Seminar information archive ~06/09Next seminarFuture seminars 06/10~

Organizer(s) ABE Noriyuki, IWAKI Kohei, KAWAZUMI Nariya (chair), KOIKE Yuta

Seminar information archive


16:30-17:30   Room #002 (Graduate School of Math. Sci. Bldg.)
SHIMOMURA Akihiro (Graduate School of mathematical Sciences, University of Tokyo)
Nonlinear dispersive evolution equations (JAPANESE)
[ Abstract ]
I will talk about the time evolution of solutions to nonlinear dispersive equations.


16:30-17:30   Room #123 (Graduate School of Math. Sci. Bldg.)
KANAI Masahiko (Graduate School of Mathematical Sciences, University of Tokyo)
Cross ratio, and all that
[ Abstract ]
Although the origin of cross ratio goes back to ancient Greek mathematics, new discoveries about it has been made even in the past few decades. It seems that our understanding as to cross ratio is still limited. I am going to show you the present status and my own concerns, as well.


16:30-17:30   Room #123 (Graduate School of Math. Sci. Bldg.)
Shihoko Ishii (Graduate School of Mathematical Sciences, University of Tokyo)
Arc spaces and algebraic geometry (JAPANESE)


16:30-17:30   Room #123 (Graduate School of Math. Sci. Bldg.)
Jun SHIHO (Graduate School of Mathematical Sciences, University of Tokyo)
On logarithmic extension of p-adic differential equations (JAPANESE)


16:30-17:30   Room #117 (Graduate School of Math. Sci. Bldg.)
Shirai Tomoyuki (Kyushu University)
Conformal invariance in probability theory (JAPANESE)


16:30-17:30   Room #117 (Graduate School of Math. Sci. Bldg.)
Yoshikazu Giga (The University of Tokyo, Graduate School of Mathematical Sciences)
Hamilton-Jacobi equations and crystal growth (JAPANESE)


16:30-17:30   Room #002 (Graduate School of Math. Sci. Bldg.)
Robin Graham (University of Washington)
Ambient metrics and exceptional holonomy (ENGLISH)
[ Abstract ]
The holonomy of a pseudo-Riemannian metric is a subgroup of the orthogonal group which measures the structure preserved by parallel translation. Construction of pseudo-Riemannian metrics whose holonomy is an exceptional Lie group has been of great interest in recent years. This talk will outline a construction of metrics in dimension 7 whose holonomy is contained in the split real form of the exceptional group $G_2$. The datum for the construction is a generic real-analytic 2-plane field on a manifold of dimension 5; the metric in dimension 7 arises as the ambient metric of a conformal structure on the 5-manifold defined by Nurowski in terms of the 2-plane field.


16:30-17:30   Room #123 (Graduate School of Math. Sci. Bldg.)
Ryu Sasaki (Yukawa Institute for Theoretical Physics, Kyoto University)
Exceptional Jacobi polynomials as solutions of a Schroedinger
(Sturm-Liouville) equation with $3 +¥ell$ ($¥ell=1,2,¥ldots) regular
singularities (JAPANESE)
[ Abstract ]
Global solutions of Fuchsian differential equations with more than 3 (hypergeometric) or four (Heun) regular singularities had been virtually unkown. Here I present a complete set of eigenfunctions of a Schroedinger (Sturm-Liouville) equation with $3 + ¥ell$ ($¥ell=1,2,¥ldots$) regular singularities. They are deformations of the Darboux-P¥" oschl-Teller potential with the Hamiltonian (Schroedinger operator) ¥[ ¥mathcal{H}=-¥frac{d^2}{dx^2}+¥frac{g(g-1)}{¥sin^2x}+¥frac{h(h-1)} {¥cos^2x}¥] The eigenfunctions consist of the {¥em exceptional Jacobi polynomials} $¥{P_{¥ell,n}(¥eta)¥}$, $n=0,1,2,¥ldots$, with deg($P_{¥ell,n}$)$=n+¥ell$. Thus the restriction due to Bochner's theorem does not apply. The confluent limit produces two sets of the exceptional Laguerre polynomials for $¥ell=1,2,¥ldots$. Similar deformation method provides the exceptional Wilson and Askey-Wilson polynomials for $¥ell=1,2,¥ldots$.


