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過去の記録
過去の記録 ～07/26｜本日 07/27 | 今後の予定 07/28～
講演会
16:30-18:00 数理科学研究科棟(駒場) 118号室
伊東一文 氏 (大学院数理科学研究科)
Fractional Evolution Equations and Applications 1
伊東一文 氏 (大学院数理科学研究科)
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
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
奥間智弘 氏 (山形大学地域教育文化学部)
スプライス商特異点について
奥間智弘 氏 (山形大学地域教育文化学部)
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
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.
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
中川淳一 氏 (新日本製鐵(株)技術開発本部)
製鐵プロセスにおける数学
中川淳一 氏 (新日本製鐵(株)技術開発本部)
製鐵プロセスにおける数学
2010年01月14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
Marius Junge 氏 (Univ. Illinois, Urbana-Champaign)
Applications of operator algebras in Quantum information theory
Marius Junge 氏 (Univ. Illinois, Urbana-Champaign)
Applications of operator algebras in Quantum information theory
2010年01月13日(水)
講演会
16:45-17:45 数理科学研究科棟(駒場) 128号室
Felix Rubin 氏 (Zurich 大学)
Scaled limit for the largest eigenvalue from the generalized Cauchy ensemble
Felix Rubin 氏 (Zurich 大学)
Scaled limit for the largest eigenvalue from the generalized Cauchy ensemble
講演会
15:30-16:30 数理科学研究科棟(駒場) 128号室
Michael Allman 氏 (Warwick 大学)
Breaking the chain: slow versus fast pulling
Michael Allman 氏 (Warwick 大学)
Breaking the chain: slow versus fast pulling
2010年01月12日(火)
Lie群論・表現論セミナー
16:30-18:00 数理科学研究科棟(駒場) 126号室
西岡斉治 氏 (東京大学大学院数理科学研究科博士課程)
代数的差分方程式の可解性と既約性
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar2010.html#20100112nishioka
西岡斉治 氏 (東京大学大学院数理科学研究科博士課程)
代数的差分方程式の可解性と既約性
[ 講演概要 ]
差分代数の理論を使って,代数的差分方程式の代数函数解や超幾
何函数解の非存在や,存在する場合の特殊解の分類をする。
[ 参考URL ]差分代数の理論を使って,代数的差分方程式の代数函数解や超幾
何函数解の非存在や,存在する場合の特殊解の分類をする。
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar2010.html#20100112nishioka
トポロジー火曜セミナー
16:30-18:30 数理科学研究科棟(駒場) 056号室
Tea: 16:00 - 16:30 コモンルーム
篠原 克寿 氏 (東京大学大学院数理科学研究科) 16:30-17:30
Index problem for generically-wild homoclinic classes in dimension three
On a generalized suspension theorem for directed Fukaya categories
Tea: 16:00 - 16:30 コモンルーム
篠原 克寿 氏 (東京大学大学院数理科学研究科) 16:30-17:30
Index problem for generically-wild homoclinic classes in dimension three
[ 講演概要 ]
In the sphere of non-hyperbolic differentiable dynamical systems, one can construct an example of a homolinic class which does not admit any kind of dominated splittings (a weak form of hyperbolicity) in a robust way. In this talk, we discuss the index (dimension of the unstable manifold) of the periodic points inside such homoclinic classes from a $C^1$-generic viewpoint.
二木 昌宏 氏 (東京大学大学院数理科学研究科) 17:30-18:30In the sphere of non-hyperbolic differentiable dynamical systems, one can construct an example of a homolinic class which does not admit any kind of dominated splittings (a weak form of hyperbolicity) in a robust way. In this talk, we discuss the index (dimension of the unstable manifold) of the periodic points inside such homoclinic classes from a $C^1$-generic viewpoint.
