## Seminar information archive

Seminar information archive ～02/18｜Today's seminar 02/19 | Future seminars 02/20～

### 2008/03/19

#### PDE Real Analysis Seminar

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

Maximal Regularity for Mixed Order Systems

**Juergen Saal**(Department of Mathematics and Statistics, University of Konstanz)Maximal Regularity for Mixed Order Systems

[ Abstract ]

In classical boundary value problems the related symbols are homogeneous in space and time. This allows for the application of a standard compactness argument in order to obtain the important maximal regularity. However, quasilinear systems arising e.g. from free boundary problems are in general of mixed order. In other words the related symbols are of intricate structure and in particular highly inhomogeneous. Therefore, the standard compactness argument fails. The purpose of this talk is to introduce the Newton polygon method, which gives a systematic approach to such mixed order systems and to demonstrate its strength by applications to the Stefan problem and a free boundary problem for the Navier-Stokes equations.

In classical boundary value problems the related symbols are homogeneous in space and time. This allows for the application of a standard compactness argument in order to obtain the important maximal regularity. However, quasilinear systems arising e.g. from free boundary problems are in general of mixed order. In other words the related symbols are of intricate structure and in particular highly inhomogeneous. Therefore, the standard compactness argument fails. The purpose of this talk is to introduce the Newton polygon method, which gives a systematic approach to such mixed order systems and to demonstrate its strength by applications to the Stefan problem and a free boundary problem for the Navier-Stokes equations.

### 2008/03/14

#### Algebraic Geometry Seminar

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

Understanding singular algebraic varieties via string theory

**David Morrison**(UC Santa Barbara)Understanding singular algebraic varieties via string theory

[ Abstract ]

String theory has helped to formulate two major new insights in the study of singular algebraic varieties. The first -- which also arose from symplectic geometry -- is that families of Kaehler metrics are an important tool in uncovering the structure of singular algebraic varieties. The second, more recent insight -- related to independent work in the representation theory of associative algebras -- is that one's understanding of a singular (affine) algebraic variety is enhanced if one can find a non-commutative ring whose center is the coordinate ring of the variety. We will describe both of these insights, and explain how they are related to string theory.

String theory has helped to formulate two major new insights in the study of singular algebraic varieties. The first -- which also arose from symplectic geometry -- is that families of Kaehler metrics are an important tool in uncovering the structure of singular algebraic varieties. The second, more recent insight -- related to independent work in the representation theory of associative algebras -- is that one's understanding of a singular (affine) algebraic variety is enhanced if one can find a non-commutative ring whose center is the coordinate ring of the variety. We will describe both of these insights, and explain how they are related to string theory.

### 2008/03/13

#### Tuesday Seminar of Analysis

15:00-17:30 Room #117 (Graduate School of Math. Sci. Bldg.)

Schr/"odinger equations on scattering manifolds and microlocal singularities

Local smoothing in the presence of lots of trapping

[ Reference URL ]

http://agusta.ms.u-tokyo.ac.jp/seminerphotos2/Zworski-abstract.pdf

**伊藤健一**(東京大学大学院数理科学研究科) 15:00-16:00Schr/"odinger equations on scattering manifolds and microlocal singularities

**Maciej ZWORSKI**(カリフォルニア大学バークレイ校) 16:30-17:30Local smoothing in the presence of lots of trapping

[ Reference URL ]

http://agusta.ms.u-tokyo.ac.jp/seminerphotos2/Zworski-abstract.pdf

### 2008/02/23

#### Infinite Analysis Seminar Tokyo

13:00-16:30 Room #270 (Graduate School of Math. Sci. Bldg.)

Solutions of hungry periodic discrete Toda equation and its ultradiscretization

A tropical analogue of Fay's trisecant identity and its application to the ultra-discrete periodic Toda equation.

