(Tambara, Numata, Gumma)

**Participants from abroad**:

Mike Eastwood, Sigurdur Helgason, Angela Pasquale, Henrik Schlichtkrull

**Organizers**:

Fulton B. Gonzalez, Tomoyuki Kakehi, Toshiyuki Kobayashi, Toshio Oshima

9:30-10:30 11:00-12:00 14:00-15:00 15:20-16:20 16:40-17:40 20:00-20:30 8/ 5 -------------------------------------------------------------- Discussion 8/ 6 Helgason1 Eastwood1 Pasquale1 Kaneyuki Yoshino 8/ 7 Schlichtkrull1 Pasquale Kakehi Nakata Takeuchi Kumahara 8/ 8 Kobayashi Oshima -- Free Discussion -- 8/ 9 Gonzalez Eastwood2 Helgason2 Hiroe Uuganbayar 8/10 Abe Schlihitkrull2Sigurdur Helgason"Support theorems and injectivity for noncompact and compact symmetric spaces" Abstract: In this lecture we prove support theorems relative to horocycles and their interior for symmetric spaces of rank one. We consider both the X-ray tranform and the horocycle Radon transform. We also discuss injectivity and and support questions for the X-ray transform for the compact symmetric spaces.Mike Eastwood"The X-ray transform on complex projective space" Abstract: The X-ray transform on complex projective space is defined by integrating a function or tensor field over the closed geodesics. I shall show how this transform may be related to the ordinary Radon transform and hence determine the kernel of the X-ray transform acting on symmetric tensor fields. The main ingredients are from representation theory and Lie algebra cohomology. This is joint work with Hubert Goldschmidt.Angela Pasquale"Analytic continuation of the resolvent of the Laplace-Beltrami operator on symmetric spaces of the noncompact type" Abstract: Let $\Delta$ be the Laplace-Beltrami operator on a symmetric space of the noncompact type $G/K$, and let $spec(\Delta)$ denote its spectrum. The resolvent $R(z)=(\Delta-z)^{-1}$ is a holomorphic function on $\mathbb C \setminus spec(\Delta)$ with values in the bounded operators on $L^2(G/K)$. We consider the meromorphic continuation of $R$ as distribution valued map on a Riemann surface above $\mathbb C \setminus spec(\Delta)$. In some examples, the singularities of the continued resolvent possess a precise description in terms of representation theory of $G$.Soji Kaneyuki"On the linearity of causal automorphisms of symmetric cones" Abstract: Let A be a simple Euclidean Jordan algebra of rank r. Then there are r+1 open orbits V_0,...,V_r under the action of the identity component of the structure group of A. V_0 is a Riemannian symmetric convex cone, and the others are affine symmetric non-convex cones. The closure C of V_0 is a causal cone. Each V_i has a natural causal structure obtained from the parallel transport of C. We will show that if dim A >2, then a diffeomorphism of V_i leaving the causal structure invariant is necessarily a linear map.Taro Yoshino"On a method to describe the topology of non-Hausdorff space" Abstract: In this talk, we introduce a method to describe the topology of non-Hausdorff space. For a given non-Hausdorff space, we define a diagram consisting of Hausdorff spaces, and see that the diagram determines the topology of the non-Hausdorf space completely.Henrik Schlichtkrull"Holomorphic extension of eigenfunctions on Riemannian symmetric spaces" Abstract: Let X=G/K be a Riemannian symmetric space of the non-compact type, and let X_C be its complexification. The COMPLEX CROWN Xi around X in X_C was introduced by Akhiezer and Gindikin. An important result in the complex analysis for X asserts that every joint eigenfunction on X admits a holomorphic extension to Xi. A new proof of this theorem will be presented, together with some generalizations (jt work with B. Kroez).Tomoyuki Kakehi"Generalized Matrix Radon transform" Abstract:Fuminori Nakata"Wave equation, Funk transform, and the LeBrun-Mason twistor theory" Abstract: We introduce integral transforms from functions on S^2 to functions on the de Sitter 3-space, and show that all the `tame' solutions of the wave equation on the de Sitter 3-space are obtained by using these integral transforms. This result is naturally arisen in the context of the LeBrun-Mason twistor theory, which is also sketched in this talk.Kiyoshi Takeuchi"Geometric Radon transforms and A-hypergeometric functions" (joint work with Y. Matsui) Abstract: Geometric (topological) Radon transforms for constructible functions are closely related to the projective duality in algebraic geometry. In particular we deduce a degree (dimension) formula for A-discriminant varieties, i.e. the singular loci of A-hypergeometric functions. As a byproduct, we prove also a formula for the monodromy at infinity of these functions.Keisaku Kumahara"On solutions of some type of ordinary linear differential equations” Abstract: We state a new approach of representing fundamental solutions of homogeneous linear differential equation $y^{(n)}=f(x)y$. Using these solutions, we get (local) expressions of soutions of linear differential equations of second order.Toshiyuki Kobayashi"Geometric Analysis on Minimal Representations. Representation" Abstract: Minimal representations are the smallest infinite dimensional unitary representations. The Weil representation for the metaplectic group, which plays a prominent role in number theory, is a classic example. We may consider that minimal representations (from the viewpoint of groups) as ''maximal symmetries (from the viewpoint of representation spaces)'', and thus propose to use minimal reprn as a guiding principle to find new interactions with other fields of mathematics. Highlighting geometric analysis on minimal representations of O(p,q), I plan to discuss conservative quantities of ultrahyperbolic equations, the generalization of the Fourier -Hankel transform on the L2-model, and its deformation.Toshio Oshima"Fractional calculus of Weyl algebra and its applications" Abstract: Restrictions of zonal spherical functions and Heckman-Opdam's hypergeometric functions on one-dimensional singular lines through the origin satisfy interesting ordinary differential equations. By a unifying study of ordinary differential equations on the Riemann sphere we have a global structure of their solutions and for example we have a new proof of Gindikin-Karpelevic formula of c-functions and the Gauss summation formula of Heckman-Opdam's hypergeometric functions.Fulton B. Gonzalez"Conical Distributions on the Space of Flat Horocycles" Abstract:Kazuki Hiroe"Generalized Whittaker functions of GL(4,R) and Horn's hypergeometric functions" Abstract:Uuganbayar Zunderiya"Generalized Gelfand hypergeometric systems" (joint work with H. Ochiai) Abstract: We give a combinatorial formula of the dimension of global solutions to a generalization of Gelfand hypergeometric system, where the quadratic differential operators are replaced by higher order operators. We also derive a polynomial estimate of the dimension of global solutions for the case in 3x3 variables. We show that the space of solutions of this generalized system near a generic point is infinite dimensional for some cases.Noriyuki Abe"Jacquet modules of parabolic induction" Abstract: The notion of Jacquet modules was introduced by Casselman. In this talk we introduce some filtration of this generalized Jacquet modules of parabolic induction and investigate this filtration. Roughly speaking, the successive quotient of this filtration is given by ``twisted induction''.Hisaichi MidorikawaTakaaki NomuraHiroyuki OchiaiToshihiko MatsukiNobukazu ShimenoKenji TaniguchiShigeru AokiTakeyoshi KogisoChifune KaiAtsushi YamamoriYoshiki Oshima８月１２日に東大数理科学研究科棟において，小集会を開催します Mini-Conference "Integral Geometry and Representation Theory" 場所：東京大学大学院数理科学研究科 002号室 10:00-11:00Sigurdur Helgason"Radon Transforms and some Applications" 10:20-12:20Fulton Gonzalez"Mutlitemporal Wave Equations : Mean Value Solutions" 14:00-15:00Angela Pasquale"Analytic continuation of the resolvent of the Laplacian in the Euclidean setting" Abstract: We discuss the analytic continuation of the resolvent of the Laplace operator on symmetric spaces of the Euclidean type and some generalizations to the rational Dunkl setting. 15:30-16:30Henrik Schlihitkrull"Decay of smooth vectors for regular representation" Abstract: Let G/H be a homogeneous space of a Lie group, and consider the regular representation L of G on E=L^p(G/H). A smooth vector for L is a function f in E such that g mapsto L(g) f is smooth, G to E. We investigate circumstances under which all such functions decay at infinity (jt with B. Krötz) -------------------------------------------------------------------------------5日の到着案東京 上野 大宮 高崎 上毛高原 セミナーハウス着 A. 11:52 11:58 12:18 12:52 13:09 14時過ぎ B. 15:32 15:38 15:58 16:27 16:44 17時40分頃 高崎線 10:44 11:09 12:38 （高崎線） 渋谷 新宿 池袋 赤羽 大宮 高崎 10:07 10:12 10:18 10:28 10:42 11:56 （湘南新宿ライン） 11:17 11:25 11:30 11:38 11:57 （A案：埼京線快速） 上毛高原から玉原へは，上の A. B. の新幹線の到着時間に合わせてタクシーを手配予定． この時期，JR沼田駅よりセンターハウス近くまで（終点です）ラベンダーバスの便があります． 東京から新幹線利用（運賃：2520円＋特急料金：2720円） 上野から新幹線利用（運賃：2520円＋特急料金：2520円） 渋谷から大宮まで湘南新宿ライン，大宮から新幹線（運賃：2520円＋特急料金：2520円） 高崎まで湘南新宿ラインを利用した場合，高崎−上毛高原間の特急料金は840円玉原国際セミナーハウス宿泊室：宿泊棟に，3名定員の部屋が9室ある（その他，スタッフ室あり. 40名程度まで宿泊可） シーツ，浴衣，タオルが用意されている 浴室 ：２室（大・小），シャンプー・リンス・ドライヤー，シャワールームあり 研修室：60名程度までのセミナーが可能 研究室：20名程度までのセミナーが可能 設備 ：黒板，ホワイトボード，OHP, 液晶プロジェクタ，プリンタ，コピー機，パソコン，無線LAN 洗濯機，乾燥機，コーヒーメーカー 自動販売機：ビール，お茶など 歯ブラシ，ひげそり購入可．バスタオルは有料で貸し出し可 食事 : 朝食 7:30 - 8:30（バイキング形式） 昼食 12:30 - 13:30 夕食 18:00 - 20:00世話人: 大島利雄

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