On a generalized suspension theorem for directed Fukaya categories
[ 講演概要 ]
The directed Fukaya category $\\mathrm{Fuk} W$ of exact Lefschetz
fibration $W : X \\to \\mathbb{C}$ proposed by Kontsevich is a
categorification of the Milnor lattice of $W$. This is defined as the
directed $A_\\infty$-category $\\mathrm{Fuk} W = \\mathrm{Fuk}^\\to
\\mathbb{V}$ generated by a distinguished basis $\\mathbb{V}$ of
vanishing cycles.
Recently Seidel has proved that this is stable under the suspension $W
+ u^2$ as a consequence of his foundational work on the directed
Fukaya category. We generalize his suspension theorem to the $W + u^d$
case by considering partial tensor product $\\mathrm{Fuk} W \\otimes'
\\mathcal{A}_{d-1}$, where $\\mathcal{A}_{d-1}$ is the category
corresponding to the $A_n$-type quiver. This also generalizes a recent
work by the author with Kazushi Ueda.
The directed Fukaya category $\\mathrm{Fuk} W$ of exact Lefschetz
fibration $W : X \\to \\mathbb{C}$ proposed by Kontsevich is a
categorification of the Milnor lattice of $W$. This is defined as the
directed $A_\\infty$-category $\\mathrm{Fuk} W = \\mathrm{Fuk}^\\to
\\mathbb{V}$ generated by a distinguished basis $\\mathbb{V}$ of
vanishing cycles.
Recently Seidel has proved that this is stable under the suspension $W
+ u^2$ as a consequence of his foundational work on the directed
Fukaya category. We generalize his suspension theorem to the $W + u^d$
case by considering partial tensor product $\\mathrm{Fuk} W \\otimes'
\\mathcal{A}_{d-1}$, where $\\mathcal{A}_{d-1}$ is the category
corresponding to the $A_n$-type quiver. This also generalizes a recent
work by the author with Kazushi Ueda.
2010年01月08日(金)
談話会・数理科学講演会
16:30-17:30 数理科学研究科棟(駒場) 123号室
お茶&Coffee&お菓子: 16:00～16:30 (コモンルーム)
大島利雄 氏 (東京大学大学院数理科学研究科)
特殊関数とFuchs型常微分方程式
お茶&Coffee&お菓子: 16:00～16:30 (コモンルーム)
大島利雄 氏 (東京大学大学院数理科学研究科)
特殊関数とFuchs型常微分方程式
[ 講演概要 ]
岩波全書の数学公式集III「特殊関数」の大部分はGaussの超幾何関数とその特殊化のBessel関数やLegendre多項式などで占められている。この超幾何関数についての最も重要な基本結果は1での値を与えるGaussの和公式とRiemann schemeによる特徴付けとであろう。この関数は一般超幾何関数やJordan-Pochhammer方程式へ、またHeun方程式からPainleve方程式へという解析、さらにAppell,Gelfand-青本,Heckman-Opdamによる多変数化という3つの方向の発展がある。講演ではこれらを含む統一的な理解、Riemann schemeの一般化とuniversal modelの存在定理(Deligne-Katz-Simpson問題)、接続公式(Gaussの和公式の一般化)、無限次元Kac-Moody Weyl群の作用について解説し、特異点の合流、積分表示、ベキ級数表示などについても述べたい。結果は構成的 でコンピュータ・プログラムで実現できる。
岩波全書の数学公式集III「特殊関数」の大部分はGaussの超幾何関数とその特殊化のBessel関数やLegendre多項式などで占められている。この超幾何関数についての最も重要な基本結果は1での値を与えるGaussの和公式とRiemann schemeによる特徴付けとであろう。この関数は一般超幾何関数やJordan-Pochhammer方程式へ、またHeun方程式からPainleve方程式へという解析、さらにAppell,Gelfand-青本,Heckman-Opdamによる多変数化という3つの方向の発展がある。講演ではこれらを含む統一的な理解、Riemann schemeの一般化とuniversal modelの存在定理(Deligne-Katz-Simpson問題)、接続公式(Gaussの和公式の一般化)、無限次元Kac-Moody Weyl群の作用について解説し、特異点の合流、積分表示、ベキ級数表示などについても述べたい。結果は構成的 でコンピュータ・プログラムで実現できる。
2010年01月07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
Luc Rey-Bellet 氏 (Univ. Massachusetts)
Large deviations, Billiards, and Non-equilibrium Statistical Mechanics
Luc Rey-Bellet 氏 (Univ. Massachusetts)
Large deviations, Billiards, and Non-equilibrium Statistical Mechanics
GCOEセミナー
16:30-18:00 数理科学研究科棟(駒場) 128号室
LucRey-Bellet 氏 (Univ. Massachusetts)
Large deviations, Billiards, and Non-equilibrium Statistical Mechanics
LucRey-Bellet 氏 (Univ. Massachusetts)
Large deviations, Billiards, and Non-equilibrium Statistical Mechanics
[ 講演概要 ]
Large deviations have applications in many aspects of statistical mechanics. New applications for the steady states of non-equilibrium statistical mechanics have emerged during the past ten years and these applications deal with the fluctuations of the entropy production. After discussing some general aspects of entropy production we turn to concrete examples, in particular billiards with and without external forces and discuss large deviations theorems for such systems.