**岩尾慎介**(東大数理) 13:00-14:30Solutions of hungry periodic discrete Toda equation and its ultradiscretization

[ Abstract ]

The hungry discrete Toda equation is a generalization of the discrete Toda

equation. Through the method of ultradiscretization, the generalized

Box-ball system (gBBS) with finitely many kinds of balls is obtained from

hungry discrete Toda eq.. It is to be expected that the general solution of

gBBS should be obtained from the solution of hungry discrete Toda eq.

through ultradiscretization. In this talk, we derive the solutions of hungry

periodic discrete Toda eq. (hpd Toda eq.), by using inverse scattering

method. Although the hpd Toda equation does not linearlized in the usual

sense on the Picard group of the spectral curve, it is possible to determine

its behavior on the Picard group.

The hungry discrete Toda equation is a generalization of the discrete Toda

equation. Through the method of ultradiscretization, the generalized

Box-ball system (gBBS) with finitely many kinds of balls is obtained from

hungry discrete Toda eq.. It is to be expected that the general solution of

gBBS should be obtained from the solution of hungry discrete Toda eq.

through ultradiscretization. In this talk, we derive the solutions of hungry

periodic discrete Toda eq. (hpd Toda eq.), by using inverse scattering

method. Although the hpd Toda equation does not linearlized in the usual

sense on the Picard group of the spectral curve, it is possible to determine

its behavior on the Picard group.

**竹縄知之**(東京海洋大・海洋工) 15:00-16:30A tropical analogue of Fay's trisecant identity and its application to the ultra-discrete periodic Toda equation.

[ Abstract ]

The ultra-discrete Toda equation is essentially equivalent to the integrable

Box and Ball system, and considered to be a fundamental object in

ultra-discrete integrable systems. In this talk, we construct the general

solution of ultra-discrete Toda equation with periodic boundary condition,

by using the tropical theta function and the bilinear form. The tropical

theta function is associated with the tropical curve defined through the Lax

matrix of (not ultra-) discrete periodic Toda equation. For the proof, we

introduce a tropical analogue of Fay's trisecant identity. (This talk is

based on the joint work with R. Inoue.)

The ultra-discrete Toda equation is essentially equivalent to the integrable

Box and Ball system, and considered to be a fundamental object in

ultra-discrete integrable systems. In this talk, we construct the general

solution of ultra-discrete Toda equation with periodic boundary condition,

by using the tropical theta function and the bilinear form. The tropical

theta function is associated with the tropical curve defined through the Lax

matrix of (not ultra-) discrete periodic Toda equation. For the proof, we

introduce a tropical analogue of Fay's trisecant identity. (This talk is

based on the joint work with R. Inoue.)

### 2008/02/20

#### Seminar on Probability and Statistics

16:20-17:30 Room #122 (Graduate School of Math. Sci. Bldg.)

A Test for Cross-sectional Dependence of Microstructure Noises and their Cross-Covariance Estimator

http://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2007/21.html

**大屋 幸輔**(大阪大学大学院経済学研究科)A Test for Cross-sectional Dependence of Microstructure Noises and their Cross-Covariance Estimator

[ Abstract ]

高頻度観測される約定データにもとづく Integrated Volatility や Integrated Covariance の推定量は Bid-Ask Bounce に代表される Market Microstructure Noise の存在により、バイアスをもち、その分散も過大なものになっている。さ まざまな推定量の改良が提案されているが、それらの多くは Microstructure Noise の dependence の構造を既知としたものである。この従属性の構造を明ら かにするために、本報告では直接観測できない Microstructure Noise の相互自 己共分散がゼロであるかどうかを検定する統計量と相互自己共分散関数の推定量 を提案する。

[ Reference URL ]高頻度観測される約定データにもとづく Integrated Volatility や Integrated Covariance の推定量は Bid-Ask Bounce に代表される Market Microstructure Noise の存在により、バイアスをもち、その分散も過大なものになっている。さ まざまな推定量の改良が提案されているが、それらの多くは Microstructure Noise の dependence の構造を既知としたものである。この従属性の構造を明ら かにするために、本報告では直接観測できない Microstructure Noise の相互自 己共分散がゼロであるかどうかを検定する統計量と相互自己共分散関数の推定量 を提案する。

http://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2007/21.html

#### Lectures

13:30-17:45 Room #123 (Graduate School of Math. Sci. Bldg.)

