## Seminar information archive

Seminar information archive ～04/19｜Today's seminar 04/20 | Future seminars 04/21～

### 2006/09/06

#### Number Theory Seminar

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

Computation of the mod l Galois representations associated to Delta

**Bas Edixhoven**(Univ. of Leiden)Computation of the mod l Galois representations associated to Delta

### 2006/09/04

#### Mathematical Finance

17:00-18:30 Room #117 (Graduate School of Math. Sci. Bldg.)

Dynamic Risk Measures and Backward Stochastic Differential Equation

**Freddy Delbaen**(ETH)Dynamic Risk Measures and Backward Stochastic Differential Equation

#### Mathematical Finance

15:45-16:45 Room #117 (Graduate School of Math. Sci. Bldg.)

リスク尺度入門及び概説

**楠岡成雄氏・梅澤祐二**(東京大)リスク尺度入門及び概説

### 2006/08/25

#### Number Theory Seminar

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

p-adic local constants

**A. Marmora**(パリ北大・東大/学振)p-adic local constants

### 2006/08/22

#### Seminar on Probability and Statistics

15:30-16:40 Room #128 (Graduate School of Math. Sci. Bldg.)

A unifying approach to inference in semimartingale and long-memory models

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2006/07.html

**Jeannette H.C. WOERNER**(University of Gottingen)A unifying approach to inference in semimartingale and long-memory models

[ Abstract ]

Over the recent years classical stochastic volatility models based on Brownian motion have been generalized in different ways, either replacing the Brownian motion by a pure jump Levy process, which leads to a pure jump model, or by a fractional Brownian motion, which makes it possible to model both long memory or turbulent behaviour. We consider robust and easily computable estimators for the inte- grated volatility, providing insight in the level of volatility, as needed for risk measurement and pricing of variance and volatility swaps. We discuss consistency and distributional results for the power and multi- power variation estimates based on high frequency data. Furthermore, we consider robustness against additive components and compare the results for the different classes of semimartingale and fractional Brow- nian motion models.

[ Reference URL ]Over the recent years classical stochastic volatility models based on Brownian motion have been generalized in different ways, either replacing the Brownian motion by a pure jump Levy process, which leads to a pure jump model, or by a fractional Brownian motion, which makes it possible to model both long memory or turbulent behaviour. We consider robust and easily computable estimators for the inte- grated volatility, providing insight in the level of volatility, as needed for risk measurement and pricing of variance and volatility swaps. We discuss consistency and distributional results for the power and multi- power variation estimates based on high frequency data. Furthermore, we consider robustness against additive components and compare the results for the different classes of semimartingale and fractional Brow- nian motion models.

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2006/07.html

#### Seminar on Probability and Statistics

16:50-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

A computation of Theta in a jump diffusion model by integration by parts

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2006/08.html

**Delphine DAVID**(Departement de Mathematiques, Universite de La Rochelle)A computation of Theta in a jump diffusion model by integration by parts

[ Abstract ]

Using Malliavin weights in a jump-diffusion model we obtain an expression for Theta (the sensitivity of an option price with respect to the time remaining until exercise), with application to non-smooth payoff functions. Optimal weights are computed by minimization of variance and numerical simulations are presented for digital and European options. Some results are also presented for Asian options. Our representation formula for Theta differs in general from the one obtained from the Black-Scholes PDE in terms of Delta and Gamma.

[ Reference URL ]Using Malliavin weights in a jump-diffusion model we obtain an expression for Theta (the sensitivity of an option price with respect to the time remaining until exercise), with application to non-smooth payoff functions. Optimal weights are computed by minimization of variance and numerical simulations are presented for digital and European options. Some results are also presented for Asian options. Our representation formula for Theta differs in general from the one obtained from the Black-Scholes PDE in terms of Delta and Gamma.

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2006/08.html

### 2006/08/03

#### Operator Algebra Seminars

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

The Cuntz semigroup as an invariant for $C^*$-algebras

**George Elliott**(University of Toronto)The Cuntz semigroup as an invariant for $C^*$-algebras

### 2006/07/31

#### Lie Groups and Representation Theory

15:00-17:30 Room #126 (Graduate School of Math. Sci. Bldg.)