Large deviations have applications in many aspects of statistical mechanics. New applications for the steady states of non-equilibrium statistical mechanics have emerged during the past ten years and these applications deal with the fluctuations of the entropy production. After discussing some general aspects of entropy production we turn to concrete examples, in particular billiards with and without external forces and discuss large deviations theorems for such systems.
2010年01月05日(火)
トポロジー火曜セミナー
16:30-18:30 数理科学研究科棟(駒場) 056号室
Tea: 16:00 - 16:30 コモンルーム
服部 広大 氏 (東京大学大学院数理科学研究科) 16:30-17:30
The volume growth of hyperkaehler manifolds of type $A_{\\infty}$
Tea: 16:00 - 16:30 コモンルーム
服部 広大 氏 (東京大学大学院数理科学研究科) 16:30-17:30
The volume growth of hyperkaehler manifolds of type $A_{\\infty}$
[ 講演概要 ]
Hyperkaehler manifolds of type $A_{\\infty}$ were constructed due to Anderson-Kronheimer-LeBrun and Goto. These manifolds are 4-demensional, noncompact and their homology groups are infinitely generated. We focus on the volume growth of these hyperkaehler metrics. Here, the volume growth is asymptotic behavior of the volume of a ball of radius $r0$ with the center fixed. There are known examples of hyperkaehler manifolds whose volume growth is $r^4$ (ALE space) or $r^3$ (Taub-NUT space). In this talk we show that there exists a hyperkaehler manifold of type $A_{\\infty}$ whose volume growth is $r^c$ for a given $3
松尾 信一郎 氏 (東京大学大学院数理科学研究科) 17:30-18:30
On the Runge theorem for instantons
Hyperkaehler manifolds of type $A_{\\infty}$ were constructed due to Anderson-Kronheimer-LeBrun and Goto. These manifolds are 4-demensional, noncompact and their homology groups are infinitely generated. We focus on the volume growth of these hyperkaehler metrics. Here, the volume growth is asymptotic behavior of the volume of a ball of radius $r0$ with the center fixed. There are known examples of hyperkaehler manifolds whose volume growth is $r^4$ (ALE space) or $r^3$ (Taub-NUT space). In this talk we show that there exists a hyperkaehler manifold of type $A_{\\infty}$ whose volume growth is $r^c$ for a given $3
On the Runge theorem for instantons
[ 講演概要 ]
A classical theorem of Runge in complex analysis asserts that a
meromorphic function on a domain in the Riemann sphere can be
approximated, over compact subsets, by rational functions, that is,
meromorphic functions on the Riemann sphere.
This theorem can be paraphrased by saying that any solution of the
Cauchy-Riemann equations on a domain in the Riemann sphere can be
approximated, over compact subsets, by global solutions.
In this talk we will present an analogous result in which the
Cauchy-Riemann equations on Riemann surfaces are replaced by the
Yang-Mills instanton equations on oriented 4-manifolds.
