is known : for any compact and regular energy surface $S$, there exists a brake orbit (a particular type of periodic solutions) on $S$. This result is first proved by S.V.Bolotin in 1978, and it is a special case of the Arnold chord conjecture. In this talk, I will explain that calculations of wrapped Floer homology (which is a variant of Lagrangian Floer homology) give a new proof of the above result.

**Atsushi Takahashi**(Osaka Univ.) 16:30-18:00Mirror Symmetry for Weighted Homogeneous Polynomials (JAPANESE)

[ Abstract ]

First we give an overview of the algebraic and the geometric aspects of the mirror symmetry conjecture for weighted homogeneous polynomials. Then we concentrate on polynomials in three variables, and show the existence of full (strongly) exceptional collection of categories of maximally graded matrix factorizations for invertible weighted homogeneous polynomials. We will also explain how the mirror symmetry naturally explains and generalizes the Arnold's strange duality between the 14 exceptional unimodal singularities.

First we give an overview of the algebraic and the geometric aspects of the mirror symmetry conjecture for weighted homogeneous polynomials. Then we concentrate on polynomials in three variables, and show the existence of full (strongly) exceptional collection of categories of maximally graded matrix factorizations for invertible weighted homogeneous polynomials. We will also explain how the mirror symmetry naturally explains and generalizes the Arnold's strange duality between the 14 exceptional unimodal singularities.

#### Number Theory Seminar

16:30-17:30 Room #117 (Graduate School of Math. Sci. Bldg.)

Finite group actions on the affine space (ENGLISH)

**Hélène Esnault**(Universität Duisburg-Essen)Finite group actions on the affine space (ENGLISH)

[ Abstract ]

If $G$ is a finite $\\ell$-group acting on an affine space $\\A^n$ over a

finite field $K$ of cardinality prime to $\\ell$, Serre shows that there

exists a rational fixed point. We generalize this to the case where $K$ is a

henselian discretely valued field of characteristic zero with algebraically

closed residue field and with residue characteristic different from $\\ell$.

We also treat the case where the residue field is finite of cardinality $q$

such that $\\ell$ divides $q-1$. To this aim, we study group actions on weak

N\\'eron models.

(Joint work with Johannes Nicaise)

If $G$ is a finite $\\ell$-group acting on an affine space $\\A^n$ over a

finite field $K$ of cardinality prime to $\\ell$, Serre shows that there

exists a rational fixed point. We generalize this to the case where $K$ is a

henselian discretely valued field of characteristic zero with algebraically

closed residue field and with residue characteristic different from $\\ell$.

We also treat the case where the residue field is finite of cardinality $q$

such that $\\ell$ divides $q-1$. To this aim, we study group actions on weak

N\\'eron models.

(Joint work with Johannes Nicaise)

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