Divergence formulae on the space of continuous functions and Malliavin calculus

Ginibre random point field and a notion of convergence of Dirichlet forms

Stochastic PDEs and infinite dimensional integration by parts formulae

ランダム環境下の確率モデルに関連する問題

(A problem arising in stochastic models in random environments)

**乙部厳己**(信州大理) 13:30-14:00Divergence formulae on the space of continuous functions and Malliavin calculus

**長田博文**(九大数理) 14:15-15:15Ginibre random point field and a notion of convergence of Dirichlet forms

**Lorenzo Zambotti**(パリ第6大学) 15:30-16:30Stochastic PDEs and infinite dimensional integration by parts formulae

**志賀徳造**(東工大理工) 16:45-17:45ランダム環境下の確率モデルに関連する問題

(A problem arising in stochastic models in random environments)

### 2008/02/19

#### Lectures

16:30-17:30 Room #118 (Graduate School of Math. Sci. Bldg.)

An overview on archimedean L-factors for G_1xG_2

**Eric Stade**(Colorado University)An overview on archimedean L-factors for G_1xG_2

[ Abstract ]

When G_1xG_2 is one of pairs GL(n)xGL(n), GL(n)xGL(n+1), GL(n)xSO(2n+1), and GL(n+1)xSO(2n+1), we have evaluation of the archimedian L-factors of automorphic L-functions obtained by Rankin-Selberg convolution.

The last two cases are joint works with Taku Ishii (Chiba Inst. of Tech) which are in progress.

When G_1xG_2 is one of pairs GL(n)xGL(n), GL(n)xGL(n+1), GL(n)xSO(2n+1), and GL(n+1)xSO(2n+1), we have evaluation of the archimedian L-factors of automorphic L-functions obtained by Rankin-Selberg convolution.

The last two cases are joint works with Taku Ishii (Chiba Inst. of Tech) which are in progress.

### 2008/02/13

#### Seminar on Probability and Statistics

16:20-17:30 Room #123 (Graduate School of Math. Sci. Bldg.)

Realized multipower variationの統計推測への応用について

http://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2007/20.html

**増田 弘毅**(九大数理)Realized multipower variationの統計推測への応用について

[ Abstract ]

確率過程からの高頻度データに基づいて定義されるMultipower variation (MPV)は,飛躍に対して頑健な累積ボラティリティ推定量や,飛躍の検出のための統 計量として,近年計量経済において脚光を浴びている.MPVはモデルの複雑さに依ら ずその計算が容易であるため,飛躍付確率過程に関する様々な統計推測問題への適用 が期待される.本報告では特に,最近Lee and Mykland (The Review of Financial Studies, to appear)によって提案された,MPVを介した飛躍時点(微小区間)の検出 手法を,複合ポアソン型飛躍付拡散過程の漸近推測へ応用することを考える.

[ Reference URL ]確率過程からの高頻度データに基づいて定義されるMultipower variation (MPV)は,飛躍に対して頑健な累積ボラティリティ推定量や,飛躍の検出のための統 計量として,近年計量経済において脚光を浴びている.MPVはモデルの複雑さに依ら ずその計算が容易であるため,飛躍付確率過程に関する様々な統計推測問題への適用 が期待される.本報告では特に,最近Lee and Mykland (The Review of Financial Studies, to appear)によって提案された,MPVを介した飛躍時点(微小区間)の検出 手法を,複合ポアソン型飛躍付拡散過程の漸近推測へ応用することを考える.

http://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2007/20.html

### 2008/02/12

#### Kavli IPMU Komaba Seminar

17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

How to lift a construction by Hiroshi Inose to conformal field theory

**Katrin Wendland**(University of Augrburg)How to lift a construction by Hiroshi Inose to conformal field theory

[ Abstract ]

The moduli space of Einstein metrics is well known to algebraic and differential geometers. Physicists have introduced the notion of conformal field theories (CFTs) associated to K3, and the moduli space of these objects is well understood as well. It can be interpreted as a generalisation of the moduli space of Einstein metrics on K3, which allows us to introduce this space without having to use background knowledge from conformal field theory. However, just as no smooth Einstein metrics on K3 are known explicitly, the explicit construction of CFTs associated to K3 in general remains an open problem. The only known constructions which allow to deal with families of CFTs give CFTs associated to K3 surfaces with orbifold singularities.

