## Seminar information archive

Seminar information archive ～02/20｜Today's seminar 02/21 | Future seminars 02/22～

#### thesis presentations

14:15-15:30 Room #126 (Graduate School of Math. Sci. Bldg.)

Singularities for Solutions to Schrödinger Equations (シュレーディンガー方程式の解の特異性)

**毛 仕寛**(東京大学大学院数理科学研究科)Singularities for Solutions to Schrödinger Equations (シュレーディンガー方程式の解の特異性)

#### thesis presentations

15:45-17:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Nevanlinna theory for holomorphic mappings and related problems (正則写像のネヴァンリンナ理論と関連する問題)

**Si, Duc Quang**(東京大学大学院数理科学研究科)Nevanlinna theory for holomorphic mappings and related problems (正則写像のネヴァンリンナ理論と関連する問題)

#### thesis presentations

11:00-12:15 Room #128 (Graduate School of Math. Sci. Bldg.)

On existence of models for the logical system MPCL (単相格論理系におけるモデルの存在について)

**高岡 洋介**(東京大学大学院数理科学研究科)On existence of models for the logical system MPCL (単相格論理系におけるモデルの存在について)

#### thesis presentations

13:00-14:15 Room #128 (Graduate School of Math. Sci. Bldg.)

Exact Solutions of Ultradiscrete Integrable Systems (超離散可積分系の厳密解)

**岩尾 慎介**(東京大学大学院数理科学研究科)Exact Solutions of Ultradiscrete Integrable Systems (超離散可積分系の厳密解)

#### thesis presentations

14:15-15:30 Room #128 (Graduate School of Math. Sci. Bldg.)

Vertex operators and background solutions for ultradiscrete soliton equations (超離散ソリトン方程式における頂点作用素と背景解)

**中田 庸一**(東京大学大学院数理科学研究科)Vertex operators and background solutions for ultradiscrete soliton equations (超離散ソリトン方程式における頂点作用素と背景解)

### 2010/02/02

#### Lie Groups and Representation Theory

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

Deformation of compact quotients of homogeneous spaces

**Fanny Kassel**(Orsay)Deformation of compact quotients of homogeneous spaces

[ Abstract ]

Let G/H be a reductive homogeneous space. In all known examples, if

G/H admits compact Clifford-Klein forms, then it admits "standard"

ones, by uniform lattices of some reductive subgroup L of G acting

properly on G/H. In order to obtain more generic Clifford-Klein forms,

we prove that for L of real rank 1, if one slightly deforms in G a

uniform lattice of L, then its action on G/H remains properly

discontinuous. As an application, we obtain compact quotients of SO(2,2n)/U(1,n) by Zariski-dense discrete subgroups of SO(2,2n) acting properly discontinuously.

Let G/H be a reductive homogeneous space. In all known examples, if

G/H admits compact Clifford-Klein forms, then it admits "standard"

ones, by uniform lattices of some reductive subgroup L of G acting

properly on G/H. In order to obtain more generic Clifford-Klein forms,

we prove that for L of real rank 1, if one slightly deforms in G a

uniform lattice of L, then its action on G/H remains properly

discontinuous. As an application, we obtain compact quotients of SO(2,2n)/U(1,n) by Zariski-dense discrete subgroups of SO(2,2n) acting properly discontinuously.

#### Tuesday Seminar on Topology

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

Deformation of compact quotients of homogeneous spaces

**Fanny Kassel**(Univ. Paris-Sud, Orsay)Deformation of compact quotients of homogeneous spaces

[ Abstract ]

Let G/H be a reductive homogeneous space. In all known examples, if

G/H admits compact Clifford-Klein forms, then it admits "standard"

ones, by uniform lattices of some reductive subgroup L of G acting

properly on G/H. In order to obtain more generic Clifford-Klein forms,

we prove that for L of real rank 1, if one slightly deforms in G a

uniform lattice of L, then its action on G/H remains properly

discontinuous. As an application, we obtain compact quotients of

SO(2,2n)/U(1,n) by Zariski-dense discrete subgroups of SO(2,2n) acting

properly discontinuously.

