Seiji Nishioka, Fanny Kassel, Yves Benoist, Shu Kato, Uuganbayar Zunderiya, Takayuki Okuda, Yoshiki Oshima, Hisayosi Matumoto, Birgit Speh #1, Kaoru Hiraga , Birgit Speh #2, Birgit Speh #3, Kanehisa Takasaki, Soji Kaneyuki, Soo Teck Lee, Bernhard Mühlherr #1, Bernhard Mühlherr #2, Daniel Sternheimer, Michael Eastwood #1, Michael Eastwood #2, Michael Eastwood #3, Katsuyuki Naoi,
Date:  January 12 (Tue), 2010, 16:3018:00 
Place:  Room 126, Graduate School of Mathematical Sciences, the University of Tokyo 
Speaker:  Seiji Nishioka (西岡斉治) (The University of Tokyo) 
Title:  代数的差分方程式の可解性と既約性 
Abstract: [ pdf ] 
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． 
(Joint seminar with Topology Seminar)  
Date:  February 2 (Tue), 2010, 16:3018:00 
Place:  Room 056, Graduate School of Mathematical Sciences, the University of Tokyo 
Speaker:  Fanny Kassel (Orsay) 
Title:  Deformation of compact quotients of homogeneous spaces 
Abstract: [ pdf ] 
Let G/H be a reductive homogeneous space. In all known examples, if G/H admits compact CliffordKlein forms, then it admits "standard" ones, by uniform lattices of some reductive subgroup L of G acting properly on G/H. In order to obtain more generic CliffordKlein forms, we prove that for L of real rank 1, if one slightly deforms in G a uniform lattice of L, then its action on G/H remains properly discontinuous. As an application, we obtain compact quotients of SO(2,2n)/U(1,n) by Zariskidense discrete subgroups of SO(2,2n) acting properly discontinuously. 
(GCOE Lectures)  
Date: 

Place:  Room 126, Graduate School of Mathematical Sciences, the University of Tokyo 
Speaker:  Yves Benoist (Orsay) 
Title:  Discrete groups acting on homogeneous spaces IV 
Abstract: [ pdf ] 
In this course I will focus on recent advances
on our understanding of discrete subgroups of Lie groups. I will first survey how ideas from semisimple algebraic groups, ergodic theory and representation theory help us to understand properties of these discrete subgroups. I will then focus on a joint work with JeanFrancois Quint studying the dynamics of these discrete subgroups on finite volume homogeneous spaces and proving the following result: We fix two integral matrices A and B of size d, of determinant 1, and such that no finite union of vector subspaces is invariant by A and B. We fix also an irrational point on the ddimensional torus. We will then prove that for n large the set of images of this point by the words in A and B of length at most n becomes equidistributed in the torus. 
Date:  April 6 (Tue), 2010, 16:3018:00 
Place:  Room 126, Graduate School of Mathematical Sciences, the University of Tokyo 
Speaker:  Shu Kato (加藤 周) (Kyoto University) 
Title:  On the characters of tempered modules of affine Hecke algebras of classical type 
Abstract: [ pdf ] 
We present an inductive algorithm to compute the characters
of tempered modules of an affine Hecke algebras of classical
types, based on a new class of representations which we call
"tempered delimits". They have some geometric origin in the
eDL correspondence. Our new algorithm has some advantage to the LusztigShoji algorithm (which also describes the characters of tempered modules via generalized Green functions) in the sense it enables us to tell how the characters of tempered modules changes as the parameters vary. This is a joint work with Dan Ciubotaru at Utah. 