We use a classical construction by Hiroshi Inose to explicitly construct a family of CFTs which are associated to a family of smooth algebraic K3 surfaces. Although these CFTs were known before, it is remarkable that they allow a description in terms of a family of smooth surfaces whose complex structure is deformed while all other geometric data remain fixed.

We also discuss possible extensions of this result to higher dimensional Calabi-Yau threefolds.

The moduli space of Einstein metrics is well known to algebraic and differential geometers. Physicists have introduced the notion of conformal field theories (CFTs) associated to K3, and the moduli space of these objects is well understood as well. It can be interpreted as a generalisation of the moduli space of Einstein metrics on K3, which allows us to introduce this space without having to use background knowledge from conformal field theory. However, just as no smooth Einstein metrics on K3 are known explicitly, the explicit construction of CFTs associated to K3 in general remains an open problem. The only known constructions which allow to deal with families of CFTs give CFTs associated to K3 surfaces with orbifold singularities.

We use a classical construction by Hiroshi Inose to explicitly construct a family of CFTs which are associated to a family of smooth algebraic K3 surfaces. Although these CFTs were known before, it is remarkable that they allow a description in terms of a family of smooth surfaces whose complex structure is deformed while all other geometric data remain fixed.

We also discuss possible extensions of this result to higher dimensional Calabi-Yau threefolds.

### 2008/02/07

#### Lectures

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

On Gabber's refined uniformization theorem and applications

**Luc Illusie**(パリ南大学)On Gabber's refined uniformization theorem and applications

[ Abstract ]

Gabber has announced the following theorem : if X is a noetherian quasi-excellent scheme, Z a nowhere dense closed subset, and l a prime number invertible on X, then, locally for the topology on schemes of finite type over X generated, up to thickenings, by proper surjective maps which are generically finite of degree prime to l and by Nisnevich covers, the pair (X,Z) can be uniformized, i. e. replaced by a pair (Y,D), where Y is regular and D a strict normal crossings divisor. The whole proof is not yet written. I will give an overview. The plan is :

1. Statement and reduction to the complete local case (techniques of approximation)

2. Refined partial algebraization of complete local noetherian rings

3. Reduction to the equivariant log regular case (de Jong's techniques)

4. Making actions very tame, end of proof.

If time permits, I will show how the above theorem provides a short proof of Gabber's finiteness theorem for higher direct images of constructible sheaves of torsion prime to the characteristics by morphisms of finite type between quasi-excellent noetherian schemes.

Gabber has announced the following theorem : if X is a noetherian quasi-excellent scheme, Z a nowhere dense closed subset, and l a prime number invertible on X, then, locally for the topology on schemes of finite type over X generated, up to thickenings, by proper surjective maps which are generically finite of degree prime to l and by Nisnevich covers, the pair (X,Z) can be uniformized, i. e. replaced by a pair (Y,D), where Y is regular and D a strict normal crossings divisor. The whole proof is not yet written. I will give an overview. The plan is :

1. Statement and reduction to the complete local case (techniques of approximation)

2. Refined partial algebraization of complete local noetherian rings

3. Reduction to the equivariant log regular case (de Jong's techniques)

4. Making actions very tame, end of proof.

If time permits, I will show how the above theorem provides a short proof of Gabber's finiteness theorem for higher direct images of constructible sheaves of torsion prime to the characteristics by morphisms of finite type between quasi-excellent noetherian schemes.

### 2008/02/06

#### Seminar on Probability and Statistics

13:30-14:40 Room #056 (Graduate School of Math. Sci. Bldg.)

Estimation of the integrated volatility in presence of microstructure noise

http://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2007/17.html

**Jean JACOD**(Universite Paris 6)Estimation of the integrated volatility in presence of microstructure noise

[ Abstract ]

The aim is to estimate the integrated volatility of a process observed discretely, in the setting of high frequency data, and when there is a microstructure noise. We use a kind of pre-averaging approach, which is rate-optimal when the noise is i.i.d., and may probably be even variance-optimal for a good choice of the kernel involved. However, the main innovative aspect is that it accommodates other types of noise, and in particular the case where the observations are rounded values of the underlying process plus an additive noise.