The Heat equation, the Segal-Bargmann transform and generalizations - II

[ Reference URL ]

http://akagi.ms.u-tokyo.ac.jp/seminar.html

Radon transforms on Grassmannians and Matrix Spaces

http://akagi.ms.u-tokyo.ac.jp/seminar.html

**Guster Olafsson**(Louisiana State University) 15:00-16:00The Heat equation, the Segal-Bargmann transform and generalizations - II

[ Reference URL ]

http://akagi.ms.u-tokyo.ac.jp/seminar.html

**Boris Rubin**(Louisiana State University) 16:30-17:30Radon transforms on Grassmannians and Matrix Spaces

[ Abstract ]

Diverse geometric problems in $R^N$ get a new flavor if a generic point $x=(x_1,...,x_N)$ is regarded as a matrix with appropriately organized entries (set, e.g., $x=(x_{i,j})_{n \\times m}$ for $N=nm$). This well known observation has led to a series of breakthrough achievements in mathematics. In integral geometry it suggests a number of the so-called ``higher-rank" problems when such traditional scalar notions as ``distance", ``angle", and ``scaling" become matrix-valued. I will be speaking about Radon transforms on Grassmann manifolds and matrix spaces and some related problems of harmonic analysis where these phenomena come into play.

[ Reference URL ]Diverse geometric problems in $R^N$ get a new flavor if a generic point $x=(x_1,...,x_N)$ is regarded as a matrix with appropriately organized entries (set, e.g., $x=(x_{i,j})_{n \\times m}$ for $N=nm$). This well known observation has led to a series of breakthrough achievements in mathematics. In integral geometry it suggests a number of the so-called ``higher-rank" problems when such traditional scalar notions as ``distance", ``angle", and ``scaling" become matrix-valued. I will be speaking about Radon transforms on Grassmann manifolds and matrix spaces and some related problems of harmonic analysis where these phenomena come into play.

http://akagi.ms.u-tokyo.ac.jp/seminar.html

### 2006/07/25

#### Lie Groups and Representation Theory

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

Radon Transforms: Basic Concepts

http://akagi.ms.u-tokyo.ac.jp/seminar.html

**Boris Rubin**(Louisiana State University)Radon Transforms: Basic Concepts

[ Abstract ]

How can we reconstruct a function on a manifold from integrals of this function over certain submanifolds?

This is one of the central problems in integral geometry and tomography, which leads to the notion of the Radon transform.

The first talk is of introductory character.

We discuss basic ideas of the original Radon's paper (1917), then proceed to the Minkowski-Funk transform and more general totally geodesic Radon transforms on the $n$-dimensional unit sphere.

The main emphasis is an intimate connection of these transforms with the relevant harmonic analysis.

We will see that Radon transforms of this type and their inverses can be regarded as members of analytic families of suitable convolution operators and successfully studied in the framework of these families.

I also plan to discuss an open problem of small divisors on the unit sphere, which arises in studying injectivity of generalized Minkowski-Funk transforms for non-central spherical sections.

[ Reference URL ]How can we reconstruct a function on a manifold from integrals of this function over certain submanifolds?

This is one of the central problems in integral geometry and tomography, which leads to the notion of the Radon transform.

The first talk is of introductory character.

We discuss basic ideas of the original Radon's paper (1917), then proceed to the Minkowski-Funk transform and more general totally geodesic Radon transforms on the $n$-dimensional unit sphere.

The main emphasis is an intimate connection of these transforms with the relevant harmonic analysis.

We will see that Radon transforms of this type and their inverses can be regarded as members of analytic families of suitable convolution operators and successfully studied in the framework of these families.

