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Seminar information archive
Seminar information archive ～02/19｜Today's seminar 02/20 | Future seminars 02/21～
2007/03/17
Infinite Analysis Seminar Tokyo
13:30-14:30 Room #117 (Graduate School of Math. Sci. Bldg.)
Paul Wiegmann (Chicago Univ.)
Calogero model and Quantum Benjamin-Ono Equation
Paul Wiegmann (Chicago Univ.)
Calogero model and Quantum Benjamin-Ono Equation
[ Abstract ]
TBA
TBA
2007/03/09
Lectures
10:30-12:00 Room #123 (Graduate School of Math. Sci. Bldg.)
Kazufumi Ito (North Carolina State University)
Nonsmooth Optimization and Applications in PDEs
Kazufumi Ito (North Carolina State University)
Nonsmooth Optimization and Applications in PDEs
[ Abstract ]
Semismooth Newton method for solving nonlinear non-smooth equations in Banach spaces is discussed.
Applications include complementarity problems, variational inequalities and optimal control problems with control or state constraints, Black Scholes model with American option and imaging analysis.
Semismooth Newton method for solving nonlinear non-smooth equations in Banach spaces is discussed.
Applications include complementarity problems, variational inequalities and optimal control problems with control or state constraints, Black Scholes model with American option and imaging analysis.
2007/03/08
Lectures
15:30-17:00 Room #123 (Graduate School of Math. Sci. Bldg.)
Kazufumi Ito (North Carolina State University)
Nonsmooth Optimization and Applications in PDEs
Kazufumi Ito (North Carolina State University)
Nonsmooth Optimization and Applications in PDEs
[ Abstract ]
Semismooth Newton method for solving nonlinear non-smooth equations in Banach spaces is discussed.
Applications include complementarity problems, variational inequalities and optimal control problems with control or state constraints, Black Scholes model with American option and imaging analysis.
Semismooth Newton method for solving nonlinear non-smooth equations in Banach spaces is discussed.
Applications include complementarity problems, variational inequalities and optimal control problems with control or state constraints, Black Scholes model with American option and imaging analysis.
2007/03/07
Seminar on Mathematics for various disciplines
14:00-15:00 Room #056 (Graduate School of Math. Sci. Bldg.)
Seung Yeal Ha (Seoul National University)
Stability theory in L^p for the space-inhomogeneous Boltzmann equation
http://coe.math.sci.hokudai.ac.jp/index.html.en
Seung Yeal Ha (Seoul National University)
Stability theory in L^p for the space-inhomogeneous Boltzmann equation
[ Abstract ]
In this talk, I will present kinetic nonlinear funtionals which are similar in sprit to Glimm type functionals in one-dimensional hyperbolic conservation laws. These functionals measures the dispersive mechanism of the Boltzmann equation near vacuum and can be used to the study of the large-time behavior and L^p-stability of the Boltzmann equation near vacuum. This is a joint work with M. Yamazaki (Univ. of Tsukuba) and Seok-Bae Yun (Seoul National Univ.)
[ Reference URL ]In this talk, I will present kinetic nonlinear funtionals which are similar in sprit to Glimm type functionals in one-dimensional hyperbolic conservation laws. These functionals measures the dispersive mechanism of the Boltzmann equation near vacuum and can be used to the study of the large-time behavior and L^p-stability of the Boltzmann equation near vacuum. This is a joint work with M. Yamazaki (Univ. of Tsukuba) and Seok-Bae Yun (Seoul National Univ.)
http://coe.math.sci.hokudai.ac.jp/index.html.en
2007/02/22
Lectures
10:30-12:00 Room #123 (Graduate School of Math. Sci. Bldg.)
Stan Osher (UCLA)
The level set method, multivalued solutions and image science
Stan Osher (UCLA)
The level set method, multivalued solutions and image science
[ Abstract ]
During the past two decades variational and partial differential based methods have greatly affected the fields of image processing, computer vision and graphics (image science in general). Almost simultaneously the level set method for computing moving interfaces has impacted many areas of mathematics, engineering and applied science, including image science. I will try to give an overview of the basics and recent advances in these topics.