http://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar2010.html#20100202kassel

Let G/H be a reductive homogeneous space. In all known examples, if

G/H admits compact Clifford-Klein forms, then it admits "standard"

ones, by uniform lattices of some reductive subgroup L of G acting

properly on G/H. In order to obtain more generic Clifford-Klein forms,

we prove that for L of real rank 1, if one slightly deforms in G a

uniform lattice of L, then its action on G/H remains properly

discontinuous. As an application, we obtain compact quotients of

SO(2,2n)/U(1,n) by Zariski-dense discrete subgroups of SO(2,2n) acting

properly discontinuously.

http://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar2010.html#20100202kassel

### 2010/02/01

#### Seminar on Geometric Complex Analysis

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Connectedness of Levi nonflat pseudoconvex hypersurfaces in Kaehler manifolds

**大沢健夫**(名古屋大学多元数理科学研究科)Connectedness of Levi nonflat pseudoconvex hypersurfaces in Kaehler manifolds

#### Algebraic Geometry Seminar

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

Extensions of two Chow stability criteria to positive characteristics

**大川 新之介**(東大数理)Extensions of two Chow stability criteria to positive characteristics

[ Abstract ]

I will talk about two results on Chow (semi-)stability of cycles in positive characteristics, which were originally known in characteristic 0. One is on the stability of non-singular projective hypersurfaces of degree greater than 2, and the other is the criterion by Y. Lee in terms of the log canonical threshold of Chow divisor. A couple of examples will be discussed in detail.

I will talk about two results on Chow (semi-)stability of cycles in positive characteristics, which were originally known in characteristic 0. One is on the stability of non-singular projective hypersurfaces of degree greater than 2, and the other is the criterion by Y. Lee in terms of the log canonical threshold of Chow divisor. A couple of examples will be discussed in detail.

#### Kavli IPMU Komaba Seminar

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

Bases in the solution space of the Mellin system

**Timur Sadykov**(Siberian Federal University)Bases in the solution space of the Mellin system

[ Abstract ]

I will present a joint work with Alicia Dickenstein.

We consider algebraic functions $z$ satisfying equations of the

form

\\begin{equation}

a_0 z^m + a_1z^{m_1} + a_2 z^{m_2} + \\ldots + a_n z^{m_n} +

a_{n+1} =0.

\\end{equation}

Here $m > m_1 > \\ldots > m_n>0,$ $m,m_i \\in \\N,$ and

$z=z(a_0,\\ldots,a_{n+1})$ is a function of the complex variables

$a_0, \\ldots, a_{n+1}.$ Solutions to such equations are

classically known to satisfy holonomic systems of linear partial

differential equations with polynomial coefficients. In the talk

I will investigate one of such systems of differential equations which

was introduced by Mellin. We compute the holonomic rank of the

Mellin system as well as the dimension of the space of its

algebraic solutions. Moreover, we construct explicit bases of

solutions in terms of the roots of initial algebraic equation and their

logarithms. We show that the monodromy of the Mellin system is

always reducible and give some factorization results in the

univariate case.

I will present a joint work with Alicia Dickenstein.

We consider algebraic functions $z$ satisfying equations of the

form

\\begin{equation}

a_0 z^m + a_1z^{m_1} + a_2 z^{m_2} + \\ldots + a_n z^{m_n} +

a_{n+1} =0.

\\end{equation}

Here $m > m_1 > \\ldots > m_n>0,$ $m,m_i \\in \\N,$ and

$z=z(a_0,\\ldots,a_{n+1})$ is a function of the complex variables

$a_0, \\ldots, a_{n+1}.$ Solutions to such equations are

classically known to satisfy holonomic systems of linear partial

differential equations with polynomial coefficients. In the talk

I will investigate one of such systems of differential equations which

was introduced by Mellin. We compute the holonomic rank of the

Mellin system as well as the dimension of the space of its

algebraic solutions. Moreover, we construct explicit bases of

solutions in terms of the roots of initial algebraic equation and their

logarithms. We show that the monodromy of the Mellin system is

always reducible and give some factorization results in the

univariate case.