We will also mention that the Runge theorem for instantons can be
applied to develop Yang-Mills gauge theory on open 4-manifolds.
A classical theorem of Runge in complex analysis asserts that a
meromorphic function on a domain in the Riemann sphere can be
approximated, over compact subsets, by rational functions, that is,
meromorphic functions on the Riemann sphere.
This theorem can be paraphrased by saying that any solution of the
Cauchy-Riemann equations on a domain in the Riemann sphere can be
approximated, over compact subsets, by global solutions.
In this talk we will present an analogous result in which the
Cauchy-Riemann equations on Riemann surfaces are replaced by the
Yang-Mills instanton equations on oriented 4-manifolds.
We will also mention that the Runge theorem for instantons can be
applied to develop Yang-Mills gauge theory on open 4-manifolds.
2009年12月25日(金)
講演会
17:00-18:00 数理科学研究科棟(駒場) 370号室
Academician T. Sh. Kalmenov 氏 (Research Centre of Physics and Mathematics Almaty, Kazakhstan)
A criterion for the strong solvability of the mixed Cauchy problem for the Laplace equation
Academician T. Sh. Kalmenov 氏 (Research Centre of Physics and Mathematics Almaty, Kazakhstan)
A criterion for the strong solvability of the mixed Cauchy problem for the Laplace equation
2009年12月24日(木)
数理人口学・数理生物学セミナー
16:00-17:30 数理科学研究科棟(駒場) 123号室
堀内 四郎 氏 (The City University of New York, Hunter College)
Decomposition分析:趨勢データ分析の新しい枠組とアプローチ
http://shiro_horiuchi.homestead.com/homepage.html
堀内 四郎 氏 (The City University of New York, Hunter College)
Decomposition分析:趨勢データ分析の新しい枠組とアプローチ
[ 講演概要 ]
A demographic measure is often expressed as a deterministic or stochastic function of multiple variables (covariates), and a general problem (the decomposition problem) is to assess contributions of individual covariates to a difference in the demographic measure (dependent variable) between two populations.
We propose a method of decomposition analysis based on an assumption that covariates change continuously along an actual or hypothetical dimension. This assumption leads to a general model that logically justifi es the additivity of covariate effects and the elimination of interaction terms, even if the dependent variable itself is a nonadditive function.
A comparison with earlier methods illustrates other practical advantages of the method: in addition to an absence of residuals or interaction terms, the method can easily handle a large number of covariates and does not require a logically meaningful ordering of covariates. Two empirical examples show that the method can be applied fl exibly to a wide variety of decomposition problems.
[ 参考URL ]A demographic measure is often expressed as a deterministic or stochastic function of multiple variables (covariates), and a general problem (the decomposition problem) is to assess contributions of individual covariates to a difference in the demographic measure (dependent variable) between two populations.
We propose a method of decomposition analysis based on an assumption that covariates change continuously along an actual or hypothetical dimension. This assumption leads to a general model that logically justifi es the additivity of covariate effects and the elimination of interaction terms, even if the dependent variable itself is a nonadditive function.
A comparison with earlier methods illustrates other practical advantages of the method: in addition to an absence of residuals or interaction terms, the method can easily handle a large number of covariates and does not require a logically meaningful ordering of covariates. Two empirical examples show that the method can be applied fl exibly to a wide variety of decomposition problems.