During the past two decades variational and partial differential based methods have greatly affected the fields of image processing, computer vision and graphics (image science in general). Almost simultaneously the level set method for computing moving interfaces has impacted many areas of mathematics, engineering and applied science, including image science. I will try to give an overview of the basics and recent advances in these topics.
Lectures
13:00-15:00 Room #123 (Graduate School of Math. Sci. Bldg.)
Dietmar Hoemberg (Berlin Technical University)
Optimal control of semilinear parabolic equations and an application to laser material treatments
Dietmar Hoemberg (Berlin Technical University)
Optimal control of semilinear parabolic equations and an application to laser material treatments
[ Abstract ]
Many technological processes can be described by partial differential equations. For many years the role of industrial mathematics was mainly to try to understand the respective process, to derive an appropriate PDE or ODE model for it and to simulate it using, e.g., a finite-element code.
However, the ultimate goal usually is to try to optimize the process. Mathematically, this requires the solution of an optimal control problem, i.e., a constrained nonlinear optimization problem in which the constraints are PDEs.
The goal of these two talks is to give an overview of the theory and numerics of optimal control of PDEs for the case of parabolic state equations including an application in laser material treatments. More specifically, I will focus on the following topics.
Many technological processes can be described by partial differential equations. For many years the role of industrial mathematics was mainly to try to understand the respective process, to derive an appropriate PDE or ODE model for it and to simulate it using, e.g., a finite-element code.
However, the ultimate goal usually is to try to optimize the process. Mathematically, this requires the solution of an optimal control problem, i.e., a constrained nonlinear optimization problem in which the constraints are PDEs.
The goal of these two talks is to give an overview of the theory and numerics of optimal control of PDEs for the case of parabolic state equations including an application in laser material treatments. More specifically, I will focus on the following topics.
2007/02/21
Lectures
10:30-12:00 Room #123 (Graduate School of Math. Sci. Bldg.)
Stan Osher (UCLA)
The level set method, multivalued solutions and image science
Stan Osher (UCLA)
The level set method, multivalued solutions and image science
[ Abstract ]
During the past two decades variational and partial differential based methods have greatly affected the fields of image processing, computer vision and graphics (image science in general). Almost simultaneously the level set method for computing moving interfaces has impacted many areas of mathematics, engineering and applied science, including image science. I will try to give an overview of the basics and recent advances in these topics.
During the past two decades variational and partial differential based methods have greatly affected the fields of image processing, computer vision and graphics (image science in general). Almost simultaneously the level set method for computing moving interfaces has impacted many areas of mathematics, engineering and applied science, including image science. I will try to give an overview of the basics and recent advances in these topics.
Lectures
13:30-15:00 Room #123 (Graduate School of Math. Sci. Bldg.)
Dietmar Hoemberg (Berlin Technical University)
Optimal control of semilinear parabolic equations and an application to laser material treatments
Dietmar Hoemberg (Berlin Technical University)
Optimal control of semilinear parabolic equations and an application to laser material treatments
[ Abstract ]
Many technological processes can be described by partial differential equations. For many years the role of industrial mathematics was mainly to try to understand the respective process, to derive an appropriate PDE or ODE model for it and to simulate it using, e.g., a finite-element code.
However, the ultimate goal usually is to try to optimize the process. Mathematically, this requires the solution of an optimal control problem, i.e., a constrained nonlinear optimization problem in which the constraints are PDEs.
The goal of these two talks is to give an overview of the theory and numerics of optimal control of PDEs for the case of parabolic state equations including an application in laser material treatments. More specifically, I will focus on the following topics.
Many technological processes can be described by partial differential equations. For many years the role of industrial mathematics was mainly to try to understand the respective process, to derive an appropriate PDE or ODE model for it and to simulate it using, e.g., a finite-element code.