[ Reference URL ]The aim is to estimate the integrated volatility of a process observed discretely, in the setting of high frequency data, and when there is a microstructure noise. We use a kind of pre-averaging approach, which is rate-optimal when the noise is i.i.d., and may probably be even variance-optimal for a good choice of the kernel involved. However, the main innovative aspect is that it accommodates other types of noise, and in particular the case where the observations are rounded values of the underlying process plus an additive noise.

http://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2007/17.html

#### Seminar on Probability and Statistics

14:50-16:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Estimating the Degree of Activity of jumps in High Frequency Data

http://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2007/18.html

**Jean JACOD**(Universite Paris 6)Estimating the Degree of Activity of jumps in High Frequency Data

[ Abstract ]

Suppose that a continuous-time process X = (X_t )_{t >= 0} is observed at finitely many times, regularly spaced, on the fixed time interval [0, T ]. We suppose that this process is an It\\^o semimartingale, with a non-vanishing diffusion coefficient, and with jumps. The aim is to estimate the so-called ”Blumenthal-Getoor” index of the (partially observed) path on [0, T ], which is the (random) infimum of all reals r such that the sum \\sum_{s\\le T} |\\Delta X_s|^r is finite (\\Delta X_s denotes the jump size at time s). When X is a L'evy process, this infimum is non-random, and also independent of T , and has been introduced by Blumenthal and Getoor. Under appropriate assumptions, unfortunately rather restrictive, we provide an estimator, which is consistent when the step size between observations goes to 0, and satisfies in addition a Central Limit Theorem. We also show the (surprising) values that this estimator takes, when applied to real financial data.

[ Reference URL ]Suppose that a continuous-time process X = (X_t )_{t >= 0} is observed at finitely many times, regularly spaced, on the fixed time interval [0, T ]. We suppose that this process is an It\\^o semimartingale, with a non-vanishing diffusion coefficient, and with jumps. The aim is to estimate the so-called ”Blumenthal-Getoor” index of the (partially observed) path on [0, T ], which is the (random) infimum of all reals r such that the sum \\sum_{s\\le T} |\\Delta X_s|^r is finite (\\Delta X_s denotes the jump size at time s). When X is a L'evy process, this infimum is non-random, and also independent of T , and has been introduced by Blumenthal and Getoor. Under appropriate assumptions, unfortunately rather restrictive, we provide an estimator, which is consistent when the step size between observations goes to 0, and satisfies in addition a Central Limit Theorem. We also show the (surprising) values that this estimator takes, when applied to real financial data.

http://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2007/18.html

#### Seminar on Probability and Statistics

16:20-17:30 Room #056 (Graduate School of Math. Sci. Bldg.)

A Hybrid Asymptotic Expansion Scheme: an Application to Long-term Currency Options

http://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2007/19.html

**竹原 浩太**(東京大学大学院経済学研究科)A Hybrid Asymptotic Expansion Scheme: an Application to Long-term Currency Options

[ Abstract ]

In this session we develop a general approximation scheme, henceforth called a hybrid asymptotic expansion scheme for the valuation of multi-factor European path-independent derivatives. Specifically, we apply it to pricing long-term currency options under a market model of interest rates and a general diffusion stochastic volatility model with jumps of spot exchange rates.

Our scheme is very effective for a type of models in which there exist correlations among all the factors whose dynamics are not necessarily affine nor even Markovian so long as the randomness is generated by Brownian motions. It can also handle models that include jump components under an assumption of their independence of the other random variables when the characteristic functions for the jump parts can be analytically obtained.

Moreover, the hybrid scheme develops Fourier transform method with an asymptotic expansion to utilize closed-form characteristic functions obtainable in parts of a model.

Our scheme also introduces a characteristic-function-based Monte Carlo simulation method with the asymptotic expansion as a control variable in order to make full use of analytical approximations by the asymptotic expansion and of closed-form characteristic functions.

Finally, a series of numerical examples shows the validity of our scheme.

(This is a collaborative research with Professor Akihiko Takahashi(Graduate School of Economics, The University of Tokyo).)