I also plan to discuss an open problem of small divisors on the unit sphere, which arises in studying injectivity of generalized Minkowski-Funk transforms for non-central spherical sections.

http://akagi.ms.u-tokyo.ac.jp/seminar.html

### 2006/07/24

#### Tuesday Seminar on Topology

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

Invariant foliations in hyperbolic dynamics:

Smoothness and smooth equivalence

http://faculty.ms.u-tokyo.ac.jp/~topology/

**Boris Hasselblatt**(Tufts University)Invariant foliations in hyperbolic dynamics:

Smoothness and smooth equivalence

[ Abstract ]

The stable and unstable leaves of a hyperbolic dynamical system are smooth and form a continuous foliation. Smoothness of this foliation sometimes constrains the topological type of the foliation, other times restricts at least the smooth equivalence class of the dynamical system, or for geodesic flows, the type of the underlying manifold. I will present a broad introduction as well as recent work, work in progress, and open problems.

[ Reference URL ]The stable and unstable leaves of a hyperbolic dynamical system are smooth and form a continuous foliation. Smoothness of this foliation sometimes constrains the topological type of the foliation, other times restricts at least the smooth equivalence class of the dynamical system, or for geodesic flows, the type of the underlying manifold. I will present a broad introduction as well as recent work, work in progress, and open problems.

http://faculty.ms.u-tokyo.ac.jp/~topology/

### 2006/07/20

#### Operator Algebra Seminars

16:30-18:00 Room #052 (Graduate School of Math. Sci. Bldg.)

Linear response theory in quantum statistical mechanics

**緒方芳子**(東大数理・学振)Linear response theory in quantum statistical mechanics

#### Lie Groups and Representation Theory

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

The Heat equation, the Segal-Bargmann transform and generalizations - I

[ Reference URL ]

http://akagi.ms.u-tokyo.ac.jp/seminar.html

**Guster Olafsson**(Louisiana State University)The Heat equation, the Segal-Bargmann transform and generalizations - I

[ Reference URL ]

http://akagi.ms.u-tokyo.ac.jp/seminar.html

### 2006/07/19

#### Seminar on Probability and Statistics

16:20-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)

Edgeworth Expansion for Likelihood Analysis on Ergodic Diffusions with applications to Bootstrap

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2006/06.html

**深澤 正彰**(東京大学大学院数理科学研究科)Edgeworth Expansion for Likelihood Analysis on Ergodic Diffusions with applications to Bootstrap

[ Abstract ]

We shall consider the maximal lilelihood estimator for the drift coefficient of a given one-dimensional diffusion. An Edgeworth expansion formula will be presented and verify a second-order correct confidence interval we shall newly propose. We are also going to mention the likelihood ratio statistic, which enjoys second-order correctness. There are Bootstrap methods closely related to the subject and introduced recently by the author. Some generalized results on those methods will be also introduced in this talk.

[ Reference URL ]We shall consider the maximal lilelihood estimator for the drift coefficient of a given one-dimensional diffusion. An Edgeworth expansion formula will be presented and verify a second-order correct confidence interval we shall newly propose. We are also going to mention the likelihood ratio statistic, which enjoys second-order correctness. There are Bootstrap methods closely related to the subject and introduced recently by the author. Some generalized results on those methods will be also introduced in this talk.

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2006/06.html

### 2006/07/13

#### Operator Algebra Seminars

16:30-18:00 Room #052 (Graduate School of Math. Sci. Bldg.)

Property (T) for universal lattices, after Y. Shalom

**小沢登高**(東大数理)Property (T) for universal lattices, after Y. Shalom

[ Abstract ]

I will talk on Shalom's recent result that

$SL_n(Z[X])$ ($n\\geq 3$) has the property (T).

The talk should be elementary.

I will talk on Shalom's recent result that

$SL_n(Z[X])$ ($n\\geq 3$) has the property (T).

The talk should be elementary.

### 2006/07/12

#### Number Theory Seminar

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

Beilinson-Drinfeld chiral algebra, geometric Langlands program and open Gromov-Witten invariants

**桜井 真**(東京大学理学系研究科)Beilinson-Drinfeld chiral algebra, geometric Langlands program and open Gromov-Witten invariants

[ Abstract ]

都合により、とりやめになりました。

都合により、とりやめになりました。

#### Mathematical Finance

18:30-20:00 Room #118 (Graduate School of Math. Sci. Bldg.)