http://shiro_horiuchi.homestead.com/homepage.html
2009年12月22日(火)
Lie群論・表現論セミナー
16:30-18:00 数理科学研究科棟(駒場) 126号室
西山享 氏 (青山学院大学)
既約表現の隨伴多様体は余次元1で連結か?--- 証明の破綻とその背景
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html
西山享 氏 (青山学院大学)
既約表現の隨伴多様体は余次元1で連結か?--- 証明の破綻とその背景
[ 講演概要 ]
既約 Harish-Chandra $ (g, K) $ 加群の原始イデアルの隨伴多様体が既約であって、ただ一つの冪零隨伴軌道 $ O^G $ の閉包になることはよく知られている(Joseph, Borho)。
一方、HC加群の隨伴多様体は必ずしも既約でないが、その既約成分は $ O^G $ の $ K $-等質ラグランジュ部分多様体の閉包になる。
それらの既約成分は余次元1で連結であることをいくつかの集会で報告したが、その証明には初等的な誤りがあった。セミナーでは、証明の元になった Vogan の定理の紹介(もちろん間違っていない)と、それを拡張する際になぜ証明が破綻するかについてお話しする。(今のところ証明修復の目処は立っていない。)
[ 参考URL ]既約 Harish-Chandra $ (g, K) $ 加群の原始イデアルの隨伴多様体が既約であって、ただ一つの冪零隨伴軌道 $ O^G $ の閉包になることはよく知られている(Joseph, Borho)。
一方、HC加群の隨伴多様体は必ずしも既約でないが、その既約成分は $ O^G $ の $ K $-等質ラグランジュ部分多様体の閉包になる。
それらの既約成分は余次元1で連結であることをいくつかの集会で報告したが、その証明には初等的な誤りがあった。セミナーでは、証明の元になった Vogan の定理の紹介(もちろん間違っていない)と、それを拡張する際になぜ証明が破綻するかについてお話しする。(今のところ証明修復の目処は立っていない。)
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html
作用素環セミナー
14:40-18:00 数理科学研究科棟(駒場) 128号室
谷本溶 氏 (Univ. Roma ``Tor Vergata'') 14:40-16:10
Symmetric representations of the group of diffeomorphisms of $\\mathbb R$
David Kerr 氏 (Texas A&M Univ.) 16:30-18:00
Topological entropy for actions of sofic groups
谷本溶 氏 (Univ. Roma ``Tor Vergata'') 14:40-16:10
Symmetric representations of the group of diffeomorphisms of $\\mathbb R$
David Kerr 氏 (Texas A&M Univ.) 16:30-18:00
Topological entropy for actions of sofic groups
トポロジー火曜セミナー
16:30-18:00 数理科学研究科棟(駒場) 056号室
Tea: 16:00 - 16:30 コモンルーム
寺杣 友秀 氏 (東京大学大学院数理科学研究科)
Relative DG-category, mixed elliptic motives and elliptic polylog
Tea: 16:00 - 16:30 コモンルーム
寺杣 友秀 氏 (東京大学大学院数理科学研究科)
Relative DG-category, mixed elliptic motives and elliptic polylog
[ 講演概要 ]
We consider a full subcategory of
mixed motives generated by an elliptic curve
over a field, which is called the category of
mixed elliptic motives. We introduce a DG
Hopf algebra such that the categroy of
mixed elliptic motives is equal to that of
comodules over it. For the construction, we
use the notion of relative DG-category with
respect to GL(2). As an application, we construct
an mixed elliptic motif associated to
the elliptic polylog. It is a joint work with
Kenichiro Kimura.
We consider a full subcategory of
mixed motives generated by an elliptic curve
over a field, which is called the category of
mixed elliptic motives. We introduce a DG
Hopf algebra such that the categroy of
mixed elliptic motives is equal to that of
comodules over it. For the construction, we
use the notion of relative DG-category with
respect to GL(2). As an application, we construct
an mixed elliptic motif associated to
the elliptic polylog. It is a joint work with
Kenichiro Kimura.