Date:  April 15 (Thu), 2010, 16:3018:00 
Place:  Room 056, Graduate School of Mathematical Sciences, the University of Tokyo 
Speaker:  Uuganbayar Zunderiya (Nagoya University) 
Title:  Generalized hypergeometric systems 
Abstract: [ pdf ] 
A new type of hypergeometric differential equations was introduced and studied by H. Sekiguchi. The investigated system of partial differential equation generalizes the GaussAomotoGelfand system which in its turn stems from the classical set of differential relations for the solutions to a generic algebraic equation introduced by K.Mayr in 1937. GaussAomotoGelfand systems can be expressed as the determinants of $2\times 2$ matrices of derivations with respect to certain variables. H. Sekiguchi generalized this construction by looking at determinations of arbitrary $l\times l$ matrices of derivations with respect to certain variables. In this talk we study the dimension of global (and local) solutions to the generalized systems of GaussAomotoGelfand hypergeometric systems. The main results in the talk are a combinatorial formula for the dimension of global (and local) solutions of the generalized GaussAomotoGelfand system and a theorem on generic holonomicity of a certain class of such systems. 
Date:  April 20 (Tue), 2010, 16:3018:00 
Place:  Room 126, Graduate School of Mathematical Sciences, the University of Tokyo 
Speaker:  Takayuki Okuda (奥田隆幸) (the University of Tokyo) 
Title:  Proper actions of SL(2,R) on semisimple symmetric spaces 
Abstract: [ pdf ] 
Complex irreducible symmetric spaces which admit proper SL(2,R)actions were classified by Katsuki Teduka. In this talk, we generalize Teduka's method and classify semisimple symmetric spaces which admit proper SL(2,R)actions. 
Date:  April 27 (Tue), 2010, 16:3018:00 
Place:  Room 126, Graduate School of Mathematical Sciences, the University of Tokyo 
Speaker:  Yoshiki Oshima (大島芳樹) (the University of Tokyo) 
Title:  Restriction of VoganZuckerman’s derived functor modules to symmetric subgroups 
Abstract: [ pdf ] 
We study the restriction of VoganZuckerman derived functor modules A_{q}(λ) to symmetric subgroups. An algebraic condition for the discrete decomposability of A_{q}(λ) was given by Kobayashi, which offers a framework for the detailed study of branching law. In this talk, when A_{q}(λ) is discretely decomposable, we construct some of irreducible components occurring in the branching law and determine their associated variety. 
Date:  May 11 (Tue), 2010, 16:3018:00 
Place:  Room 126, Graduate School of Mathematical Sciences, the University of Tokyo 
Speaker:  Hisayosi Matumoto (松本久義) (the University of Tokyo) 
Title:  On a finite $W$algebra module structure on the space of continuous Whittaker vectors for an irreducible HarishChandra module 
Abstract: [ pdf ] 
Let $G$ be a real reductive Lie group. The space of continuous Whittaker vectors for an irreducible HarishChandra module has a structure of a module over a finite $W$algebra. We have seen such modules are irreducible for groups of type A. However, there is a counterexample to the naive conjecture. We discuss a refined version of the conjecture and further examples in this talk. 
Date:  May 18 (Tue), 2010, 16:3018:00 
Place:  Room 126, Graduate School of Mathematical Sciences, the University of Tokyo 
Speaker:  Birgit Speh (Cornel University) 
Title:  On the eigenvalues of the Laplacian on locally symmetric hyperbolic spaces 
Abstract: [ pdf ] 
A famous Theorem of Selberg says that the nonzero eigenvalues of the Laplacian acting on functions on quotients of the upper half plane H by congruence subgroups of the integral modular group, are bounded away from zero, as the congruence subgroup varies. Analogous questions on Laplacians acting on differential forms of higher degree on locally symmetric spaces (functions may be thought of as differential forms of degree zero) have geometric implications for the cohomology of the locally symmetric space.
Let $X$ be the real hyperbolic nspace and $\Gamma \subset $ SO(n, 1) a congruence arithmetic subgroup. Bergeron conjectured that the eigenvalues of the Laplacian acting on the differential forms on $ X / \Gamma $ are bounded from the below by a constant independent of the congruence subgroup. In the lecture I will show how one can use representation theory to show that this conjecture is true provided that it is true in the middle degree. This is joint work with T.N. Venkataramana. 