### 2010/01/29

#### Colloquium

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

Extension of Functions and Interpolation of Data

**Charles Fefferman**(Princeton University)Extension of Functions and Interpolation of Data

[ Abstract ]

Let $f$ be a given real-valued function defined on a subset of $\\mathbb{R}^n$. We explain how to decide whether $f$ extends to a function $F$ in $C^m(\\mathbb{R}^n)$. If such an $F$ exists, we show how to construct one.

Let $f$ be a given real-valued function defined on a subset of $\\mathbb{R}^n$. We explain how to decide whether $f$ extends to a function $F$ in $C^m(\\mathbb{R}^n)$. If such an $F$ exists, we show how to construct one.

### 2010/01/28

#### Applied Analysis

16:00-17:30 Room #002 (Graduate School of Math. Sci. Bldg.)

相転移を伴う非圧縮性2相流の線形化問題について

**清水扇丈**(静岡大学理学部)相転移を伴う非圧縮性2相流の線形化問題について

[ Abstract ]

氷が常圧で0度以上になると水になるなどの相転移を伴う非圧縮性2相流に対し,質量保存則, 運動量保存則, エネルギー保存則を界面を含む系全体に適用し, 線形化した方程式系について考察する. 本講演では, 線形化方程式系のL_p-L_q 最大正則性定理について述べる.

密度が異なる場合は, 法線方向の高さ関数は表面張力つき2相Stokes問題の高さ関数と同じ正則性をもち, 系は流速が支配するのに対し,密度が等しい場合は, Gibbs-Thomson補正された表面張力つき2相Stefan問題の高さ関数と同じ正則性をもち, 系は温度が支配する.

氷が常圧で0度以上になると水になるなどの相転移を伴う非圧縮性2相流に対し,質量保存則, 運動量保存則, エネルギー保存則を界面を含む系全体に適用し, 線形化した方程式系について考察する. 本講演では, 線形化方程式系のL_p-L_q 最大正則性定理について述べる.

密度が異なる場合は, 法線方向の高さ関数は表面張力つき2相Stokes問題の高さ関数と同じ正則性をもち, 系は流速が支配するのに対し,密度が等しい場合は, Gibbs-Thomson補正された表面張力つき2相Stefan問題の高さ関数と同じ正則性をもち, 系は温度が支配する.

#### Lectures

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

Partial differential equations in Finance I

**Olivier Alvarez**(Head of quantitative research, IRFX options Asia, BNP Paribas)Partial differential equations in Finance I

[ Abstract ]

1. Markov processes and Partial differential equations (PDE)

- Markov processes, stochastic differential equations and infinitesimal generator

- The Feynman Kac formula and the backward Kolmogorov equation

- The maximum principle

- Exit time problems and Dirichlet boundary conditions

- Optimal time problems and obstacle problems

2. Application to the pricing of exotic options

- The model equation

- The Black-Scholes equation : absence of arbitrage and dynamical hedging

- Recovering the Black-Scholes formula

- Pricing exotic options : Knock-out / knock-in, american, Asian, lookback

- Overview of affine models and semi-closed formulae

- Heston model : valuing European options

- The Hull White model for IR exotics : valuing zero-coupons, caplets and swaptions.

