## 過去の記録

#### GCOEレクチャーズ

10:00-12:10   数理科学研究科棟(駒場) 123号室
Haim Brezis 氏 (Rutgers University / Technion)
Sobolev maps with values into the circle (ENGLISH)
[ 講演概要 ]
Sobolev functions with values into R are very well understood and play an immense role in many branches of Mathematics. By contrast, the theory of Sobolev maps with values into the unit circle is still under construction. Such maps occur e.g. in the asymptotic analysis of the Ginzburg-Landau model. The reason one is interested in Sobolev maps, rather than smooth maps is to allow singularities such as x/|x| in 2D or line singularities 3D which appear in physical problems. Our focus in these lectures is not the Ginzburg-Landau equation per se, but rather the intrinsic study of the function space W^{1,p} of maps from a smooth domain in R^N taking their values into the unit circle. Such classes of maps have an amazingly rich structure. Geometrical and Topological effects are already noticeable in this simple framework, since S^1 has nontrivial topology. Moreover the fact that the target space is the circle (as opposed to higher-dimensional manifolds) offers the option to introduce a lifting. We'll see that "optimal liftings" are in one-to-one correspondence with minimal connections (resp. minimal surfaces) spanned by the topological singularities of u.
I will also discuss the question of uniqueness of lifting . A key ingredient in some of the proofs is a formula (due to myself, Bourgain and Mironescu) which provides an original way of approximating Sobolev norms (or the total variation) by nonlocal functionals. Nonconvex versions of these functionals raise very challenging questions recently tackled together with H.-M. Nguyen. Comparable functionals also occur in Image Processing and suggest exciting interactions with this field.

### 2012年11月28日(水)

#### 幾何コロキウム

10:30-12:00   数理科学研究科棟(駒場) 128号室

リッチ曲率と角度 (JAPANESE)
[ 講演概要 ]
リッチ曲率が下に有界なリーマン多様体の極限空間(これは距離空間)を考える.この極限空間を調べること,特にその regularity を調べることは様々な幾何と接点を持ち,多くの応用を持つ.この講演ではそのような regularity に関する一結果を紹介する.具体的には,そのような空間の上で角度が定義できること,そしてその応用として,極限空間は必ず弱い意味で二階微分可能構造を持つことを紹介する.また,時間が許せばその後の進展についても述べたい.

#### 作用素環セミナー

16:30-18:00   数理科学研究科棟(駒場) 118号室

Fundamental group of simple $C^*$-algebras with unique trace (JAPANESE)

#### 講演会

10:45-11:45   数理科学研究科棟(駒場) 002号室

Pascal Chossat 氏 (CNRS / University of Nice)
Pattern formation in the hyperbolic plane (ENGLISH)
[ 講演概要 ]
Initially motivated by a model for the visual perception of textures by the cortex, the problem of pattern formation in the hyperbolic plane, or equivalently the Poincaré disc D, shows some similar but mostly quite different features from the same problem posed on the Euclidean plane. The hyperbolic structure induces a large variety of possible periodic patterns and even the bifurcation of "hyperbolic" traveling waves. We call these patterns "H-planforms". I shall show how H-planforms are determined by the means of equivariant bifurcation theory and Helgason-Fourier analysis in D. However the question of their observability is still open. The talk will be illustrated with pictures of H-planforms that have been computed using non trivial algorithms based on harmonic analysis in D.

#### GCOEレクチャーズ

13:30-14:30   数理科学研究科棟(駒場) 123号室
Haim Brezis 氏 (Rutgers University / Technion)
How Poincare became my hero (ENGLISH)
[ 講演概要 ]
I recently discovered little-known texts of Poincare which include fundamental results on PDEs together with prophetic insights into their future impact on various branches of modern mathematics.

#### GCOEレクチャーズ

14:50-17:30   数理科学研究科棟(駒場) 123号室
Haim Brezis 氏 (Rutgers University / Technion)
Can you hear the degree of a map from the circle into itself? An intriguing story which is not yet finished (ENGLISH)
[ 講演概要 ]
A few years ago - following a suggestion by I. M. Gelfand - I discovered an intriguing connection between the topological degree of a map from the circle into itself and its Fourier coefficients. This relation is easily justified when the map is smooth. However, the situation turns out to be much more delicate if one assumes only continuity, or even Holder continuity. I will present recent developments and open problems. The initial motivation for this direction of research came from the analysis of the Ginzburg-Landau model.