Date:  May 25 (Tue), 2010, 17:0018:00 
Place:  Room 126, Graduate School of Mathematical Sciences, the University of Tokyo 
Speaker:  Kaoru Hiraga (平賀 郁) (Kyoto University) 
Title:  On endoscopy, packets, and invariants 
Abstract: [ pdf ] 
The theory of endoscopy came out of the Langlands functoriality and the trace formula. In this talk, I will briefly explain what the endoscopy is, and talk about packet, formal degree and Whittaker normalization of transfer. I would like to talk about the connection between these topics and the endoscopy. 
(GCOE Lectures)  
Date:  June 1 (Tue), 2010, 16:3018:00 
Place:  Room 126, Graduate School of Mathematical Sciences, the University of Tokyo 
Speaker:  Birgit Speh (Cornel University) 
Title:  Introduction to the cohomology of locally symmetric spaces 
Abstract: [ pdf ] 
I will give an introduction to the cohomology of noncompact locally symmetric spaces $X_\Gamma =K \backslash G / \Gamma $. If $X_\Gamma $ is cocompact this cohomology can be expressed as the $(g,K)$cohomology of automorphic representations. I will explain how representation theory and automorphic forms can be used to study the cohomology in this case. 
(GCOE Lectures)  
Date:  June 3 (Thu), 2010, 16:3018:00 
Place:  Room 470, Graduate School of Mathematical Sciences, the University of Tokyo 
Speaker:  Birgit Speh (Cornel University) 
Title:  Introduction to the cohomology of locally symmetric spaces 2 
Abstract: [ pdf ] 
I will give an introduction to the cohomology of noncompact locally symmetric spaces $X_\Gamma =K \backslash G / \Gamma $. If $X_\Gamma $ is cocompact this cohomology can be expressed as the $(g,K)$cohomology of automorphic representations. I will explain how representation theory and automorphic forms can be used to study the cohomology in this case. 
（集中講義：数理科学特別講義XI）  
Date:  June 7 (Mon)11 (Fri), 2010, 14:4016:40 
Place:  Room 123, Graduate School of Mathematical Sciences, the University of Tokyo 
Speaker:  Kanehisa Takasaki (高崎金久) (Kyoto University) 
Title:  ツイスターの数理 
Abstract: 
ツイスターは数理物理学者Ｒ．ペンローズによって1960年代
に導入された概念である．ペンローズの本来の意図は４次元の相対論的
時空とその上の粒子・場を複素数や複素解析函数の言葉で記述すること
にあったが，その後の研究は本来の物理学よりもむしろ数学のさまざま
な分野（幾何学，表現論，超函数，超幾何方程式，可積分系など）にお
いて豊かな成果を生んでいる．
この講義ではツイスター理論の数学的な側面を解説する．取り上げる おもな項目は以下の通りである．
参考書等

Date:  June 8 (Tue), 2010, 17:0018:30 
Place:  Room 126, Graduate School of Mathematical Sciences, the University of Tokyo 
Speaker:  Soji Kaneyuki (金行壮二) (Sophia University) 
Title:  Automorphism groups of causal Makarevich spaces 
Abstract: [ pdf ] 
Let G^ be a simple Lie group of Hermitian type and U^ be a maximal parabolic subgroup of G^ with abelian nilradical. The flag manifold M^= G^/ U^ is the Shilov boundary of an irreducible bounded symmetric domain of tube type. M^ has the Ginvariant causal structure. A causal Makarevich space is, by definition, an open symmetric Gorbit M in M^, endowed with the causal structure induced from that of the ambient space M^, G being a reductive subgroup of G^. All symmetric cones fall in the class of causal Makarevich spaces. In this talk, we determine the causal automorphism groups of all causal Makarevich spaces. 