[ Reference URL ]In this session we develop a general approximation scheme, henceforth called a hybrid asymptotic expansion scheme for the valuation of multi-factor European path-independent derivatives. Specifically, we apply it to pricing long-term currency options under a market model of interest rates and a general diffusion stochastic volatility model with jumps of spot exchange rates.

Our scheme is very effective for a type of models in which there exist correlations among all the factors whose dynamics are not necessarily affine nor even Markovian so long as the randomness is generated by Brownian motions. It can also handle models that include jump components under an assumption of their independence of the other random variables when the characteristic functions for the jump parts can be analytically obtained.

Moreover, the hybrid scheme develops Fourier transform method with an asymptotic expansion to utilize closed-form characteristic functions obtainable in parts of a model.

Our scheme also introduces a characteristic-function-based Monte Carlo simulation method with the asymptotic expansion as a control variable in order to make full use of analytical approximations by the asymptotic expansion and of closed-form characteristic functions.

Finally, a series of numerical examples shows the validity of our scheme.

(This is a collaborative research with Professor Akihiko Takahashi(Graduate School of Economics, The University of Tokyo).)

http://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2007/19.html

#### Mathematical Finance

18:00-19:30 Room #128 (Graduate School of Math. Sci. Bldg.)

Fast calibration of some Affine and Quadratic models with applications to derivatives on variance swaps

**Daniel Bloch**( )Fast calibration of some Affine and Quadratic models with applications to derivatives on variance swaps

### 2008/01/31

#### Operator Algebra Seminars

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

A generalization of property (T) of SL(n,R)

**見村万佐人**(東大数理)A generalization of property (T) of SL(n,R)

#### Lectures

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

On Gabber's refined uniformization theorem and applications

**Luc Illusie**(パリ南大学)On Gabber's refined uniformization theorem and applications

[ Abstract ]

Gabber has announced the following theorem : if X is a noetherian quasi-excellent scheme, Z a nowhere dense closed subset, and l a prime number invertible on X, then, locally for the topology on schemes of finite type over X generated, up to thickenings, by proper surjective maps which are generically finite of degree prime to l and by Nisnevich covers, the pair (X,Z) can be uniformized, i. e. replaced by a pair (Y,D), where Y is regular and D a strict normal crossings divisor. The whole proof is not yet written. I will give an overview. The plan is :

1. Statement and reduction to the complete local case (techniques of approximation)

2. Refined partial algebraization of complete local noetherian rings

3. Reduction to the equivariant log regular case (de Jong's techniques)

4. Making actions very tame, end of proof.

If time permits, I will show how the above theorem provides a short proof of Gabber's finiteness theorem for higher direct images of constructible sheaves of torsion prime to the characteristics by morphisms of finite type between quasi-excellent noetherian schemes.

Gabber has announced the following theorem : if X is a noetherian quasi-excellent scheme, Z a nowhere dense closed subset, and l a prime number invertible on X, then, locally for the topology on schemes of finite type over X generated, up to thickenings, by proper surjective maps which are generically finite of degree prime to l and by Nisnevich covers, the pair (X,Z) can be uniformized, i. e. replaced by a pair (Y,D), where Y is regular and D a strict normal crossings divisor. The whole proof is not yet written. I will give an overview. The plan is :

1. Statement and reduction to the complete local case (techniques of approximation)

2. Refined partial algebraization of complete local noetherian rings

3. Reduction to the equivariant log regular case (de Jong's techniques)

4. Making actions very tame, end of proof.

If time permits, I will show how the above theorem provides a short proof of Gabber's finiteness theorem for higher direct images of constructible sheaves of torsion prime to the characteristics by morphisms of finite type between quasi-excellent noetherian schemes.

### 2008/01/30

#### Number Theory Seminar

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

Odds and ends on finite group actions and traces

**Luc Illusie**(Universite Paris-Sud 11)Odds and ends on finite group actions and traces

[ Abstract ]

Suppose a finite group G acts on a scheme X separated and of finite type over a field k. This raises several questions about the traces of elements s of G (or more generally products sg, for g in the Galois group of k) on cohomology groups of various types associated with X/k (with compact support or no support, Betti if k = C, l-adic, rigid). Some were considered and solved long ago, others only recently. I will in particular discuss an equivariant generalization of a theorem of Laumon on Euler-Poincar¥'e characteristics.