### 2012年11月27日(火)

#### 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

[ 講演概要 ]
Thurstonの例により、葉層構造の二次特性類は有界でないことが知られている。本講演では、横断的な共形平坦構造などを持つ葉層構造に対しては(例外的な場合を除き)二次特性類が有限性を持つことを、非有界性や葉層構造の剛性との関連と共に説明する。
(本講演はSantiago de Compostela大学のJesús Antonio Álvarez
López氏との共同研究 arXiv:1205.3375に基づく。)

#### 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

[ 講演概要 ]
In this talk we will discuss geometric quantization of a flag manifold. In particular, we construct a family of complex structures on a flag manifold that converge 'at the quantum level' to the real polarization coming from the Gelfand-Cetlin integrable system.
Our construction is based on a toric degeneration of flag varieties and a deformation of K¥"ahler structure on toric varieties by symplectic potentials.
This is a joint work with Mark Hamilton.

### 2012年11月26日(月)

#### 代数幾何学セミナー

15:30-17:00   数理科学研究科棟(駒場) 122号室

[ 講演概要 ]

#### 複素解析幾何セミナー

10:30-12:00   数理科学研究科棟(駒場) 126号室

[ 講演概要 ]

### 2012年11月22日(木)

#### 講演会

13:30-14:15   数理科学研究科棟(駒場) 002号室

Danielle Hilhorst 氏 (CNRS / Univ. Paris-Sud)
A multiple scale pattern formation cascade in reaction-diffusion systems of activator-inhibitor type (ENGLISH)
[ 講演概要 ]
A family of singular limits of reaction-diffusion systems of activator-inhibitor type in which stable stationary sharp-interface patterns may form is investigated. For concreteness, the analysis is performed for the FitzHugh-Nagumo model on a suitably rescaled bounded domain in $\\R^N$, with $N \\geq 2$. It is proved that when the system is sufficiently close to the limit the dynamics starting from the appropriate smooth initial data breaks down into five distinct stages on well-separated time scales, each of which can be approximated by a suitable reduced problem. The analysis allows to follow fully the progressive refinement of spatiotemporal patterns forming in the systems under consideration and provides a framework for understanding the pattern formation scenarios in a large class of physical, chemical, and biological systems modeled by the class of reaction-diffusion equations, which we consider. This is joint work with Marie Henry and Cyrill Muratov.
[ 講演参考URL ]
https://www.ms.u-tokyo.ac.jp/gcoe/index_007.html

#### 講演会

14:25-15:10   数理科学研究科棟(駒場) 002号室

Thanh Nam Ngyuen 氏 (University of Paris-Sud)
Formal asymptotic limit of a diffuse interface tumor-growth model (ENGLISH)
[ 講演概要 ]
We consider a diffuse interface tumor-growth model, which has the form of a phase-field system. We discuss the singular limit of this problem. More precisely, we formally prove that as the reaction coefficient tends to zero, the solution converges to the solution of a free boundary problem.

This is a joint work with Danielle Hilhorst, Johannes Kampmann and Kristoffer G. van der Zee.
[ 講演参考URL ]
https://www.ms.u-tokyo.ac.jp/gcoe/index_007.html

#### 講演会

15:30-16:15   数理科学研究科棟(駒場) 002号室

Peter Gordon 氏 (Akron University)
Gelfand type problem for two phase porous media (ENGLISH)
[ 講演概要 ]
In this talk I will introduce a generalization of well known Gelfand problem arising in a Frank-Kamenetskii theory of thermal explosion. This generalization is a natural extension of the Gelfand problem to two phase materials, where, in contrast to classical Gelfand problem which utilizes single temperature approach, the state of the system is described by two different temperatures. As a result the problem is modeled by a system of two coupled nonlinear heat equations. The new ingredient in such a generalized Gelfand problem is a presence of inter-phase heat exchange which can be viewed as a strength of coupling for the system.

I will show that similar to classical Gelfand problem the thermal explosion (blow up of solution) for generalized Gelfand problem occurs exclusively due to the absence of stationary temperature distribution, that is non-existence of solution of corresponding elliptic problem. I also will show that the presence of inter-phase heat exchange delays a thermal explosion. Moreover, in the limit of infinite heat exchange between phases the problem of thermal explosion in two phase porous media reduces to classical Gelfand problem with re-normalized constants. The latter result partially justifies a single temperature approach to two phase systems often used in a physical literature.