Date:  July 15 (Thu), 2010, 14:3016:00 
Place:  Room 122, Graduate School of Mathematical Sciences, the University of Tokyo 
Speaker:  Soo Teck Lee (Singapore National University) 
Title:  Pieri rule and Pieri algebras for the orthogonal groups 
Abstract: [ pdf ] 
The irreducible rational representations of the complex orthogonal group $\mathrm{O}_n$ are labeled by a certain set of Young diagrams, and we denote the representation corresponding to the Young diagram $D$ by $\sigma^D_n$. Consider the tensor product $\sigma^D_n\otimes\sigma^E_n$ of two such representations. It can be decomposed as \[\sigma^D_n\otimes\sigma^E_n=\bigoplus_Fm_F\sigma^F_n,\] where for each Young diagram $F$ in the sum, $m_F$ is the multiplicity of $\sigma^F_n$ in $\sigma^D_n\otimes\sigma^E_n$. In the case when the Young diagram $E$ consists of only one row, a description of the multiplicities in $\sigma^D_n\otimes\sigma^E_n$ is called the {\em Pieri Rule}. In this talk, I shall describe a family of algebras whose structure encodes a generalization of the Pieri Rule. 
Date:  September 1 (Wed), 2010, 16:3018:00 
Place:  Room 002, Graduate School of Mathematical Sciences, the University of Tokyo 
Speaker:  Bernhard Mühlherr (JustusLiebigUniversität Giessen) 
Title:  Groups of KacMoody type 
Abstract: [ pdf ] 
Groups of KacMoody type are natural generalizations of KacMoody groups over fields in the sense that they have an RGDsystem. This is a system of subgroups indexed by the roots of a root system and satisfying certain commutation relations.
Roughly speaking, there is a onetoone correspondence between groups of KacMoody type and Moufang twin buildings. This correspondence was used in the last decade to prove several group theoretic results on RGDsystems and in particular on KacMoody groups over fields. In my talk I will explain RGDsystems and how they provide twin buildings in a natural way. I will then present some of the group theoretic applications mentioned above and describe how twin buildings are used as a main tool in their proof. 
Date:  1. September 4 (Sat), 2010, 09:3011:00 2. September 4 (Sat), 2010, 11:1012:40 3. September 9 (Thu), 2010, 16:3018:00 
Place:  Room 126, Graduate School of Mathematical Sciences, the University of Tokyo 
Speaker:  Bernhard Mühlherr (JustusLiebigUniversität Giessen) 
Title:  Minicourse on buildings 
Abstract: 
The goal of this course is to provide an overview on the theory of
buildings which was developed by Jacques Tits. The idea is to start
with basic examples, then to explain the classification of spherical
buildings and to finish with a survey on affine buildings and twin
buildings. 1st Lecture: We recall the basic facts about Coxeter groups which are important for the theory of buildings and we explain the class of rank 2 buildings which are generalized polygons or trees. Finally we give a general definition of a building and some characterizations. 2nd Lecture: We start with the basic facts about spherical buildings and the Moufang property. We explain Tits' extension theorem for spherical buildings an how it implies the Moufang property. We then give an outline of the classification. 3rd Lecture: We give examples of affine buildings and a survey on the main ideas in their theory and their classification. We then move on to the theory of twin buildings and explain how they can be used to study groups of KacMoody type. We will finish with some remarks on the classification problem for arbitrary buildings. 
Date:  October 26 (Tue), 2010, 16:3018:00 
Place:  Room 126, Graduate School of Mathematical Sciences, the University of Tokyo 
Speaker:  Daniel Sternheimer (Keio University and Institut de Mathématiques de Bourgogne) 
Title:  Some instances of the reasonable effectiveness (and limitations) of symmetries and deformations in fundamental physics 
Abstract: [ pdf ] 
In this talk we survey some applications of group theory and deformation theory (including quantization) in mathematical physics. We start with sketching applications of rotation and discrete groups representations in molecular physics ("dynamical" symmetry breaking in crystals, RacahFlatoKibler; chains of groups and symmetry breaking). These methods led to the use of "classification Lie groups" ("internal symmetries") in particle physics. Their relation with spacetime symmetries will be discussed. Symmetries are naturally deformed, which eventually brought to Flato's deformation philosophy and the realization that quantization can be viewed as a deformation, including the many avatars of deformation quantization (such as quantum groups and quantized spaces). Nonlinear representations of Lie groups can be viewed as deformations (of their linear part), with applications to covariant nonlinear evolution equations. Combining all these suggests an Ansatz based on Anti de Sitter spacetime and group, a deformation of the Poincare group of Minkowski spacetime, which could eventually be quantized, with possible implications in particle physics and cosmology. Prospects for future developments between mathematics and physics will be indicated. 
