Seminar information archive ~05/26Next seminarFuture seminars 05/27~

Organizer(s) ASUKE Taro, TERADA Itaru, HASEGAWA Ryu, MIYAMOTO Yasuhito (chair)

Seminar information archive


16:30-17:30   Room #002 (Graduate School of Math. Sci. Bldg.)
S.R.Srinivasa Varadhan (Courant Institute of Mathematical Sciences, New York University)
Large Deviations of Random Graphs and Random Matrices (ENGLISH)
[ Abstract ]
A random graph with $n$ vertices is a random symmetric matrix of $0$'s and $1$'s and they share some common aspects in their large deviation behavior. For random matrices it is the question of having large eigenvalues. For random graphs it is having too many or too few subgraph counts, like the number of triangles etc. The question that we will try to answer is what would a random matrix or a random graph conditioned to exhibit such a large deviation look like. Since the randomness is of size $n^2$ large deviation rates of order $n^2$ are possible.


16:30-17:30   Room #002 (Graduate School of Math. Sci. Bldg.)
Harald Niederreiter (RICAM, Austrian Academy of Sciences)
Quasi-Monte Carlo methods: deterministic is often better than random (ENGLISH)
[ Abstract ]
Quasi-Monte Carlo (QMC) methods are deterministic analogs of statistical Monte Carlo methods in computational mathematics. QMC methods employ evenly distributed low-discrepancy sequences instead of the random samples used in Monte Carlo methods. For many types of computational problems, QMC methods are more efficient than Monte Carlo methods. After a general introduction to QMC methods, the talk focuses on the problem of constructing low-discrepancy sequences which has fascinating links with subjects such as finite fields, error-correcting codes, and algebraic curves.

This talk also serves as the first talk of the four lecture series. The other three are on 5/28, 5/29, 5/30, 14:50-16:20 at room 123.
[ Reference URL ]


16:30-17:30   Room #002 (Graduate School of Math. Sci. Bldg.)
SAKASAI Takuya (University of Tokyo)
Moduli spaces and symplectic derivation Lie algebras (JAPANESE)
[ Abstract ]
First we overview Kontsevich's theorem describing a deep connection between homology of certain infinite dimensional Lie algebras (symplectic derivation Lie algebras) and cohomology of various moduli spaces. Then we discuss some computational results on the Lie algebras together with their applications (joint work with Shigeyuki Morita and Masaaki Suzuki).


16:30-17:30   Room #123 (Graduate School of Math. Sci. Bldg.)
Chun LIU (University of Tokyo / Penn State University)
On Complex Fluids (ENGLISH)
[ Abstract ]
The talk is on the mathematical theories, in particular the energetic variational approaches, of anisotropic complex fluids, such as viscoelastic materials, liquid crystals and ionic fluids in proteins and bio-solutions.

Complex fluids, including mixtures and solutions, are abundant in our daily life. The complicated phenomena and properties exhibited by these materials reflects the coupling and competition between the microscopic interactions and the macroscopic dynamics. We study the underlying energetic variational structures that is common among all these multiscale-multiphysics systems.

In this talk, I will demonstrate the modeling as well as analysis and numerical issues arising from various complex fluids.


15:00-16:00   Room #050 (Graduate School of Math. Sci. Bldg.)
Aleksandar Ivic (University of Belgrade, the Serbian Academy of Science and Arts)
Problems and results on Hardy's Z-function (JAPANESE)
[ Abstract ]
The title is self-explanatory: G.H. Hardy first used the function
$Z(t)$ to show that there are infinitely many zeta-zeros on the
critical line $\\Re s = 1/2$. In recent years there is a revived
interest in this function, with many results and open problems.


16:30-17:30   Room #123 (Graduate School of Math. Sci. Bldg.)
Taro YOSHINO (Graduate School of Mathematical Sciences, University of Tokyo)
On Topological Blow-up (JAPANESE)
[ Abstract ]
Topological blow-up is a method to understand non-Hausdorff spaces. We define an obstruction to Hausdorffness, and call it crack. By `blowing up' the points in crack, we can obtain better space.


16:30-17:30   Room #123 (Graduate School of Math. Sci. Bldg.)
TAKAGI, Shunsuke (Graduate School of Mathematical Sciences, University of Tokyo)
Application of positive characteristic methods to singularity theory (JAPANESE)
[ Abstract ]
As an application of positive characteristic methods to singularity theory, I will talk about a characterization of singularities in characteristic zero using Frobenius maps. Log canonical singularities form a class of singularities associated to the minimal model program. It is conjectured that they correspond to $F$-pure singularities, which form a class of singularities defined via splitting of Frobenius maps. In this talk, I will explain a recent progress on this conjecture, especially its connection to another conjecture on ordinary reductions of Calabi-Yau varieties defined over number fields.


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。

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