This is a joint work with Vitaly Moroz (Swansea University).
[ 講演参考URL ]
https://www.ms.u-tokyo.ac.jp/gcoe/index_007.html

#### 講演会

16:25-17:10   数理科学研究科棟(駒場) 002号室

Cyrill Muratov 氏 (New Jersey Institute of Technology)
On the shape of charged drops: an isoperimetric problem with a competing non-local term (ENGLISH)
[ 講演概要 ]
In this talk I will give an overview of my recent work with H. Knuepfer on the analysis of a class of geometric problems in the calculus of variations. I will discuss the basic questions of existence and non-existence of energy minimizers for the isoperimetric problem with a competing non-local term. A complete answer will be given for the case of slowly decaying kernels in two space dimensions, and qualitative properties of the minimizers will be established for general Riesz kernels.
[ 講演参考URL ]
https://www.ms.u-tokyo.ac.jp/gcoe/index_007.html

### 2012年11月21日(水)

#### PDE実解析研究会

10:30-11:30   数理科学研究科棟(駒場) 056号室

Giovanni Pisante 氏 (Seconda Università degli Studi di Napoli)
Shape Optimization And Asymptotic For The Twisted Dirichlet Eigenvalue (ENGLISH)
[ 講演概要 ]
Aim of the talk is to discuss some recent results obtained with G. Croce and A. Henrot on a generalization of the functional defining the first twisted eigenvalue.
We look at the set functional defined by minimizing a Rayleigh quotient involving Lebesgue norms with different exponents p and q among functions satisfying a zero boundary condition as well as a zero mean condition of order q.
First under suitable conditions on p and q, that ensure the existence of a minimizing function, we investigate the validity of an isoperimetric type inequality of the Reyleigh-Faber-Krahn type.
Then we study the limit of the functional for p and q tending to 1 and to infinity and discuss the relation with the limits of the second eigenvalues of the p-laplacian operator.

#### 古典解析セミナー

16:00-17:30   数理科学研究科棟(駒場) 128号室
Philip Boalch 氏 (ENS-DMA & CNRS Paris)
Beyond the fundamental group (ENGLISH)
[ 講演概要 ]
Moduli spaces of representations of the fundamental group of a Riemann surface have been studied from numerous points of view and appear in many parts of mathematics and theoretical physics. They form an interesting class of symplectic manifolds, they often have Kahler or hyperkahler metrics (in which case they are diffeomorphic to spaces of Higgs bundles, i.e. Hitchin integrable systems), and they admit nonlinear actions of braid groups and mapping class groups with fascinating dynamical properties. The aim of this talk is to describe some aspects of this story and sketch their extension to the context of the "wild fundamental group", which naturally appears when one considers {\\em meromorphic} connections on Riemann surfaces. In particular some new examples of hyperkahler manifolds appear in this way, some of which are familiar from classical work on the Painleve equations.

### 2012年11月20日(火)

#### 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

3次元双曲幾何と団代数 (JAPANESE)
[ 講演概要 ]
クラスター代数は2000年にFomin-Zelevinskyによって発見された代数系である.

#### Lie群論・表現論セミナー

16:30-17:30   数理科学研究科棟(駒場) 126号室
Ali Baklouti 氏 (Sfax University)
On the geometry of discontinuous subgroups acting on some homogeneous spaces (ENGLISH)
[ 講演概要 ]
Let $G$ be a Lie group, $H$ a closed subgroup of $G$ and \\Gamma$a discontinuous subgroup for the homogeneous space$G/H$. I first introduce the deformation space${\\mathcal{T}}^{K_o}(\\Gamma, G, H)$of the action of$\\Gamma$on$G/H$in the sense of Kobayashi and some of its refined versions, namely the Clifford--Klein space of deformations of the form${\\mathcal{X}}=\\Gamma \\backslash G/H$. The deformation space${\\mathcal{T}}^{G_o}(\\Gamma, G,H)$of marked$(G,H)$-structures on${\\mathcal{X}}$in the sense of Goldman is also introduced. As an important motivation, I will explain the connection between the spaces${\\mathcal{T}}^{K_o}(\\Gamma, G, H)$and${\\mathcal{T}}^{G_o}(\\Gamma, G, H)$and study some of their topological features, namely the rigidity in the sense of Selberg--Weil--Kobayashi and the stability in the sense of Kobayashi--Nasrin. The latter appears to be of major interest to write down the connection explicitly. ### 2012年11月19日(月) #### 講演会 16:45-17:45 数理科学研究科棟(駒場) 126号室 Hendrik Weber 氏 (University of Warwick) Invariant measure of the stochastic Allen-Cahn equation: the regime of small noise and large system size (ENGLISH) [ 講演概要 ] We study the invariant measure of the one-dimensional stochastic Allen-Cahn equation for a small noise strength and a large but finite system. We endow the system with inhomogeneous Dirichlet boundary conditions that enforce at least one transition from -1 to 1. We are interested in the competition between the energy'' that should be minimized due to the small noise strength and the entropy'' that is induced by the large system size. Our methods handle system sizes that are exponential with respect to the inverse noise strength, up to the critical'' exponential size predicted by the heuristics. We capture the competition between energy and entropy through upper and lower bounds on the probability of extra transitions between$\\pm 1\$. These bounds are sharp on the exponential scale and imply in particular that the probability of having one and only one transition from -1 to +1 is exponentially close to one. In addition, we show that the position of the transition layer is uniformly distributed over the system on scales larger than the logarithm of the inverse noise strength.
Our arguments rely on local large deviation bounds, the strong Markov property, the symmetry of the potential, and measure-preserving reflections.
This is a joint work with Felix Otto and Maria Westdickenberg.