16:30-17:30   Room #123 (Graduate School of Math. Sci. Bldg.)
Mitsuhiro Shishikura (Kyoto University)
Hausdorff dimension and measure of conformal fractals (JAPANESE)


17:00-18:00   Room #123 (Graduate School of Math. Sci. Bldg.)
Masaaki Umehara (Osaka University)
The Gauss-Bonnet Theorem and singular points on surfaces (JAPANESE)
[ Abstract ]
We generalize the classical Gauss-Bonnet formula for closed surfaces as wave fronts. Using it, we can find a new view point of inflection points and the topology of immersed surfaces in Euclidean 3-space via the singularities of their Gauss maps.


16:30-17:30   Room #123 (Graduate School of Math. Sci. Bldg.)
Jean-Pierre Puel (The University of Tokyo, Universite de Versailles Saint-Quentin)
Why to study controllability problems and the mathematical tools involved (ENGLISH)
[ Abstract ]
We will give some examples of controllability problems and the underlying applications to practical situations. This includes vibrations of membranes or plates, motion of incompressible fluids or quantum systems occuring in quantum chemistry or in quantum logic information theory. These examples correspond to different types of partial differential equations for which specific analysis has to be done. Of course, at the moment, very few results are known and the domain is widely open. We will describe very briefly the mathematical tools used for each type of PDE, in particular microlocal analysis, global Carleman estimates or some specific real analysis estimates.These methods appear to be also useful to study some inverse problems and, if time permits, we will give a few elements on some examples.


16:30-17:30   Room #123 (Graduate School of Math. Sci. Bldg.)
松本 眞 (東京大学大学院数理科学研究科)
疑似乱数発生に用いられる数学:メルセンヌ・ツイスターを例に (JAPANESE)
[ Abstract ]
[ Reference URL ]


15:00-17:30   Room #050 (Graduate School of Math. Sci. Bldg.)
岡本和夫 (東京大学大学院数理科学研究科) 15:00-16:00
[ Abstract ]
森田茂之 (東京大学大学院数理科学研究科) 16:30-17:30
[ Abstract ]


16:30-17:30   Room #002 (Graduate School of Math. Sci. Bldg.)
Charles Fefferman (Princeton University)
Extension of Functions and Interpolation of Data
[ Abstract ]
Let $f$ be a given real-valued function defined on a subset of $\\mathbb{R}^n$. We explain how to decide whether $f$ extends to a function $F$ in $C^m(\\mathbb{R}^n)$. If such an $F$ exists, we show how to construct one.


16:30-17:30   Room #123 (Graduate School of Math. Sci. Bldg.)
大島利雄 (東京大学大学院数理科学研究科)
[ Abstract ]
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。


16:30-17:30   Room #002 (Graduate School of Math. Sci. Bldg.)
小澤登高 (東京大学大学院数理科学研究科)
[ Abstract ]
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.


16:30-17:30   Room #002 (Graduate School of Math. Sci. Bldg.)
Louis Nirenberg
(New York University)
On solving fully nonlinear elliptic Partial Differential Equations

[ Abstract ]
The talk will present some results in recent work by R.Harvey and B. Lawson: Dirichlet duality and the nonlinear Dirichlet problem, Comm. Pure Appl. Math. 62 (2009), 396-443. It concerns solving boundary value problems for elliptic equations of the form F(D'2u) = 0. They find generalized solutions which are merely continuous . The talk will be expository. No knowledge of Partial Differential Equations will be necessary.


16:30-17:30   Room #002 (Graduate School of Math. Sci. Bldg.)
辻 雄 (東京大学大学院数理科学研究科)
[ Abstract ]
複素や実の多様体の特異コホモロジーを微分形式の言葉で記述する理論として、de Rhamの定理やHodge理論が良く知られている。p進Hodge理論は、これらの類似をp進体上の代数多様体のp進エタール・コホモロジーで考える理論である。p進エタール・コホモロジーにはp進体の絶対ガロア群が非常に複雑に作用しており、この作用を分かりやすい別の言葉で記述する理論の構築が、p進Hodge理論における大きな課題となっている。前半でp進Hodge理論の研究の歴史や背景について概観した後、後半ではp進体の絶対ガロア群のp進表現の相対版である、p進体上定義された代数多様体上のp進エタール層についての最近の研究を紹介する。


16:30-17:30   Room #123 (Graduate School of Math. Sci. Bldg.)
Carlos Simpson (CNRS, University of Nice)
Differential equations and the topology of algebraic varieties
[ Abstract ]
The study of the topology of complex algebraic varieties makes use of differential equations in several different ways. The classical notion of variation of Hodge structure contains, on the one hand, the Gauss-Manin differential equations, on the other hand Hodge metric data which satisfy harmonic bundle equations. These two aspects persist in the study of arbitrary representations of the fundamental group. Combining them leads to a notion of ``Hodge structure'' on the space of representations. This can be extended to the higher homotopical structure of a variety, by using ideas of ``shape'' and nonabelian cohomology.