However, the ultimate goal usually is to try to optimize the process. Mathematically, this requires the solution of an optimal control problem, i.e., a constrained nonlinear optimization problem in which the constraints are PDEs.
The goal of these two talks is to give an overview of the theory and numerics of optimal control of PDEs for the case of parabolic state equations including an application in laser material treatments. More specifically, I will focus on the following topics.
2007/02/20
Lectures
10:30-17:20 Room #123 (Graduate School of Math. Sci. Bldg.)
Erwin Bolthausen (University of Zurich) 10:30-12:00
Exit distributions for random walks in random environments
Erwin Bolthausen (University of Zurich) 14:00-15:30
Quasi one-dimensional random walks in random environments
田村要造 (慶応大理工) 15:50-16:30
Large deviation principle for currents generated by stochasticline integrals
on compact Riemannian manifolds (joint work with S. Kusuoka and K. Kuwada)
長田博文 (九大数理) 16:40-17:20
Interacting Brownian motions related to Ginibre random point field
Erwin Bolthausen (University of Zurich) 10:30-12:00
Exit distributions for random walks in random environments
Erwin Bolthausen (University of Zurich) 14:00-15:30
Quasi one-dimensional random walks in random environments
田村要造 (慶応大理工) 15:50-16:30
Large deviation principle for currents generated by stochasticline integrals
on compact Riemannian manifolds (joint work with S. Kusuoka and K. Kuwada)
長田博文 (九大数理) 16:40-17:20
Interacting Brownian motions related to Ginibre random point field
Tuesday Seminar of Analysis
16:30-18:00 Room #117 (Graduate School of Math. Sci. Bldg.)
Patrick G¥'erard (パリ南大学)
On the dynamics of the Gross-Pitaevskii equation
Patrick G¥'erard (パリ南大学)
On the dynamics of the Gross-Pitaevskii equation
2007/02/17
Infinite Analysis Seminar Tokyo
13:30-16:00 Room #117 (Graduate School of Math. Sci. Bldg.)
阿部 友紀 (上智理工数学) 13:30-14:30
Finite-dimensional representations of the small quantum algebras
Combinatorics of Young walls and crystal bases
阿部 友紀 (上智理工数学) 13:30-14:30
Finite-dimensional representations of the small quantum algebras
[ Abstract ]
量子代数は定義にパラメーターを一つ含み、量子代数の表現論は、
そのパラメーターが1のべき根であるか、そうでないかによって大きく異なる。
さらに、1のべき根の場合は、Lusztig氏によって定義された「制限型量子代数」と、
De Conini-Kac氏らによって定義された「非制限型量子代数」の2種類が存在し、
それぞれ表現論が異なる。
また、制限型量子代数は、「小型量子代数(=small quantum algebra)」と
呼ばれる真部分代数を含み、その表現論は、制限型量子代数と非制限型量子代数の
どちらの表現論においても重要な役割を果たしている。
今回の講演では、主に以下の3点について説明したい:
●小型量子代数が、非制限型量子代数のある商代数と同型になることを、
有限型とループ型の場合に示す。
●A, B, C, D, G型の小型量子代数の有限次元既約表現を、
Schnizer表現の部分表現として構成する。
●A型の小型ループ量子代数のevaluation表現の性質を調べる。
Seok-Jin Kang (Seoul National University) 15:00-16:00量子代数は定義にパラメーターを一つ含み、量子代数の表現論は、
そのパラメーターが1のべき根であるか、そうでないかによって大きく異なる。
さらに、1のべき根の場合は、Lusztig氏によって定義された「制限型量子代数」と、
De Conini-Kac氏らによって定義された「非制限型量子代数」の2種類が存在し、
それぞれ表現論が異なる。
また、制限型量子代数は、「小型量子代数(=small quantum algebra)」と
呼ばれる真部分代数を含み、その表現論は、制限型量子代数と非制限型量子代数の
どちらの表現論においても重要な役割を果たしている。
今回の講演では、主に以下の3点について説明したい:
●小型量子代数が、非制限型量子代数のある商代数と同型になることを、
有限型とループ型の場合に示す。
●A, B, C, D, G型の小型量子代数の有限次元既約表現を、
Schnizer表現の部分表現として構成する。
●A型の小型ループ量子代数のevaluation表現の性質を調べる。
Combinatorics of Young walls and crystal bases
[ Abstract ]
We introduce combinatorics of Young walls and give a realization of crystal bases in terms of reduced Young walls. We also discuss their connection with representation theory of Hecke algebras.