東京無限可積分系セミナー
10:00-14:00 数理科学研究科棟(駒場) 056号室
岩尾 慎介 氏 (東大数理) 10:00-11:00
離散周期KP方程式の簡約と、初期値問題の解の構成
Laplacian on graphs: Examples from physics
岩尾 慎介 氏 (東大数理) 10:00-11:00
離散周期KP方程式の簡約と、初期値問題の解の構成
[ 講演概要 ]
様々に簡約された離散周期KP方程式に対して、スペクトル曲線を用いた逆散乱解法を考える。 このとき、簡約の種類によっては、超楕円とは限らない代数曲線が多数あらわれてくる。 本講演では、簡約周期KP方程式の初期値問題の解を構成する方法を紹介する。この方法はFayの恒等式を用いない構成的なもので、わかりやすいものである。
Y. Avishai 氏 (Ben Gurion University) 13:00-14:00様々に簡約された離散周期KP方程式に対して、スペクトル曲線を用いた逆散乱解法を考える。 このとき、簡約の種類によっては、超楕円とは限らない代数曲線が多数あらわれてくる。 本講演では、簡約周期KP方程式の初期値問題の解を構成する方法を紹介する。この方法はFayの恒等式を用いない構成的なもので、わかりやすいものである。
Laplacian on graphs: Examples from physics
[ 講演概要 ]
When the Laplacian operator is written as a second order difference operator the physicists refer to it as a tight-binding model. In two dimensions the eigenvalue problem connects the function at a given point to the sum of its values on its nearest neighbors. In numerous physical problems, some of the coefficients are multiplied by phase factors. This problem is amazingly rich and the pattern of eigenvalues E(φ) has a fractal nature known as the Hofstadter butterfly.
I will discuss some of these models and especially concentrate on two problems, which I solved recently, where the vertices of the graphs are located on the sphere S2. The first one corresponds to the famous problem of the Dirac magnetic monopole, while in the second one, the eigenfunctions are two component vectors and the phase factors are replaced by unitary 2x2 matrices. This is relevant to the spin-orbit problem in Physics. In both cases the solutions can be obtained in closed form, and exhibit a beautiful symmetry pattern. Their elucidation requires some special techniques in graph theory. Quite surprisingly, the spectra of the two systems coincide at one symmetry point.
When the Laplacian operator is written as a second order difference operator the physicists refer to it as a tight-binding model. In two dimensions the eigenvalue problem connects the function at a given point to the sum of its values on its nearest neighbors. In numerous physical problems, some of the coefficients are multiplied by phase factors. This problem is amazingly rich and the pattern of eigenvalues E(φ) has a fractal nature known as the Hofstadter butterfly.
I will discuss some of these models and especially concentrate on two problems, which I solved recently, where the vertices of the graphs are located on the sphere S2. The first one corresponds to the famous problem of the Dirac magnetic monopole, while in the second one, the eigenfunctions are two component vectors and the phase factors are replaced by unitary 2x2 matrices. This is relevant to the spin-orbit problem in Physics. In both cases the solutions can be obtained in closed form, and exhibit a beautiful symmetry pattern. Their elucidation requires some special techniques in graph theory. Quite surprisingly, the spectra of the two systems coincide at one symmetry point.
2009年12月21日(月)
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
源 泰幸 氏 (京都大学理学部数学教室)
Ampleness of two-sided tilting complexes
源 泰幸 氏 (京都大学理学部数学教室)
Ampleness of two-sided tilting complexes
[ 講演概要 ]
From the view point of noncommutative algebraic geometry (NCAG),
a two-sided tilting complex is an analog of a line bundle.
In this talk we introduce the notion of ampleness for two-sided
tilting complexes over finite dimensional algebras.
From the view point of NCAG, the Serre functors are considered to be
shifted canonical bundles. We show by examples that the property
of shifted canonical bundle captures some representation theoretic
property of algebras.
From the view point of noncommutative algebraic geometry (NCAG),
a two-sided tilting complex is an analog of a line bundle.
In this talk we introduce the notion of ampleness for two-sided
tilting complexes over finite dimensional algebras.
From the view point of NCAG, the Serre functors are considered to be
shifted canonical bundles. We show by examples that the property
of shifted canonical bundle captures some representation theoretic
property of algebras.
統計数学セミナー
15:00-16:10 数理科学研究科棟(駒場) 128号室
Thomas Simon 氏 (Universite de Lille 1)
Absolute continuity of Ornstein-Uhlenbeck processes
https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2009/13.html
Thomas Simon 氏 (Universite de Lille 1)
Absolute continuity of Ornstein-Uhlenbeck processes
[ 講演概要 ]
Let X be a multidimensional Ornstein-Uhlenbeck process, solution to the S.D.E.
dX = AX + dB
where A is a real nxn matrix and B a Lévy process. We show that when A is non-singular, the law of X_1 is absolutely continuous if and only if the jumping measure of B fulfils a certain geometric condition with respect to A and the Gaussian part of B, which we call the exhaustion property. This optimal criterion is much weaker than for B, which might be very singular and genuinely one-dimensional. The proof uses a certain time derivation procedure and basic arguments from controllability theory.
