## Lectures

Seminar information archive ～09/15｜Next seminar｜Future seminars 09/16～

**Seminar information archive**

### 2010/03/17

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

方向依存性を持つ長距離パーコレーションの臨界曲線

**三角 淳**(東大数理)方向依存性を持つ長距離パーコレーションの臨界曲線

### 2010/02/24

15:00-16:30 Room #370 (Graduate School of Math. Sci. Bldg.)

Protein Moduli Space

**Robert Penner**(Aarhus University / University of Southern California)Protein Moduli Space

[ Abstract ]

Recent joint works with J. E. Andersen and others

provide explicit discrete and continuous models

of protein geometry. These models are inspired

by corresponding constructions in the study of moduli

spaces of flat G-connections on surfaces, in particular,

for G=PSL(2,R) and G=SO(3). These models can be used

for protein classification as well as for folding prediction,

and computer experiments towards these ends will

be discussed.

Recent joint works with J. E. Andersen and others

provide explicit discrete and continuous models

of protein geometry. These models are inspired

by corresponding constructions in the study of moduli

spaces of flat G-connections on surfaces, in particular,

for G=PSL(2,R) and G=SO(3). These models can be used

for protein classification as well as for folding prediction,

and computer experiments towards these ends will

be discussed.

### 2010/02/23

14:00-15:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Homogenization Limit and Singular Limit of the Allen-Cahn equation

**Bendong LOU**(同済大学)Homogenization Limit and Singular Limit of the Allen-Cahn equation

[ Abstract ]

We consider the Allen-Cahn equation in a cylinder with periodic undulating boundaries in the plane. Our problem involves two small parameters $\\delta$ and $\\epsilon$, where $\\delta$ appears in the equation to denote the scale of the singular limit and $\\epsilon$ appears in the boundary conditions to denote the scale of the homogenization limit. We consider the following two limiting processes:

(I): taking homogenization limit first and then taking singular limit;

(II) taking singular limit first and then taking homogenization limit.

We formally show that they both result in the same mean curvature flow equation, but with different boundary conditions.

We consider the Allen-Cahn equation in a cylinder with periodic undulating boundaries in the plane. Our problem involves two small parameters $\\delta$ and $\\epsilon$, where $\\delta$ appears in the equation to denote the scale of the singular limit and $\\epsilon$ appears in the boundary conditions to denote the scale of the homogenization limit. We consider the following two limiting processes:

(I): taking homogenization limit first and then taking singular limit;

(II) taking singular limit first and then taking homogenization limit.

We formally show that they both result in the same mean curvature flow equation, but with different boundary conditions.

### 2010/01/28

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

### 2010/01/28

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

### 2010/01/26

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

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.

### 2010/01/22

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

### 2010/01/20

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

Fractional Evolution Equations and Applications 2

**伊東一文**(大学院数理科学研究科)Fractional Evolution Equations and Applications 2

[ 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.

Existence and Uniqueness by C_0 semigroup theory, dissipative linear

operator

and Hille-Yoshida, Trotter-Kato theory.

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.

Existence and Uniqueness by C_0 semigroup theory, dissipative linear

operator

and Hille-Yoshida, Trotter-Kato theory.

### 2010/01/19

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

Fractional Evolution Equations and Applications 1

**伊東一文**(大学院数理科学研究科)Fractional Evolution Equations and Applications 1

[ 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.

Motivation: Continuous time random walk (CTRW) process

Fractional differential equations in time and Mittag-Leffler functions

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.

Motivation: Continuous time random walk (CTRW) process

Fractional differential equations in time and Mittag-Leffler functions

### 2010/01/13

16:45-17:45 Room #128 (Graduate School of Math. Sci. Bldg.)

Scaled limit for the largest eigenvalue from the generalized Cauchy ensemble

**Felix Rubin**(Zurich 大学)Scaled limit for the largest eigenvalue from the generalized Cauchy ensemble

### 2010/01/13

15:30-16:30 Room #128 (Graduate School of Math. Sci. Bldg.)

Breaking the chain: slow versus fast pulling

**Michael Allman**(Warwick 大学)Breaking the chain: slow versus fast pulling

### 2009/12/25

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

A criterion for the strong solvability of the mixed Cauchy problem for the Laplace equation

**Academician T. Sh. Kalmenov**(Research Centre of Physics and Mathematics Almaty, Kazakhstan)A criterion for the strong solvability of the mixed Cauchy problem for the Laplace equation

### 2009/12/10

10:40-12:10 Room #128 (Graduate School of Math. Sci. Bldg.)

冨田竹崎理論とその応用 (9)

**竹崎正道**(UCLA)冨田竹崎理論とその応用 (9)

### 2009/12/09

14:40-16:10 Room #128 (Graduate School of Math. Sci. Bldg.)

冨田竹崎理論とその応用 (8)

**竹崎正道**(UCLA)冨田竹崎理論とその応用 (8)

### 2009/12/08

14:40-16:10 Room #128 (Graduate School of Math. Sci. Bldg.)

冨田竹崎理論とその応用 (7)

**竹崎正道**(UCLA)冨田竹崎理論とその応用 (7)

### 2009/11/19

15:00-16:00 Room #123 (Graduate School of Math. Sci. Bldg.)

数論的D加群の特性サイクルと分岐理論

**阿部知行**(東京大学大学院数理科学研究科)数論的D加群の特性サイクルと分岐理論

### 2009/11/18

15:30-17:00 Room #128 (Graduate School of Math. Sci. Bldg.)

The stochastic Burgers equation and its discretization

**Herbert Spohn**(ミュンヘン工科大学・九州大学)The stochastic Burgers equation and its discretization

### 2009/11/12

10:40-12:10 Room #128 (Graduate School of Math. Sci. Bldg.)

冨田竹崎理論とその応用 (6)

**竹崎正道**(UCLA)冨田竹崎理論とその応用 (6)

### 2009/11/11

14:40-16:10 Room #128 (Graduate School of Math. Sci. Bldg.)

冨田竹崎理論とその応用 (5)

**竹崎正道**(UCLA)冨田竹崎理論とその応用 (5)

### 2009/11/10

14:40-16:10 Room #128 (Graduate School of Math. Sci. Bldg.)

冨田竹崎理論とその応用 (4)

**竹崎正道**(UCLA)冨田竹崎理論とその応用 (4)

### 2009/10/29

16:30-17:30 Room #270 (Graduate School of Math. Sci. Bldg.)

Spectral properties of Nikolaevskiy chaos

**Michael I. Tribelsky**(MIREA (Technical University), Moscow, Russia)Spectral properties of Nikolaevskiy chaos

### 2009/10/28

16:30-17:30 Room #370 (Graduate School of Math. Sci. Bldg.)

Dispersive and Strichartz estimates for hyperbolic equations of general form

**Michael Ruzhansky**(Imperial College, London)Dispersive and Strichartz estimates for hyperbolic equations of general form

### 2009/10/22

10:40-12:10 Room #128 (Graduate School of Math. Sci. Bldg.)

冨田竹崎理論とその応用 (3)

**竹崎正道**(UCLA)冨田竹崎理論とその応用 (3)

### 2009/10/21

14:40-16:10 Room #128 (Graduate School of Math. Sci. Bldg.)

冨田竹崎理論とその応用 (2)

**竹崎正道**(UCLA)冨田竹崎理論とその応用 (2)