We introduce combinatorics of Young walls and give a realization of crystal bases in terms of reduced Young walls. We also discuss their connection with representation theory of Hecke algebras.
2007/02/16
Colloquium
16:30-17:30 Room #123 (Graduate School of Math. Sci. Bldg.)
松本幸夫 (東京大学・大学院数理科学研究科)
トポロジーと私の思い出
松本幸夫 (東京大学・大学院数理科学研究科)
トポロジーと私の思い出
[ Abstract ]
大学院に入ったのが1967年ですから、ちょうど40年前の
ことになります。それから「多様体のトポロジー」の分野で研究を
してきましたが、この40年間にトポロジーもずいぶん変化した
ように思います。自分の思い出話を交えて、その変化の様子をお話
できればと思います。私的な観点のものですので、それほど大所
高所からの話ではありません。
大学院に入ったのが1967年ですから、ちょうど40年前の
ことになります。それから「多様体のトポロジー」の分野で研究を
してきましたが、この40年間にトポロジーもずいぶん変化した
ように思います。自分の思い出話を交えて、その変化の様子をお話
できればと思います。私的な観点のものですので、それほど大所
高所からの話ではありません。
Applied Analysis
15:00-16:00 Room #056 (Graduate School of Math. Sci. Bldg.)
Ratnasingham SHIVAJI (ミシシッピ州立大学)
Multiple positive solutions for classes of elliptic systems with combined nonlinear effects
Ratnasingham SHIVAJI (ミシシッピ州立大学)
Multiple positive solutions for classes of elliptic systems with combined nonlinear effects
[ Abstract ]
We study the existence of multiple positive solutions to systems of the form
-\\Delta u = \\lambda f(v)
-\\Delta v = \\lambda g(u)
in a bounded domain in R^N under the Dirichlet boundary conditions. Here f, g belong to a class of positive functions having a combined sublinear effect at infinity. Our result also easily extends to the corresponding p-Laplacian systems. We prove our results by the method of sub and super solutions.
We study the existence of multiple positive solutions to systems of the form
-\\Delta u = \\lambda f(v)
-\\Delta v = \\lambda g(u)
in a bounded domain in R^N under the Dirichlet boundary conditions. Here f, g belong to a class of positive functions having a combined sublinear effect at infinity. Our result also easily extends to the corresponding p-Laplacian systems. We prove our results by the method of sub and super solutions.
2007/02/01
Lectures
15:00-16:00 Room #118 (Graduate School of Math. Sci. Bldg.)
Lassi Paivarinta (Helsinki University of Technology, Finland)
On Calderon's inverse conductivity problem in the plane.
Lassi Paivarinta (Helsinki University of Technology, Finland)
On Calderon's inverse conductivity problem in the plane.
Lectures
16:15-17:15 Room #118 (Graduate School of Math. Sci. Bldg.)
Nuuti Huyvonen (Helsinki University of Technology,
Finland)
Locating transparent cavities in optical absorption and scattering
tomography
Nuuti Huyvonen (Helsinki University of Technology,
Finland)
Locating transparent cavities in optical absorption and scattering
tomography
2007/01/31
Seminar on Mathematics for various disciplines
10:30-11:30 Room #123 (Graduate School of Math. Sci. Bldg.)