A complete-market generalization of the Black-Scholes model

**高岡 浩一郎**(一橋大)A complete-market generalization of the Black-Scholes model

#### PDE Real Analysis Seminar

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

Analysis of a crystal growth model

http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html

**Piotr Rybka**(Warsaw University)Analysis of a crystal growth model

[ Abstract ]

We are concerned with mathematical model of a single crystal growing from vapor. Mathematically this is an exterior, one-phase Stefan-type problem with Gibbs-Thomson law. We restrict our attention to an idealization of a ice crystal, i.e. our evolving free boundary is a circular cylinder. The system under consideration consists of an equation for the motion of the free boundary (the crystal surface) coupled to the quasi-steady approximation of the diffusion equation for the supersaturation of vapor. We present analysis of the system, we show well-posedness and draw the phase portrait, we use here the fact that we need just to variable to describe evolution of a cylinder.

We are mostly concerned with the shape-persitency problem of the

evolution. The problem is, the Gibbs-Thomson relation is in fact a

driven, weighted, mean, singular curvature flow and it is not obvious that the shape of the initial interface will persists throughout the evolution or even for some time. In order to solve this problem we show existence of the region in the phase plane which is a neighborhood of a unique steady state, such that in this region the shape of the cylinder is preserved. However, this set is not invariant with respect to dynamics of the problem.

It is a very interesting question what happens to surface of our crystal at the boundary of the shape-persitency (or shape stability) region. This problem in its full generality is open. However, we give some insight by studying the Gibbs-Thomson relation with a given driving, which inherits properties of the coupling to the diffusion field. We study the resulting driven weighted mean curvature flow for graphs and some special closed Lipschitz curves. We show well-posedness of the problem, but mainly we exhibit the phenomenon of bending flat parts of the curve, which grow ``too big''.

[ Reference URL ]We are concerned with mathematical model of a single crystal growing from vapor. Mathematically this is an exterior, one-phase Stefan-type problem with Gibbs-Thomson law. We restrict our attention to an idealization of a ice crystal, i.e. our evolving free boundary is a circular cylinder. The system under consideration consists of an equation for the motion of the free boundary (the crystal surface) coupled to the quasi-steady approximation of the diffusion equation for the supersaturation of vapor. We present analysis of the system, we show well-posedness and draw the phase portrait, we use here the fact that we need just to variable to describe evolution of a cylinder.

We are mostly concerned with the shape-persitency problem of the

evolution. The problem is, the Gibbs-Thomson relation is in fact a

driven, weighted, mean, singular curvature flow and it is not obvious that the shape of the initial interface will persists throughout the evolution or even for some time. In order to solve this problem we show existence of the region in the phase plane which is a neighborhood of a unique steady state, such that in this region the shape of the cylinder is preserved. However, this set is not invariant with respect to dynamics of the problem.

It is a very interesting question what happens to surface of our crystal at the boundary of the shape-persitency (or shape stability) region. This problem in its full generality is open. However, we give some insight by studying the Gibbs-Thomson relation with a given driving, which inherits properties of the coupling to the diffusion field. We study the resulting driven weighted mean curvature flow for graphs and some special closed Lipschitz curves. We show well-posedness of the problem, but mainly we exhibit the phenomenon of bending flat parts of the curve, which grow ``too big''.

http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html

### 2006/07/11

#### Tuesday Seminar on Topology

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

全葉層の存在について(浅岡正幸,Emmanuel Dufraineとの共同研究)

http://faculty.ms.u-tokyo.ac.jp/~topology/

**野田健夫**(秋田大学工学資源学部)全葉層の存在について(浅岡正幸,Emmanuel Dufraineとの共同研究)

[ Abstract ]

n次元多様体上のn個の余次元1葉層構造の組で、n個の葉層構造の接空間の共通部分が各点で0になるものを全葉層と呼ぶ。3次元の場合においては任意の有向閉多様体上に全葉層が存在することが Hardorpによって示されていた。3次元多様体上の全葉層をなす各々の葉層構造の接平面場は互いにホモトピックでありオイラー類が0になることが容易に分かるが、逆にオイラー類が0の平面場を与えたときそれを実現する全葉層が存在するかという問題が自然に生じる。