3. Finite difference methods in Finance

- Basic concepts for numerical schemes : consistency, stability, accuracy and

convergence; the Lax equivalence theorem

- Finite difference methods in dimension 1 : Explicit, implicit, Crank-Nicholson methods for the heat equation : overview, accuracy and convergence

Incorporating first-order derivatives : upwind derivative, stability

- Finite difference methods in dimension 2 : presentation of various schemes :explicit, implicit, alternating direction implicit (ADI), Hopscotch method

- Solving high dimensional linear systems :

LU decomposition, iterative methods

- Finite difference and Monte Carlo methods

4. Optimal control in finance

- Introduction to optimal control

- The dynamic programming principle and the Hamilton-Jacobi-Bellman equation

- The verification theorem and the determination of the optimal control policy

- Utility maximization and Merton's problem

- Pricing with uncertain parameters

- Pricing with transaction costs

- Finite difference methods for optimal control

1. Markov processes and Partial differential equations (PDE)

- Markov processes, stochastic differential equations and infinitesimal generator

- The Feynman Kac formula and the backward Kolmogorov equation

- The maximum principle

- Exit time problems and Dirichlet boundary conditions

- Optimal time problems and obstacle problems

2. Application to the pricing of exotic options

- The model equation

- The Black-Scholes equation : absence of arbitrage and dynamical hedging

- Recovering the Black-Scholes formula

- Pricing exotic options : Knock-out / knock-in, american, Asian, lookback

- Overview of affine models and semi-closed formulae

- Heston model : valuing European options

- The Hull White model for IR exotics : valuing zero-coupons, caplets and swaptions.

3. Finite difference methods in Finance

- Basic concepts for numerical schemes : consistency, stability, accuracy and

convergence; the Lax equivalence theorem

- Finite difference methods in dimension 1 : Explicit, implicit, Crank-Nicholson methods for the heat equation : overview, accuracy and convergence

Incorporating first-order derivatives : upwind derivative, stability

- Finite difference methods in dimension 2 : presentation of various schemes :explicit, implicit, alternating direction implicit (ADI), Hopscotch method

- Solving high dimensional linear systems :

LU decomposition, iterative methods

- Finite difference and Monte Carlo methods

4. Optimal control in finance

- Introduction to optimal control

- The dynamic programming principle and the Hamilton-Jacobi-Bellman equation

- The verification theorem and the determination of the optimal control policy

- Utility maximization and Merton's problem

- Pricing with uncertain parameters

- Pricing with transaction costs

- Finite difference methods for optimal control

#### Lectures

13:00-14:10 Room #122 (Graduate School of Math. Sci. Bldg.)

Partial differential equations in Finance II

**Olivier Alvarez**(Head of quantitative research, IRFX options Asia, BNP Paribas)Partial differential equations in Finance II

[ Abstract ]

1. Markov processes and Partial differential equations (PDE)

- Markov processes, stochastic differential equations and infinitesimal generator

- The Feynman Kac formula and the backward Kolmogorov equation

- The maximum principle

- Exit time problems and Dirichlet boundary conditions

- Optimal time problems and obstacle problems

2. Application to the pricing of exotic options

- The model equation

- The Black-Scholes equation : absence of arbitrage and dynamical hedging

- Recovering the Black-Scholes formula

- Pricing exotic options : Knock-out / knock-in, american, Asian, lookback

- Overview of affine models and semi-closed formulae

- Heston model : valuing European options

- The Hull White model for IR exotics : valuing zero-coupons, caplets and swaptions.

3. Finite difference methods in Finance

- Basic concepts for numerical schemes : consistency, stability, accuracy and

convergence; the Lax equivalence theorem

- Finite difference methods in dimension 1 : Explicit, implicit, Crank-Nicholson methods for the heat equation : overview, accuracy and convergence

Incorporating first-order derivatives : upwind derivative, stability

- Finite difference methods in dimension 2 : presentation of various schemes :explicit, implicit, alternating direction implicit (ADI), Hopscotch method

- Solving high dimensional linear systems :

LU decomposition, iterative methods

- Finite difference and Monte Carlo methods

4. Optimal control in finance

- Introduction to optimal control

- The dynamic programming principle and the Hamilton-Jacobi-Bellman equation

- The verification theorem and the determination of the optimal control policy

- Utility maximization and Merton's problem

- Pricing with uncertain parameters

- Pricing with transaction costs

- Finite difference methods for optimal control

1. Markov processes and Partial differential equations (PDE)

- Markov processes, stochastic differential equations and infinitesimal generator

- The Feynman Kac formula and the backward Kolmogorov equation

- The maximum principle

- Exit time problems and Dirichlet boundary conditions

- Optimal time problems and obstacle problems

2. Application to the pricing of exotic options

- The model equation

- The Black-Scholes equation : absence of arbitrage and dynamical hedging

- Recovering the Black-Scholes formula

- Pricing exotic options : Knock-out / knock-in, american, Asian, lookback

- Overview of affine models and semi-closed formulae

- Heston model : valuing European options

- The Hull White model for IR exotics : valuing zero-coupons, caplets and swaptions.