小谷正博 (学習院大学)
ガラスに吸着した色素分子の一分子観察――ランダムウォークと拡散――
http://coe.math.sci.hokudai.ac.jp/sympo/various/index.html
小谷正博 (学習院大学)
ガラスに吸着した色素分子の一分子観察――ランダムウォークと拡散――
[ Abstract ]
夜空の星を望遠鏡で観察できる。星の望遠鏡観察から星の一生、ひいては宇宙の成因まで議論できる。同様に蛍光を使えば色素分子一匹を顕微鏡で見ることができる。一分子観察は個々の分子の挙動が見えるので、分子レベルでの確率的な過程を調べる手段、分子のおかれた環境の不均一を研究する手段になることが認識されてきた。
ガラスの上に希薄に吸着した蛍光性の色素を使って分子の表面拡散をしらべた。平均自乗偏位は時間に比例して増大するようにみえ、これから拡散係数を見積もることができる。
このようにして求めた拡散係数は測定環境の湿度に大きく依存することがわかった。こうして、問題はガラス表面にある数ナノメートルの吸着水のなかでの色素分子の運動の議論になってきた。拡散係数に場所ムラはあるのか、時間依存性はあるのか、実験は進行中である。
[ Reference URL ]夜空の星を望遠鏡で観察できる。星の望遠鏡観察から星の一生、ひいては宇宙の成因まで議論できる。同様に蛍光を使えば色素分子一匹を顕微鏡で見ることができる。一分子観察は個々の分子の挙動が見えるので、分子レベルでの確率的な過程を調べる手段、分子のおかれた環境の不均一を研究する手段になることが認識されてきた。
ガラスの上に希薄に吸着した蛍光性の色素を使って分子の表面拡散をしらべた。平均自乗偏位は時間に比例して増大するようにみえ、これから拡散係数を見積もることができる。
このようにして求めた拡散係数は測定環境の湿度に大きく依存することがわかった。こうして、問題はガラス表面にある数ナノメートルの吸着水のなかでの色素分子の運動の議論になってきた。拡散係数に場所ムラはあるのか、時間依存性はあるのか、実験は進行中である。
http://coe.math.sci.hokudai.ac.jp/sympo/various/index.html
Number Theory Seminar
15:15-18:45 Room #117 (Graduate School of Math. Sci. Bldg.)
Dennis Eriksson (東大数理/Paris) 15:15-16:15
Towards a proof of a metrized Deligne-Riemann-Roch theorem
小林 真一 (名古屋大学多元数理) 16:30-17:30
CM楕円曲線の超特異点における2変数p進L関数
(A two variable p-adic L-function for CM elliptic curves at supersingular primes)
Frans Oort (Utrecht) 17:45-18:45
Irreducibility of strata and leaves in the moduli space of abelian varieties
Dennis Eriksson (東大数理/Paris) 15:15-16:15
Towards a proof of a metrized Deligne-Riemann-Roch theorem
小林 真一 (名古屋大学多元数理) 16:30-17:30
CM楕円曲線の超特異点における2変数p進L関数
(A two variable p-adic L-function for CM elliptic curves at supersingular primes)
Frans Oort (Utrecht) 17:45-18:45
Irreducibility of strata and leaves in the moduli space of abelian varieties
Lectures
16:30-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)
Li Daqian (復旦大学)
Exact Controllability and Exact Observability for Quasilinear Hyperbolic Systems
Li Daqian (復旦大学)
Exact Controllability and Exact Observability for Quasilinear Hyperbolic Systems
Seminar on Probability and Statistics
16:20-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)
西山 慶彦 (京都大学経済研究所)
A Sequential Unit Root Test
http://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2006/18.html
西山 慶彦 (京都大学経済研究所)
A Sequential Unit Root Test
[ Abstract ]
It is well known that conventional unit root tests such as Dickey=Fuller and its variants do not have good power properties when sample size is not large. Lai and Siegmund (1983, AS) proved that OLS estimator of the AR(1) coefficient is asymptotically normally distributed in a sequential framework even if the time series has a unit root unlike the OLS estimator under a standard sampling scheme. We pursue this direction to propose a unit root test under a sequential sampling. The proposed test uses not only the OLS estimator of the AR(1) coefficient, which is asymptotically normal, but also the stopping time to construct the critical region, anticipating a better power property. We obtain analytic expressions of the joint distribution of the two statistics as well as its marginals under the null. We also consider the distribution of the statistics under local alternatives. The properties of the stopping time, to the best of our knowledge, have not been studied in the unit root literature. We calculate its expectation and variance.