本講演ではこの問題に肯定的な解決をあたえる。

また、この結果の応用として双接触構造、すなわち横断的に交わる正と負の接触構造の組の存在問題にも触れたい。

[ Reference URL ]n次元多様体上のn個の余次元1葉層構造の組で、n個の葉層構造の接空間の共通部分が各点で0になるものを全葉層と呼ぶ。3次元の場合においては任意の有向閉多様体上に全葉層が存在することが Hardorpによって示されていた。3次元多様体上の全葉層をなす各々の葉層構造の接平面場は互いにホモトピックでありオイラー類が0になることが容易に分かるが、逆にオイラー類が0の平面場を与えたときそれを実現する全葉層が存在するかという問題が自然に生じる。

本講演ではこの問題に肯定的な解決をあたえる。

また、この結果の応用として双接触構造、すなわち横断的に交わる正と負の接触構造の組の存在問題にも触れたい。

http://faculty.ms.u-tokyo.ac.jp/~topology/

#### Lie Groups and Representation Theory

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

On the isomorphism problem of Coxeter groups and related topics

http://akagi.ms.u-tokyo.ac.jp/seminar.html

**縫田 光司**(産業技術総合研究所)On the isomorphism problem of Coxeter groups and related topics

[ Abstract ]

コクセター群について、そのコクセター系としての同型類がコクセターグラフと一対一対応することは周知の事実であるが、一方で抽象群としての同型類は(ワイル群の場合に限っても)そのような対応をしていない。今回は、この同型類の決定問題について、その歴史のあらまし(特に、無限群も含めた一般の場合について、10年ほど前まで殆ど何の結果も得られていなかったことは特筆に価する)と、近年の研究の進展状況を、具体例や関連する結果を交えつつ紹介する。

[ Reference URL ]コクセター群について、そのコクセター系としての同型類がコクセターグラフと一対一対応することは周知の事実であるが、一方で抽象群としての同型類は(ワイル群の場合に限っても)そのような対応をしていない。今回は、この同型類の決定問題について、その歴史のあらまし(特に、無限群も含めた一般の場合について、10年ほど前まで殆ど何の結果も得られていなかったことは特筆に価する)と、近年の研究の進展状況を、具体例や関連する結果を交えつつ紹介する。

http://akagi.ms.u-tokyo.ac.jp/seminar.html

### 2006/07/10

#### Seminar on Geometric Complex Analysis

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

Characterization of domains in $C^n$ by their noncompact automorphism groups

**Do Duc Thai**(Hanoi教育大)Characterization of domains in $C^n$ by their noncompact automorphism groups

[ Abstract ]

In this talk, the characterization of domains in $C^n$ by their noncompact automorphism groups are given. By this characterization, the Bedford-Pinchuk theorem is true for any domain (not necessary bounded) in $C^n$.

In this talk, the characterization of domains in $C^n$ by their noncompact automorphism groups are given. By this characterization, the Bedford-Pinchuk theorem is true for any domain (not necessary bounded) in $C^n$.

### 2006/07/08

#### Monthly Seminar on Arithmetic of Automorphic Forms

13:30-15:45 Room #123 (Graduate School of Math. Sci. Bldg.)

Rankin-Cohen-Ibukiyama operators for holomorphic automorphic forms on type I symmetric domains

On Dirichlet series counting cubic alegebras

**伴 克馬**(東京大学大学院数理科学研究科) 13:30-14:30Rankin-Cohen-Ibukiyama operators for holomorphic automorphic forms on type I symmetric domains

**谷口 隆**(東京大学大学院数理科学研究科) 14:45-15:45On Dirichlet series counting cubic alegebras

### 2006/07/07

#### Colloquium

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

周期的変動環境下における侵入生物の時空間パターンと伝播速度

**重定 南奈子**(同志社大学)周期的変動環境下における侵入生物の時空間パターンと伝播速度

[ Abstract ]

侵入生物の空間的な伝播に関する数理的研究は,Fisher (1937)の先駆的研究以来,外来植物や昆虫,伝染病などの侵入を中心に,主として一様な空間における拡散増殖モデルを用いて進められてきた.しかし,実際の自然環境は,森,林,河川,道路などの,生物にとって好適な環境と不適な環境がパッチ状に入り混じっており,決して一様な空間とはいえない.