3. Finite difference methods in Finance

- Basic concepts for numerical schemes : consistency, stability, accuracy and

convergence; the Lax equivalence theorem

- Finite difference methods in dimension 1 : Explicit, implicit, Crank-Nicholson methods for the heat equation : overview, accuracy and convergence

Incorporating first-order derivatives : upwind derivative, stability

- Finite difference methods in dimension 2 : presentation of various schemes :explicit, implicit, alternating direction implicit (ADI), Hopscotch method

- Solving high dimensional linear systems :

LU decomposition, iterative methods

- Finite difference and Monte Carlo methods

4. Optimal control in finance

- Introduction to optimal control

- The dynamic programming principle and the Hamilton-Jacobi-Bellman equation

- The verification theorem and the determination of the optimal control policy

- Utility maximization and Merton's problem

- Pricing with uncertain parameters

- Pricing with transaction costs

- Finite difference methods for optimal control

#### GCOE lecture series

16:30-17:30 Room #999 (Graduate School of Math. Sci. Bldg.)

Extension of Functions and Interpolation of Data

**Charles Fefferman**(Princeton University)Extension of Functions and Interpolation of Data

[ Abstract ]

This series of three lectures will discuss the following questions. No special background will be assumed, and the third lecture will not assume familiarity with the first two.

Fix positive integers $m, n$. Let $f$ be a real-valued function on a subset $E$ of $\\mathbf{R}^n$. How can we tell whether $f$ extends to a $C^m$ function $F$ on the whole $\\mathbf{R}^n$?

If $F$ exists, how small can we take its $C^m$ norm? Can we take $F$ to depend linearly on $f$? What can we say about the derivatives of $F$ at a given point of $E$?

Suppose $E$ is finite. Can we then compute an $F$ with $C^m$ norm close to least-possible? How many operations does it take? What if we ask merely that $F$ and $f$ agree approximately on $E$? What if we are allowed to delete a few points of $E$?

What can be said about the above problems for function spaces other than $C^m(\\mathbf{R}^n)$?

This series of three lectures will discuss the following questions. No special background will be assumed, and the third lecture will not assume familiarity with the first two.

Fix positive integers $m, n$. Let $f$ be a real-valued function on a subset $E$ of $\\mathbf{R}^n$. How can we tell whether $f$ extends to a $C^m$ function $F$ on the whole $\\mathbf{R}^n$?

If $F$ exists, how small can we take its $C^m$ norm? Can we take $F$ to depend linearly on $f$? What can we say about the derivatives of $F$ at a given point of $E$?

Suppose $E$ is finite. Can we then compute an $F$ with $C^m$ norm close to least-possible? How many operations does it take? What if we ask merely that $F$ and $f$ agree approximately on $E$? What if we are allowed to delete a few points of $E$?

What can be said about the above problems for function spaces other than $C^m(\\mathbf{R}^n)$?

### 2010/01/27

#### GCOE lecture series

14:40-16:10 Room #002 (Graduate School of Math. Sci. Bldg.)

Extension of Functions and Interpolation of Data

**Charles Fefferman**(Princeton University)Extension of Functions and Interpolation of Data

[ Abstract ]

This series of three lectures will discuss the following questions. No special background will be assumed, and the third lecture will not assume familiarity with the first two.

Fix positive integers $m, n$. Let $f$ be a real-valued function on a subset $E$ of $\\mathbf{R}^n$. How can we tell whether $f$ extends to a $C^m$ function $F$ on the whole $\\mathbf{R}^n$?

