## 過去の記録

#### 講演会

10:40-12:10   数理科学研究科棟(駒場) 123号室
Olivier Alvarez 氏 (Head of quantitative research, IRFX options Asia, BNP Paribas)
Partial differential equations in Finance I
[ 講演概要 ]
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

#### 講演会

13:00-14:10   数理科学研究科棟(駒場) 122号室
Olivier Alvarez 氏 (Head of quantitative research, IRFX options Asia, BNP Paribas)
Partial differential equations in Finance II
[ 講演概要 ]
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レクチャーズ

16:30-17:30   数理科学研究科棟(駒場) 999号室
Charles Fefferman 氏 (Princeton University)
Extension of Functions and Interpolation of Data
[ 講演概要 ]
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年01月27日(水)

#### GCOEレクチャーズ

14:40-16:10   数理科学研究科棟(駒場) 002号室
Charles Fefferman 氏 (Princeton University)
Extension of Functions and Interpolation of Data
[ 講演概要 ]
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年01月26日(火)

#### 解析学火曜セミナー

16:30-18:00   数理科学研究科棟(駒場) 128号室
Jacob S. Christiansen 氏 (コペンハーゲン大学)
Finite gap Jacobi matrices (joint work with Barry Simon and Maxim Zinchenko)

#### トポロジー火曜セミナー

17:00-18:00   数理科学研究科棟(駒場) 056号室
Tea: 16:40 - 17:00 コモンルーム

On the (co)chain type levels of spaces
[ 講演概要 ]
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.

#### 講演会

16:30-18:00   数理科学研究科棟(駒場) 118号室

Fractional Evolution Equations and Applications 5
[ 講演概要 ]
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年01月25日(月)

#### GCOEレクチャーズ

14:40-16:10   数理科学研究科棟(駒場) 002号室
Charles Fefferman 氏 (Princeton University)
Extension of Functions and Interpolation of Data
[ 講演概要 ]
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 $C^m(\\mathbf{R}^n)$?

#### 講演会

16:30-18:00   数理科学研究科棟(駒場) 056号室

Fractional Evolution Equations and Applications 4
[ 講演概要 ]
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.

#### 複素解析幾何セミナー

10:30-12:00   数理科学研究科棟(駒場) 128号室
Colin Guillarmou 氏 (Ecole Normale Superieure)
Eta invariant and Selberg Zeta function of odd type over convex co-compact hyperbolic manifolds

#### 代数幾何学セミナー

16:40-18:10   数理科学研究科棟(駒場) 126号室

On weak Fano varieties with log canonical singularities
[ 講演概要 ]
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年01月22日(金)

#### GCOE社会数理講演シリーズ

16:20-17:50   数理科学研究科棟(駒場) 117号室

#### 講演会

16:30-18:00   数理科学研究科棟(駒場) 118号室

Fractional Evolution Equations and Applications 3
[ 講演概要 ]
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年01月21日(木)

#### 作用素環セミナー

16:30-18:00   数理科学研究科棟(駒場) 128号室

On Subfactors Arising from Asymptotic Representations of Symmetric Groups

#### 応用解析セミナー

16:00-17:30   数理科学研究科棟(駒場) 122号室
Danielle Hilhorst 氏 (パリ南大学 / CNRS)
A finite volume method on general meshes for a degenerate parabolic convection-reaction-diffusion equation
[ 講演概要 ]
We propose a finite volume method on general meshes for degenerate parabolic convection-reaction-diffusion equations. Such equations arise for instance in the modeling of contaminant transport in groundwater. After giving a convergence proof, we present the results of numerical tests.

### 2010年01月20日(水)

#### 東京幾何セミナー

17:00-18:30   数理科学研究科棟(駒場) 122号室

「今後の予定」欄には、東工大で行われるセミナーは表��

Craig Van Coevering 氏 (MIT)
Asymptotically conical manifolds and the Monge-Ampere equation
[ 講演概要 ]
Some analysis is considered on manifolds with a conical end. Then we show that in the Kahler case the complex Monge-Ampere equation can be solved with the same regularity as is known in the ALE case. By considering resolutions of toric singularities and hypersurface singularities this can easily be used to produce many Calabi-Yau manifolds with a conical end.

#### 講演会

16:30-18:00   数理科学研究科棟(駒場) 118号室

Fractional Evolution Equations and Applications 2
[ 講演概要 ]
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.

#### 数理人口学・数理生物学セミナー

14:40-16:10   数理科学研究科棟(駒場) 052号室

[ 講演概要 ]

はない.そこで本研究では,individual based modelに東京都市圏パーソント
リップ調査を組み合わせることにより感染拡大モデルを構築し,数値シミュレー
ションによって外出規制および施設閉鎖の効果を検討した.外出規制に関して
は,規制日数が12日以上と長い場合には効果が大きいことがわかった.また,施

わらないものの,累積罹患率は低下することがわかった.

### 2010年01月19日(火)

#### 解析学火曜セミナー

16:30-18:00   数理科学研究科棟(駒場) 128号室

#### 作用素環セミナー

16:30-18:00   数理科学研究科棟(駒場) 126号室

Entire Cyclic Cohomology of Noncommutative Spheres

#### トポロジー火曜セミナー

17:00-18:00   数理科学研究科棟(駒場) 056号室
Tea: 16:40 - 17:00 コモンルーム

Localization via group action and its application to
the period condition of algebraic minimal surfaces
[ 講演概要 ]
The optimal estimate for the number of exceptional
values of the Gauss map of algebraic minimal surfaces is a long
standing problem. In this lecture, I will introduce new ideas
toward the solution of this problem. The collective Cohn-Vossen
inequality" is the key idea. From this we have effective
Nevanlinna's lemma on logarithmic derivative for a certain class
of meromorphic functions on the disk. On the other hand, we can
construct a family holomorphic functions on the disk from the
Weierstrass data of the algebraic minimal surface under
consideration, which encodes the period condition.
Applying effective Lemma on logarithmic derivative to these
functions, we can extract an intriguing inequality.

#### 講演会

16:30-18:00   数理科学研究科棟(駒場) 118号室

Fractional Evolution Equations and Applications 1
[ 講演概要 ]
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年01月18日(月)

#### 複素解析幾何セミナー

10:30-12:00   数理科学研究科棟(駒場) 128号室

スプライス商特異点について

#### 代数幾何学セミナー

16:40-18:10   数理科学研究科棟(駒場) 126号室
Anne-Sophie Kaloghiros 氏 (RIMS)
The divisor class group of terminal Gorenstein Fano 3-folds and rationality questions
[ 講演概要 ]
Let Y be a quartic hypersurface in CP^4 with mild singularities, e.g. no worse than ordinary double points.
If Y contains a surface that is not a hyperplane section, Y is not Q-factorial and the divisor class group of Y, Cl Y, contains divisors that are not Cartier. However, the rank of Cl Y is bounded.

In this talk, I will show that in most cases, it is possible to describe explicitly the divisor class group Cl Y by running a Minimal Model Program (MMP) on X, a small Q-factorialisation of Y. In this case, the generators of Cl Y/ Pic Y are topological traces " of K-negative extremal contractions on X.
This has surprising consequences: it is possible to conclude that a number of families of non-factorial quartic 3-folds are rational.
In particular, I give some examples of rational quartic hypersurfaces Y_4\\subset CP^4 with rk Cl Y=2 and show that when the divisor class group of Y has sufficiently high rank, Y is always rational.

### 2010年01月15日(金)

#### GCOE社会数理講演シリーズ

16:20-17:50   数理科学研究科棟(駒場) 117号室