本研究では、帯状の好適生息地と不適な生息地が交互に配列する2次元縞状 分断環境の中を、侵入生物が分布拡大する過程を拡散係数と増殖率が好適生息 地と不適生息地で異なる拡張Fisher modelを用いて記述し、それを heuristicな方法を用いて解くことにより,侵入種の分布拡大パターン,ならびに,伝播速度の数学公式を導いた.

侵入生物の空間的な伝播に関する数理的研究は,Fisher (1937)の先駆的研究以来,外来植物や昆虫,伝染病などの侵入を中心に,主として一様な空間における拡散増殖モデルを用いて進められてきた.しかし,実際の自然環境は,森,林,河川,道路などの,生物にとって好適な環境と不適な環境がパッチ状に入り混じっており,決して一様な空間とはいえない.

本研究では、帯状の好適生息地と不適な生息地が交互に配列する2次元縞状 分断環境の中を、侵入生物が分布拡大する過程を拡散係数と増殖率が好適生息 地と不適生息地で異なる拡張Fisher modelを用いて記述し、それを heuristicな方法を用いて解くことにより,侵入種の分布拡大パターン,ならびに,伝播速度の数学公式を導いた.

### 2006/07/06

#### Seminar for Mathematical Past of Asia

17:00-18:30 Room #123 (Graduate School of Math. Sci. Bldg.)

ユークリッドをめぐる最新の研究動向

[ Reference URL ]

https://www.ms.u-tokyo.ac.jp/~kawazumi/asia.html

**斎藤 憲**(大阪府立大学 人間社会学部)ユークリッドをめぐる最新の研究動向

[ Reference URL ]

https://www.ms.u-tokyo.ac.jp/~kawazumi/asia.html

#### Operator Algebra Seminars

16:30-18:00 Room #052 (Graduate School of Math. Sci. Bldg.)

Endomorphisms of half-sided modular inclusions

**Rolf Dyre Svegstrup**(東大数理)Endomorphisms of half-sided modular inclusions

### 2006/07/05

#### Seminar on Mathematics for various disciplines

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

Level Set Methods and Multi-valued solutions

http://coe.math.sci.hokudai.ac.jp/sympo/various/index.html

**Y. H. Richard Tsai**(University of Texas)Level Set Methods and Multi-valued solutions

[ Abstract ]

We review the level set methods for computing multi-valued

solutions to a class of nonlinear first order partial differential

equations, including Hamilton-Jacobi equations, quasi-linear

hyperbolic equations, and conservative transport equations with

multi-valued transport speeds.

The multivalued solutions are embedded as the zeros of a set of scalar functions that solve the initial value problems of a time dependent partial differential equation in an augmented space.

We discuss the essential ideas behind the techniques, the coupling of these techniques to the projection of the interaction of zero level sets and a collection of applications including the omputation of the semiclassical limit for Schr\\"{o}dinger quations and the high frequency geometrical optics limits of linear wave equations.

[ Reference URL ]We review the level set methods for computing multi-valued

solutions to a class of nonlinear first order partial differential

equations, including Hamilton-Jacobi equations, quasi-linear

hyperbolic equations, and conservative transport equations with

multi-valued transport speeds.

The multivalued solutions are embedded as the zeros of a set of scalar functions that solve the initial value problems of a time dependent partial differential equation in an augmented space.

We discuss the essential ideas behind the techniques, the coupling of these techniques to the projection of the interaction of zero level sets and a collection of applications including the omputation of the semiclassical limit for Schr\\"{o}dinger quations and the high frequency geometrical optics limits of linear wave equations.

http://coe.math.sci.hokudai.ac.jp/sympo/various/index.html

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