If $F$ exists, how small can we take its $C^m$ norm? Can we take $F$ to depend linearly on $f$? What can we say about the derivatives of $F$ at a given point of $E$?

Suppose $E$ is finite. Can we then compute an $F$ with $C^m$ norm close to least-possible? How many operations does it take? What if we ask merely that $F$ and $f$ agree approximately on $E$? What if we are allowed to delete a few points of $E$?

What can be said about the above problems for function spaces other than $C^m(\\mathbf{R}^n)$?

This series of three lectures will discuss the following questions. No special background will be assumed, and the third lecture will not assume familiarity with the first two.

Fix positive integers $m, n$. Let $f$ be a real-valued function on a subset $E$ of $\\mathbf{R}^n$. How can we tell whether $f$ extends to a $C^m$ function $F$ on the whole $\\mathbf{R}^n$?

If $F$ exists, how small can we take its $C^m$ norm? Can we take $F$ to depend linearly on $f$? What can we say about the derivatives of $F$ at a given point of $E$?

Suppose $E$ is finite. Can we then compute an $F$ with $C^m$ norm close to least-possible? How many operations does it take? What if we ask merely that $F$ and $f$ agree approximately on $E$? What if we are allowed to delete a few points of $E$?

What can be said about the above problems for function spaces other than $C^m(\\mathbf{R}^n)$?

### 2010/01/26

#### Tuesday Seminar of Analysis

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

Finite gap Jacobi matrices (joint work with Barry Simon and Maxim Zinchenko)

**Jacob S. Christiansen**(コペンハーゲン大学)Finite gap Jacobi matrices (joint work with Barry Simon and Maxim Zinchenko)

#### Tuesday Seminar on Topology

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

On the (co)chain type levels of spaces

**栗林 勝彦**(信州大学)On the (co)chain type levels of spaces

[ Abstract ]

Avramov, Buchweitz, Iyengar and Miller have introduced

the notion of the level for an object of a triangulated category.

The invariant measures the number of steps to build the given object

out of some fixed object with triangles.

Using this notion in the derived category of modules over a (co)chain

algebra,

we define a new topological invariant, which is called

the (co)chain type level of a space.

In this talk, after explaining fundamental properties of the invariant,

I describe the chain type level of the Borel construction

of a homogeneous space as a computational example.

I will also relate the chain type level of a space to algebraic

approximations of the L.-S. category due to Kahl and to

the original L.-S. category of a map.

Avramov, Buchweitz, Iyengar and Miller have introduced

the notion of the level for an object of a triangulated category.

The invariant measures the number of steps to build the given object

out of some fixed object with triangles.

Using this notion in the derived category of modules over a (co)chain

algebra,

we define a new topological invariant, which is called

the (co)chain type level of a space.

In this talk, after explaining fundamental properties of the invariant,

I describe the chain type level of the Borel construction

of a homogeneous space as a computational example.

I will also relate the chain type level of a space to algebraic

approximations of the L.-S. category due to Kahl and to

the original L.-S. category of a map.

#### Lectures

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

Fractional Evolution Equations and Applications 5

**伊東一文**(大学院数理科学研究科)Fractional Evolution Equations and Applications 5

[ Abstract ]

In recent years increasing interests and considerable

researches have been given to the fractional differential equations both

in time and space variables.

These are due to the applications of the fractional calculus

to problems in a wide areas of physics and engineering science and a rapid

development of the corresponding theory. A motivating example includes

the so-called continuous time random walk process

and the Levy process model for the mathematical finance.

In this lecture we develop solution techniques based on the linear and

nonlinear semigroup theory and apply it to solve the associated inverse

and optimal control problems. The property and stability of the solutions

as well as numerical integration methods

are discussed. The lecture also covers the basis and application of the

so-called Crandall-Ligget theory and the locally quasi-dissipative

operator method developed by Kobayashi-Kobayashi-Oharu.