[ Reference URL ]It is well known that conventional unit root tests such as Dickey=Fuller and its variants do not have good power properties when sample size is not large. Lai and Siegmund (1983, AS) proved that OLS estimator of the AR(1) coefficient is asymptotically normally distributed in a sequential framework even if the time series has a unit root unlike the OLS estimator under a standard sampling scheme. We pursue this direction to propose a unit root test under a sequential sampling. The proposed test uses not only the OLS estimator of the AR(1) coefficient, which is asymptotically normal, but also the stopping time to construct the critical region, anticipating a better power property. We obtain analytic expressions of the joint distribution of the two statistics as well as its marginals under the null. We also consider the distribution of the statistics under local alternatives. The properties of the stopping time, to the best of our knowledge, have not been studied in the unit root literature. We calculate its expectation and variance.
http://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2006/18.html
2007/01/30
Tuesday Seminar on Topology
16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)
John F. Duncan (Harvard University)
Elliptic genera and some finite groups
John F. Duncan (Harvard University)
Elliptic genera and some finite groups
[ Abstract ]
Recent developments in the representation theory of sporadic groups
suggest new formulations of `moonshine' in which Jacobi forms take on the
role played by modular forms in the monstrous case. On the other hand,
Jacobi forms arise naturally in the study of elliptic genera. We review
the use of vertex algebra as a tool for constructing the elliptic genus of
a suitable vector bundle, and illustrate connections between this and
vertex algebraic representations of certain sporadic simple groups.
Recent developments in the representation theory of sporadic groups
suggest new formulations of `moonshine' in which Jacobi forms take on the
role played by modular forms in the monstrous case. On the other hand,
Jacobi forms arise naturally in the study of elliptic genera. We review
the use of vertex algebra as a tool for constructing the elliptic genus of
a suitable vector bundle, and illustrate connections between this and
vertex algebraic representations of certain sporadic simple groups.
Lectures
16:30-18:00 Room #118 (Graduate School of Math. Sci. Bldg.)
Jerome Le Rousseau (Laboratoire d'Analyse Topologie Probabilit\'es
Universit\'e de Provence / CNRS)
Controllability of parabolic equations with non-smooth coefficients by means of global Carleman estimates
Jerome Le Rousseau (Laboratoire d'Analyse Topologie Probabilit\'es
Universit\'e de Provence / CNRS)
Controllability of parabolic equations with non-smooth coefficients by means of global Carleman estimates
[ Abstract ]
We shall review the different concepts of controllability for parabolic equations and a fix-point method to achieve null-controllability of classes of semilinear equations. It is mainly based on observability inequalities and a precise knowledge of the observability constant. These inequalities are obtained by means of global Carleman estimates. We shall review their derivations and how they can be obtained in the case of non-smooth coefficients. We shall also present some open questions.
Part of this work is in collaboration with Assia Benabdallah and Yves Dermenjian (also from Universit\\'e de Provence).
We shall review the different concepts of controllability for parabolic equations and a fix-point method to achieve null-controllability of classes of semilinear equations. It is mainly based on observability inequalities and a precise knowledge of the observability constant. These inequalities are obtained by means of global Carleman estimates. We shall review their derivations and how they can be obtained in the case of non-smooth coefficients. We shall also present some open questions.