Suppose a finite group G acts on a scheme X separated and of finite type over a field k. This raises several questions about the traces of elements s of G (or more generally products sg, for g in the Galois group of k) on cohomology groups of various types associated with X/k (with compact support or no support, Betti if k = C, l-adic, rigid). Some were considered and solved long ago, others only recently. I will in particular discuss an equivariant generalization of a theorem of Laumon on Euler-Poincar¥'e characteristics.

#### PDE Real Analysis Seminar

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

Carleman estimates for parabolic equations, a Stokes system and the Navier-Stokes equations and applications to the control problem

**Oleg Yu. Emanouilov**(Colorado State University)Carleman estimates for parabolic equations, a Stokes system and the Navier-Stokes equations and applications to the control problem

[ Abstract ]

We prove a new Carleman estimate for general linear second order parabolic equation with nonhomogeneous boundary conditions. On basis of this estimate we obtain an improved Carleman estimate for the Stokes system and a system of parabolic equations with a parameter which can be viewed as an approximation of the Stokes system. We will discuss the applications to the control problem for these systems.

We prove a new Carleman estimate for general linear second order parabolic equation with nonhomogeneous boundary conditions. On basis of this estimate we obtain an improved Carleman estimate for the Stokes system and a system of parabolic equations with a parameter which can be viewed as an approximation of the Stokes system. We will discuss the applications to the control problem for these systems.

### 2008/01/29

#### Algebraic Geometry Seminar

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

Homological methods in non-commutative geometry, part 11 (last lecture)

[ Reference URL ]

http://imperium.lenin.ru/~kaledin/math/tokyo/

**Dmitry KALEDIN**(Steklov研究所, 東大数理)Homological methods in non-commutative geometry, part 11 (last lecture)

[ Reference URL ]

http://imperium.lenin.ru/~kaledin/math/tokyo/

#### Tuesday Seminar on Topology

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

The rotation number function on groups of circle diffeomorphisms

A Diagrammatic Construction of Third Homology Classes of Knot Quandles

**松田 能文**(東京大学大学院数理科学研究科) 16:30-17:30The rotation number function on groups of circle diffeomorphisms

[ Abstract ]

ポアンカレは、円周の向きを保つ同相写像に対して、回転数の有理性と有限軌道の存在が

同値であることを示した。この講演では、この事実が円周の向きを保つ同相写像のなすあ

る種の群に対して一般化できることを説明する。特に、円周の向きを保つ実解析的微分同

相のなす非離散的な群に対して、回転数関数による像の有限性と有限軌道の存在が同値で

あることを示す。

ポアンカレは、円周の向きを保つ同相写像に対して、回転数の有理性と有限軌道の存在が

同値であることを示した。この講演では、この事実が円周の向きを保つ同相写像のなすあ

る種の群に対して一般化できることを説明する。特に、円周の向きを保つ実解析的微分同

相のなす非離散的な群に対して、回転数関数による像の有限性と有限軌道の存在が同値で

あることを示す。

**木村 康人**(東京大学大学院数理科学研究科) 17:30-18:30A Diagrammatic Construction of Third Homology Classes of Knot Quandles

[ Abstract ]

There exists a family of third (quandle / rack) homology classes,

called the shadow (fundamental / diagram) classes,

of the knot quandle, which are obtained from

the shadow colourings of knot diagrams.

We will show the construction of these homology classes,

and also show their relation to the shadow quandle cocycle

invariants of knots and that to other third homology classes.

There exists a family of third (quandle / rack) homology classes,

called the shadow (fundamental / diagram) classes,

of the knot quandle, which are obtained from

the shadow colourings of knot diagrams.

We will show the construction of these homology classes,

and also show their relation to the shadow quandle cocycle

invariants of knots and that to other third homology classes.