In recent years increasing interests and considerable

researches have been given to the fractional differential equations both

in time and space variables.

These are due to the applications of the fractional calculus

to problems in a wide areas of physics and engineering science and a rapid

development of the corresponding theory. A motivating example includes

the so-called continuous time random walk process

and the Levy process model for the mathematical finance.

In this lecture we develop solution techniques based on the linear and

nonlinear semigroup theory and apply it to solve the associated inverse

and optimal control problems. The property and stability of the solutions

as well as numerical integration methods

are discussed. The lecture also covers the basis and application of the

so-called Crandall-Ligget theory and the locally quasi-dissipative

operator method developed by Kobayashi-Kobayashi-Oharu.

### 2010/01/25

#### GCOE lecture series

14:40-16:10 Room #002 (Graduate School of Math. Sci. Bldg.)

Extension of Functions and Interpolation of Data

**Charles Fefferman**(Princeton University)Extension of Functions and Interpolation of Data

[ Abstract ]

This series of three lectures will discuss the following questions. No special background will be assumed, and the third lecture will not assume familiarity with the first two.

Fix positive integers $m, n$. Let $f$ be a real-valued function on a subset $E$ of $\\mathbf{R}^n$. How can we tell whether $f$ extends to a $C^m$ function $F$ on the whole $\\mathbf{R}^n$?

If $F$ exists, how small can we take its $C^m$ norm? Can we take $F$ to depend linearly on $f$? What can we say about the derivatives of $F$ at a given point of $E$?

Suppose $E$ is finite. Can we then compute an $F$ with $C^m$ norm close to least-possible? How many operations does it take? What if we ask merely that $F$ and $f$ agree approximately on $E$? What if we are allowed to delete a few points of $E$?

What can be said about the above problems for function spaces other than $C^m(\\mathbf{R}^n)$?

This series of three lectures will discuss the following questions. No special background will be assumed, and the third lecture will not assume familiarity with the first two.

Fix positive integers $m, n$. Let $f$ be a real-valued function on a subset $E$ of $\\mathbf{R}^n$. How can we tell whether $f$ extends to a $C^m$ function $F$ on the whole $\\mathbf{R}^n$?

If $F$ exists, how small can we take its $C^m$ norm? Can we take $F$ to depend linearly on $f$? What can we say about the derivatives of $F$ at a given point of $E$?

Suppose $E$ is finite. Can we then compute an $F$ with $C^m$ norm close to least-possible? How many operations does it take? What if we ask merely that $F$ and $f$ agree approximately on $E$? What if we are allowed to delete a few points of $E$?

What can be said about the above problems for function spaces other than $C^m(\\mathbf{R}^n)$?

#### Lectures

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

Fractional Evolution Equations and Applications 4

**伊東一文**(大学院数理科学研究科)Fractional Evolution Equations and Applications 4

[ Abstract ]

In recent years increasing interests and considerable

researches have been given to the fractional differential equations both

in time and space variables.

These are due to the applications of the fractional calculus

to problems in a wide areas of physics and engineering science and a rapid

development of the corresponding theory. A motivating example includes

the so-called continuous time random walk process

and the Levy process model for the mathematical finance.

In this lecture we develop solution techniques based on the linear and

nonlinear semigroup theory and apply it to solve the associated inverse

and optimal control problems. The property and stability of the solutions

as well as numerical integration methods

are discussed. The lecture also covers the basis and application of the

so-called Crandall-Ligget theory and the locally quasi-dissipative

operator method developed by Kobayashi-Kobayashi-Oharu.

In recent years increasing interests and considerable

researches have been given to the fractional differential equations both

in time and space variables.

These are due to the applications of the fractional calculus

to problems in a wide areas of physics and engineering science and a rapid

development of the corresponding theory. A motivating example includes

the so-called continuous time random walk process

and the Levy process model for the mathematical finance.