Part of this work is in collaboration with Assia Benabdallah and Yves Dermenjian (also from Universit\\'e de Provence).
Infinite Analysis Seminar Tokyo
14:00-15:00 Room #123 (Graduate School of Math. Sci. Bldg.)
Michael Lashkevich (Landau Institute)
Scaling limits for the SOS models and bosonization
Michael Lashkevich (Landau Institute)
Scaling limits for the SOS models and bosonization
[ Abstract ]
Two different scaling limits in the SOS models are considered. The scaling limits of the bosonic construction for form factors provide form factors of some classes of operators in the scaling SOS/RSOS models and the sine-Gordon model.
Two different scaling limits in the SOS models are considered. The scaling limits of the bosonic construction for form factors provide form factors of some classes of operators in the scaling SOS/RSOS models and the sine-Gordon model.
2007/01/29
Lectures
16:30-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)
Li Daqian (復旦大学)
Exact Controllability and Exact Observability for Quasilinear Hyperbolic Systems
Li Daqian (復旦大学)
Exact Controllability and Exact Observability for Quasilinear Hyperbolic Systems
Seminar on Geometric Complex Analysis
10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)
松島敏夫 (石川工業高専)
Radial cluster set of a bounded holomorphic map in the unit ball of C^n
松島敏夫 (石川工業高専)
Radial cluster set of a bounded holomorphic map in the unit ball of C^n
2007/01/27
Infinite Analysis Seminar Tokyo
13:30-16:00 Room #117 (Graduate School of Math. Sci. Bldg.)
清水 寧 (立命館理工物理) 13:30-14:30
マイクロクラスターの特異なダイナミクス
例外型アフィンリー環$D_4^{(3)}$に付随するキリロフ・レシェティヒン加群の結晶基底に関する話題
清水 寧 (立命館理工物理) 13:30-14:30
マイクロクラスターの特異なダイナミクス
[ Abstract ]
数十個から数千個の原子からなる有限多体系であるマイクロクラスターは、表面原子と内部の原子という異なる環境にある構成原子からなる空間的に不均一な系である。これが原因となり、マイクロクラスターは静的な面においても動的な面においても結晶やアモルファスのバルクとは大きく異なる特異な振る舞いを見せることが知られている。その一例として、神戸大学保田らの実験グループにより確認されているナノ金属マイクロクラスター内における構成原子の非常に速い拡散現象(急速合金化)を取り上げ、このダイナミクスに関する我々の数値シミュレーションに基づく結果を紹介する。得られたいくつかの数値結果の解釈を通じ、「動的に維持されている物質」としてのマイクロクラスターの一側面を示す。