#### 代数幾何学セミナー

15:30-17:00   数理科学研究科棟(駒場) 122号室

Stability conditions and birational geometry (JAPANESE)
[ 講演概要 ]
I propose a conjecture which claims that MMP for a smooth projective variety is realized as a variation of Bridgeland moduli spaces of semistable objects in the derived category of coherent sheaves. I will discuss the surface case and extremal contractions for 3-folds. In the former case, the conjecture is completely solved. In the latter case, I will construct the perverse t-structure associated to the extremal contraction, and construct a candidate of the desired stability condition as a double tilting of the perverse heart.

#### GCOEセミナー

15:30-17:00   数理科学研究科棟(駒場) 056号室
Alfred Ramani 氏 (Ecole Polytechnique)
Linearisable mappings (ENGLISH)
[ 講演概要 ]
We present a series of results on linearisable second-order mappings.
Three distinct families of such mappings do exist: projective, mappings of Gambier type and mappings which we have dubbed "of third kind".
Our starting point are the linearisable mappings belonging to the QRT family. We show how they can be linearised and how in some cases their explicit solution can be constructed. We discuss also the growth property of these mappings, a property intimately related to linearisability.
In the second part of the talk we address the question of the extension of these mappings to a non-autonomous form.
We show that the QRT invariant can also be extended (to a quantity which depends explicitly on the independent variable). Using this non-autonomous form we show that it is possible to construct the explicit solution of all third-kind mappings. We discuss also the relation of mappings of the third kind to Gambier-type mappings. We show that a large subclass of third-kind mappings can be considered as the discrete derivative of Gambier-type ones.

#### 複素解析幾何セミナー

10:30-12:00   数理科学研究科棟(駒場) 126号室

トーラス上の種数２のリーマン面の退化族について (JAPANESE)
[ 講演概要 ]
ある固定したトーラスの2重分岐被覆として得られる種数2のリーマン面を一般ファイバーに持つ、リーマン面の退化族を構成し、その特異ファイバーと正則切断を決定する。

### 2012年11月16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) 002号室

お茶&Coffee&お菓子: 16:00～16:30 (コモンルーム)。

[ 講演概要 ]
コンパクト複素多様体のもっともなじみ深い不変量は Chern 類であろう.この講演ではその secondary classes にあたる正則ベクトル場を含んだ積分不変量の族で,次のような3つを含むものについて紹介する.
(1) 各 k に対し,k 次 Chern 形式が調和形式であるようなケーラー計量が存在するための障害となる不変量.
(2) 非ケーラー多様体でも定義される不変量で,横断的正則葉層構造の特性類やLefchetz 数などから自然に得られる不変量.
(3) 代数多様体に対し,漸近的 Chow 半安定性の障害となる不変量.
これらの3つの族の共通部分にケーラー・アインシュタイン計量が存在するための障害がある.

#### GCOEセミナー

16:30-18:00   数理科学研究科棟(駒場) 122号室
Dietmar Bisch 氏 (Vanderbilt University)
Subfactors with small Jones index (ENGLISH)

### 2012年11月14日(水)

#### 代数学コロキウム

18:00-19:00   数理科学研究科棟(駒場) 002号室
いつもと時間・場所が異なりますのでご注意ください.
Pierre Berthelot 氏 (Université de Rennes 1)
De Rham-Witt complexes with coefficients and rigid cohomology
(ENGLISH)
[ 講演概要 ]
For a smooth scheme over a perfect field of characteristic p, we will explain a generalization of the classical comparison theorem between crystalline cohomology and de Rham-Witt cohomology to the case of cohomologies with coefficients in a p-torsion free crystal. This provides in particular a description of the rigid cohomology of a proper singular scheme in terms of a de Rham-Witt complex built from a closed immersion into a smooth scheme.
(本講演は「東京パリ数論幾何セミナー」として、インターネットによる東大数理とIHESとの双方向同時中継で行います.)