[ 参考URL ]Let X be a multidimensional Ornstein-Uhlenbeck process, solution to the S.D.E.
dX = AX + dB
where A is a real nxn matrix and B a Lévy process. We show that when A is non-singular, the law of X_1 is absolutely continuous if and only if the jumping measure of B fulfils a certain geometric condition with respect to A and the Gaussian part of B, which we call the exhaustion property. This optimal criterion is much weaker than for B, which might be very singular and genuinely one-dimensional. The proof uses a certain time derivation procedure and basic arguments from controllability theory.
https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2009/13.html
2009年12月18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
山下 浩 氏 (数理システム代表取締役)
数理科学をビジネスに - 最適化とデータマイニングの周辺でⅡ
山下 浩 氏 (数理システム代表取締役)
数理科学をビジネスに - 最適化とデータマイニングの周辺でⅡ
談話会・数理科学講演会
16:30-17:30 数理科学研究科棟(駒場) 002号室
お茶&Coffee&お菓子: 16:00～16:30 (コモンルーム)
小澤登高 氏 (東京大学大学院数理科学研究科)
Dixmierの相似問題
お茶&Coffee&お菓子: 16:00～16:30 (コモンルーム)
小澤登高 氏 (東京大学大学院数理科学研究科)
Dixmierの相似問題
[ 講演概要 ]
群のユニタリ表現に関しては美しい理論があるが, ユニタリでない無限次元表現はまったくとらえがたい対象である. そこで, 群のヒルベルト空間上の表現がいつユニタリ表現と相似(共役ともいう)になるかを問うのがDixmierの相似問題である. この問題は従順性という概念と深い関わりを持ち, 従って群の従順性の代数的な特徴づけを問うたvon Neumannの問題とも関わっている. von Neumannの問題は, 1980年代に否定的に解かれたものの, 近年の測度論的群論の発展により予想外の展開を見た. 講演では, これらのストーリ ーと測度論的群論の相似問題への応用(Monod氏との共同研究)を話す予定である. 予備知識はほとんど仮定しないので, 学部生にも聞きに来てもらいたい.
群のユニタリ表現に関しては美しい理論があるが, ユニタリでない無限次元表現はまったくとらえがたい対象である. そこで, 群のヒルベルト空間上の表現がいつユニタリ表現と相似(共役ともいう)になるかを問うのがDixmierの相似問題である. この問題は従順性という概念と深い関わりを持ち, 従って群の従順性の代数的な特徴づけを問うたvon Neumannの問題とも関わっている. von Neumannの問題は, 1980年代に否定的に解かれたものの, 近年の測度論的群論の発展により予想外の展開を見た. 講演では, これらのストーリ ーと測度論的群論の相似問題への応用(Monod氏との共同研究)を話す予定である. 予備知識はほとんど仮定しないので, 学部生にも聞きに来てもらいたい.
2009年12月17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
佐藤康彦 氏 (北海道大理)
Almost commuting unitaries and ${\\mathbb{Z}}^2$-action
佐藤康彦 氏 (北海道大理)
Almost commuting unitaries and ${\\mathbb{Z}}^2$-action
複素解析幾何セミナー
16:30-18:00 数理科学研究科棟(駒場) 126号室
時間と部屋が通常と異なります
Alan Huckleberry 氏 (Ruhr-Universität Bochum)
Hyperbolicity of cycle spaces and automorphism groups of flag domains
時間と部屋が通常と異なります
Alan Huckleberry 氏 (Ruhr-Universität Bochum)
Hyperbolicity of cycle spaces and automorphism groups of flag domains
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