In this lecture we develop solution techniques based on the linear and

nonlinear semigroup theory and apply it to solve the associated inverse

and optimal control problems. The property and stability of the solutions

as well as numerical integration methods

are discussed. The lecture also covers the basis and application of the

so-called Crandall-Ligget theory and the locally quasi-dissipative

operator method developed by Kobayashi-Kobayashi-Oharu.

#### Seminar on Geometric Complex Analysis

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Eta invariant and Selberg Zeta function of odd type over convex co-compact hyperbolic manifolds

**Colin Guillarmou**(Ecole Normale Superieure)Eta invariant and Selberg Zeta function of odd type over convex co-compact hyperbolic manifolds

#### Algebraic Geometry Seminar

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

On weak Fano varieties with log canonical singularities

**權業 善範**(東大数理)On weak Fano varieties with log canonical singularities

[ Abstract ]

We prove that the anti-canonical divisors of weak Fano

3-folds with log canonical singularities are semiample. Moreover, we consider

semiampleness of the anti-log canonical divisor of any weak log Fano pair

with log canonical singularities. We show semiampleness dose not hold in

general by constructing several examples. Based on those examples, we propose

sufficient conditions which seem to be the best possible and we prove

semiampleness under such conditions. In particular we derive semiampleness of the

anti-canonical divisors of log canonical weak Fano 4-folds whose lc centers

are at most 1-dimensional. We also investigate the Kleiman-Mori cones of

weak log Fano pairs with log canonical singularities.

We prove that the anti-canonical divisors of weak Fano

3-folds with log canonical singularities are semiample. Moreover, we consider

semiampleness of the anti-log canonical divisor of any weak log Fano pair

with log canonical singularities. We show semiampleness dose not hold in

general by constructing several examples. Based on those examples, we propose

sufficient conditions which seem to be the best possible and we prove

semiampleness under such conditions. In particular we derive semiampleness of the

anti-canonical divisors of log canonical weak Fano 4-folds whose lc centers

are at most 1-dimensional. We also investigate the Kleiman-Mori cones of

weak log Fano pairs with log canonical singularities.

### 2010/01/22

#### Lecture Series on Mathematical Sciences in Soceity

16:20-17:50 Room #117 (Graduate School of Math. Sci. Bldg.)

数学者と企業研究者との連携

**中川淳一**(新日本製鐵(株)技術開発本部)数学者と企業研究者との連携

#### Lectures

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

Fractional Evolution Equations and Applications 3

**伊東一文**(大学院数理科学研究科)Fractional Evolution Equations and Applications 3

[ Abstract ]

In recent years increasing interests and considerable

researches have been given to the fractional differential equations both

in time and space variables.

These are due to the applications of the fractional calculus

to problems in a wide areas of physics and engineering science and a rapid

development of the corresponding theory. A motivating example includes

the so-called continuous time random walk process

and the Levy process model for the mathematical finance.

In this lecture we develop solution techniques based on the linear and

nonlinear semigroup theory and apply it to solve the associated inverse

and optimal control problems. The property and stability of the solutions

as well as numerical integration methods

are discussed. The lecture also covers the basis and application of the

so-called Crandall-Ligget theory and the locally quasi-dissipative

operator method developed by Kobayashi-Kobayashi-Oharu.

Nonlinear evolution equations, Crandall-Ligget theory,

Locally quasi-dissipative operators approach

In recent years increasing interests and considerable

researches have been given to the fractional differential equations both

in time and space variables.

These are due to the applications of the fractional calculus

to problems in a wide areas of physics and engineering science and a rapid

development of the corresponding theory. A motivating example includes

the so-called continuous time random walk process

and the Levy process model for the mathematical finance.

In this lecture we develop solution techniques based on the linear and

nonlinear semigroup theory and apply it to solve the associated inverse

and optimal control problems. The property and stability of the solutions

as well as numerical integration methods

are discussed. The lecture also covers the basis and application of the

so-called Crandall-Ligget theory and the locally quasi-dissipative

operator method developed by Kobayashi-Kobayashi-Oharu.

Nonlinear evolution equations, Crandall-Ligget theory,

Locally quasi-dissipative operators approach

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