16:30-17:30   Room #002 (Graduate School of Math. Sci. Bldg.)
Nessim Sibony (Universite Paris-Sud)
Holomorphic dynamics in several variables: equidistribution problems and statistical properties
[ Abstract ]
The main problem in the dynamical study of a map is to understand the long term behavior of orbits. The abstract theory of non uniformly hyperbolic systems is well understood but it is very difficult to decide when a given system is non uniformly hyperbolic and to study it's sharp ergodic properties.
Holomorphic dynamics in several variables provide large classes of examples of non uniformly hyperbolic systems. One can compute the entropy, construct a measure of maximal entropy and study the sharp statistical properties: central limit theorem, large deviations and exponential decay of correlations. It is also possible to prove sharp equidistribution results for preimages of analytic sets of arbitrary dimension. The main tools are: pluripotential theory, analytic geometry, and good estimates from PDE.
These systems appear naturally if we apply Newton's method to localise the common zeros of of polynomial equations in several variables. In the study of polynomial automorphisms of complex Euclidean spaces, or automorphisms of compact K\\"ahler manifolds.


16:30-17:30   Room #002 (Graduate School of Math. Sci. Bldg.)
緒方芳子 (東京大学大学院数理科学研究科)
[ Abstract ]
量子スピン系における大偏差原理の研究について紹介する。特に、Gibbs State, Finitely Correlated State と呼ばれる状態についてお話しする。一次元量子スピン系の大偏差原理、Gibbs State の大偏差原理のレート関数の特徴づけ、さらに統計力学における分布の同値性の問題との関連について述べる。


15:00-17:30   Room #050 (Graduate School of Math. Sci. Bldg.)
菊地文雄 (東京大学大学院数理科学研究科) 15:00-16:00
[ Abstract ]
桂 利行 (東京大学大学院数理科学研究科) 16:30-17:30
[ Abstract ]


16:00-17:00   Room #002 (Graduate School of Math. Sci. Bldg.)
神保道夫 (東京大学大学院数理科学研究科)
[ Abstract ]


16:30-17:30   Room #123 (Graduate School of Math. Sci. Bldg.)
川又雄二郎 (東京大学大学院数理科学研究科)
[ Abstract ]
代数多様体上に載っている曲線と因子の交点数を使うと、互いに双対な有限次元実ベクトル空間内の、互いに双対な閉凸錐体 -- 曲線の錐体と因子の錐体が定義される。極小モデル理論では、曲線の錐体の端射線から収縮写像が構成されるが、双有理同値な代数多様体をたくさん同時に考えるためには、因子の錐体のほうが便利である。標準環の有限生成定理は、因子の錐体の集まりの間の壁越えの様子を詳しく調べることによって証明された。一般の代数多様体に対する極小モデルの存在は未解決問題であるが、そのためには因子の錐体についてのより深い理解が必要と思われる。この講演ではそのあたりの事情を解説する。
[ Reference URL ]


16:30-17:30   Room #002 (Graduate School of Math. Sci. Bldg.)
平地健吾 (東京大学大学院数理科学研究科)
What is Q-curvature?
[ Abstract ]
共形幾何は次元の偶奇におうじて著しく異なった性質をもちます。その多くは n次元球面の共形自己同型群SO(n+1,1)が奇数次元ならB型,偶数次元ならD型になるといことから説明できます。この講演では偶数次元にのみ現れるQ-曲率とよばれる局所不変量とその周辺に現れる共形不変量および不変作用素の理論を紹介します。Q-曲率はAdS/CFT対応にも自然に現れることもあり,最近の共形幾何の主要テーマになっていますが,その定義は簡単ではありません。Q-曲率の(短い)歴史と表現論の結果をふまえて,なっとくのできる定義を与えることを目指します。
[ Reference URL ]
次回開催日は11月28日(金)(講演者:川又雄二郎 氏)です。ご注意下さい。

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