### 2008/01/28

#### Seminar on Geometric Complex Analysis

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

閉リーマン面の$\mathbf{C}^{*}$-作用付き退化族と$\mathbf{C}^{*}$-作用付き複素2次元特異点

**都丸 正**(群馬大学)閉リーマン面の$\mathbf{C}^{*}$-作用付き退化族と$\mathbf{C}^{*}$-作用付き複素2次元特異点

### 2008/01/25

#### Colloquium

17:00-18:00 Room #123 (Graduate School of Math. Sci. Bldg.)

Keller-Segel系に対する保存的上流有限要素法

**齊藤宣一**(東京大学数理科学)Keller-Segel系に対する保存的上流有限要素法

[ Abstract ]

非線形放物型偏微分方程式系に対して、有限要素法による数値解法を考え、スキーム構成の勘所と誤差解析の最近の動向についてお話したい。具体的な例としては、細胞性粘菌の凝集現象を記述するモデルとして広く知られるKeller-Segel(KS)系とその保存的上流有限要素法を取り上げる。このスキームは、KS系の解の基本性質である正値性保存と質量保存を厳密に再現し、解が凝集による集中化を起こしても安定に計算が遂行できる。さらに、離散 $L^p$ 空間における離散的解析半群の理論を応用して、陽的な誤差評価が導出される。

非線形放物型偏微分方程式系に対して、有限要素法による数値解法を考え、スキーム構成の勘所と誤差解析の最近の動向についてお話したい。具体的な例としては、細胞性粘菌の凝集現象を記述するモデルとして広く知られるKeller-Segel(KS)系とその保存的上流有限要素法を取り上げる。このスキームは、KS系の解の基本性質である正値性保存と質量保存を厳密に再現し、解が凝集による集中化を起こしても安定に計算が遂行できる。さらに、離散 $L^p$ 空間における離散的解析半群の理論を応用して、陽的な誤差評価が導出される。

### 2008/01/24

#### Applied Analysis

16:00-17:30 Room #126 (Graduate School of Math. Sci. Bldg.)

A compactness result in micromagnetics

**Radu IGNAT**(パリ南大学(オルセー))A compactness result in micromagnetics

[ Abstract ]

We study a model for the magnetization in thin ferromagnetic films. It comes as a variational problem, depending on two parameters, for maps with values into the unit sphere. There is a physical prediction for the optimal configuration of the magnetization called the Landau state. Our goal is to prove compactness of the Landau state. This is a joint work with Felix Otto.

We study a model for the magnetization in thin ferromagnetic films. It comes as a variational problem, depending on two parameters, for maps with values into the unit sphere. There is a physical prediction for the optimal configuration of the magnetization called the Landau state. Our goal is to prove compactness of the Landau state. This is a joint work with Felix Otto.

### 2008/01/23

#### Number Theory Seminar

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

Integrality, Rationality, and Independence of l in l-adic Cohomology over Local Fields

**Weizhe Zheng**(Universite Paris-Sud 11)Integrality, Rationality, and Independence of l in l-adic Cohomology over Local Fields

[ Abstract ]

I will discuss two problems on traces in l-adic cohomology over local fields with finite residue field. In the first part, I will describe the behavior of integral complexes of l-adic sheaves under Grothendieck's six operations and the nearby cycle functor. In the second part, I will talk about rationality and independence of l. More precisely, I will introduce a notion of compatibility for systems of l-adic complexes and explain the proof of its stability by the above operations, in a slightly more general context (equivariant under finite groups). The main tool in this talk is a theorem of de Jong on

alterations.

I will discuss two problems on traces in l-adic cohomology over local fields with finite residue field. In the first part, I will describe the behavior of integral complexes of l-adic sheaves under Grothendieck's six operations and the nearby cycle functor. In the second part, I will talk about rationality and independence of l. More precisely, I will introduce a notion of compatibility for systems of l-adic complexes and explain the proof of its stability by the above operations, in a slightly more general context (equivariant under finite groups). The main tool in this talk is a theorem of de Jong on

alterations.

#### Mathematical Finance

17:30-19:00 Room #128 (Graduate School of Math. Sci. Bldg.)

確率微分方程式に対するRunge-Kutta法を用いた新たな弱近似手法

**二宮 真理子**(東京大)確率微分方程式に対するRunge-Kutta法を用いた新たな弱近似手法

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