山田 大輔 (東大数理) 15:00-16:00数十個から数千個の原子からなる有限多体系であるマイクロクラスターは、表面原子と内部の原子という異なる環境にある構成原子からなる空間的に不均一な系である。これが原因となり、マイクロクラスターは静的な面においても動的な面においても結晶やアモルファスのバルクとは大きく異なる特異な振る舞いを見せることが知られている。その一例として、神戸大学保田らの実験グループにより確認されているナノ金属マイクロクラスター内における構成原子の非常に速い拡散現象(急速合金化)を取り上げ、このダイナミクスに関する我々の数値シミュレーションに基づく結果を紹介する。得られたいくつかの数値結果の解釈を通じ、「動的に維持されている物質」としてのマイクロクラスターの一側面を示す。
例外型アフィンリー環$D_4^{(3)}$に付随するキリロフ・レシェティヒン加群の結晶基底に関する話題
[ Abstract ]
可解格子模型の1点関数を計算するために、Kang-柏原-Misra-三輪-中島-中屋敷らにより、``完全結晶"という概念が導入された。これはアフィンリー環$\\mathfrak{g}$の量子展開代数$U'_q(\\mathfrak{g})$に付随する結晶基底の中で、非常に良い性質をもつものである。完全結晶の存在性は、幾つかの場合に証明されたが、その後の研究の中で新たに発見され続けている。ところが、任意の既約な有限次元$U'_q(\\mathfrak{g})$-加群が必ずしも結晶基底をもつとは限らない。そこで次の問題を考えたい。
問題:「結晶基底をもつ既約な有限次元$U'_q(\\mathfrak{g})$-加群を全て見つけよ。」
この問題にアプローチするために、キリロフ・レシェティヒン加群$W_s^{(r)}$ (以下略してKR加群)を研究したい。これはアフィンリー環のディンキン図形の頂点$0$を除く頂点の番号$r$と、任意の正整数$s$の組によってパラメトライズされる。KR加群に関して、``フェルミ型公式''に起源をもつ以下の予想がある。尚, 現在までにこの予想の反例は見つかっていない。
予想:「KR加群$W_s^{(r)}$は結晶基底をもつ。
さらに$s$が$t_r:=max(1,2/(\\alpha_r \\vert \\alpha_r))$の倍数ならば、KR加群$W_s^{(r)}$の結晶基底$B^{r,s}$は、レベル$s/t_r$の完全結晶である。ただし, $(\\cdot \\vert \\cdot)$はウェイト格子上の標準線形形式。」
我々は, 例外型アフィンリー環$D_4^{(3)}$のKR加群$W_s^{(1)}$と$W_1^{(2)}$について、上の予想が正しいことを示した。その応用として、超離散可積分系の重要な例である「箱玉系」を構成し、そこに現れるソリトンの散乱則を表現論的に記述した。
前回の講演では、$U'_q(D_4^{(3)})$-加群の結晶基底に関する組合せ論的な部分を話した。今回の講演ではその表現論的な部分を解説する。
可解格子模型の1点関数を計算するために、Kang-柏原-Misra-三輪-中島-中屋敷らにより、``完全結晶"という概念が導入された。これはアフィンリー環$\\mathfrak{g}$の量子展開代数$U'_q(\\mathfrak{g})$に付随する結晶基底の中で、非常に良い性質をもつものである。完全結晶の存在性は、幾つかの場合に証明されたが、その後の研究の中で新たに発見され続けている。ところが、任意の既約な有限次元$U'_q(\\mathfrak{g})$-加群が必ずしも結晶基底をもつとは限らない。そこで次の問題を考えたい。
問題:「結晶基底をもつ既約な有限次元$U'_q(\\mathfrak{g})$-加群を全て見つけよ。」
この問題にアプローチするために、キリロフ・レシェティヒン加群$W_s^{(r)}$ (以下略してKR加群)を研究したい。これはアフィンリー環のディンキン図形の頂点$0$を除く頂点の番号$r$と、任意の正整数$s$の組によってパラメトライズされる。KR加群に関して、``フェルミ型公式''に起源をもつ以下の予想がある。尚, 現在までにこの予想の反例は見つかっていない。
予想:「KR加群$W_s^{(r)}$は結晶基底をもつ。
さらに$s$が$t_r:=max(1,2/(\\alpha_r \\vert \\alpha_r))$の倍数ならば、KR加群$W_s^{(r)}$の結晶基底$B^{r,s}$は、レベル$s/t_r$の完全結晶である。ただし, $(\\cdot \\vert \\cdot)$はウェイト格子上の標準線形形式。」
我々は, 例外型アフィンリー環$D_4^{(3)}$のKR加群$W_s^{(1)}$と$W_1^{(2)}$について、上の予想が正しいことを示した。その応用として、超離散可積分系の重要な例である「箱玉系」を構成し、そこに現れるソリトンの散乱則を表現論的に記述した。
前回の講演では、$U'_q(D_4^{(3)})$-加群の結晶基底に関する組合せ論的な部分を話した。今回の講演ではその表現論的な部分を解説する。
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