Date:  November 2 (Tue), 2010, 16:3018:00 
Place:  Room 126, Graduate School of Mathematical Sciences, the University of Tokyo 
Speaker:  Michael Eastwood (Australian National University) 
Title:  Twistor theory and the harmonic hull 
Abstract: [ pdf ] 
Harmonic functions are realanalytic and so automatically extend from being functions of real variables to being functions of complex variables. But how far do they extend? This question may be answered by twistor theory, the Penrose transform, and associated geometry. I shall base the constructions on a formula of Bateman from 1904. This is joint work with Feng Xu. 
(GCOE Lectures)  
Date:  November 5 (Fri), 2010, 16:3018:00 
Place:  Room 123, Graduate School of Mathematical Sciences, the University of Tokyo 
Speaker:  Michael Eastwood (Australian National University) 
Title:  How to recognise the geodesics of a metric connection 
Abstract: [ pdf ] 
The geodesics on a Riemannian manifold form a distinguished family of curves, one in every direction through every point. Sometimes two metrics can provide the same family of curves: the Euclidean metric and the round metric on the hemisphere have this property. It is also possible that a family of curves does not arise from a metric at all. Following a classical procedure due to Roger Liouville, I shall explain how to tell these cases apart on a surface. This is joint work with Robert Bryant and Maciej Dunajski. 
(GCOE Lectures)  
Date:  November 8 (Mon), 2010, 16:3018:00 
Place:  Room 128, Graduate School of Mathematical Sciences, the University of Tokyo 
Speaker:  Michael Eastwood (Australian National University) 
Title:  Invariant differential operators on the sphere 
Abstract: [ pdf ] 
The circle is acted upon by the rotation group SO(2) and there are plenty of differential operators invariant under this action. But the circle is also acted upon by SL(2,R) and this larger symmetry group cuts down the list of invariant differential operators to something smaller but more interesting! I shall explain what happens and how this phenomenon generalises to spheres. These constructions are part of a general theory but have numerous unexpected applications, for example in suggesting a new stable finiteelement scheme in linearised elasticity (due to Arnold, Falk, and Winther). 
Date:  December 21 (Tue), 2010, 16:3018:00 
Place:  Room 126, Graduate School of Mathematical Sciences, the University of Tokyo 
Speaker:  Katsuyuki Naoi (直井克之) (The University of Tokyo) 
Title:  Some relation between the Weyl module and the crystal basis of the tensor product of fudamental representations 
Abstract: [ pdf ] 
The Lie algebra defined by the tensor product of a simple Lie algebra and a polynomial ring is called the current algebra, and the Weyl module is defined by a finite dimensional module of the current algebra having some universal property. The fundamental representation is a irreducible, finite dimensional, level zero integrable representation of the quantized affine algebra, and it is known that this module has a crystal basis. If the simple Lie algebra is of ADE type, Fourier and Littelamnn has shown that the Weyl module is isomorphic to a module called the Demazure module. Using this fact, we can easily see that the (Zgraded) characters of the Weyl module and the crystal basis of the tensor product of fundamental representations coincides. In my talk, I will introduce the generalization of this result in the nonsimply laced case. In this case, the result of Fourier and Littelmann does not necessarily true, but we can show the characters of two objects also coincide in this case. This fact is shown using the Demazure modules and its ''crystal basis'' called the Demazure crystals. 
© Toshiyuki Kobayashi