Mathematisches Forschungsinstitut Oberwolfach Arithmetic Geometry
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Raport De Activitate Pe Anul 2009
Raport de Activitate - 2009 Colectivul IMAR January 26, 2010 1 Lucrari publicate la finele lui 2008 si necontinute in Raportul pe 2008 1.1 In reviste cotate ISI 1. Belinschi, Serban; Nica, Alexandru: On a remarkable semigroup of homomorphisms with respect to free multiplicative convolution, Indiana University Mathematics Journal, volum 57, No.4 (2008), pag. 1679 – 1713 2. D. Beltit¸˘a,K.-H. Neeb: A non-smooth continuous unitary representation of a Banach-Lie group, Journal of Lie Theory 18 (2008), no. 4, pag. 933–936. 3. C. Calinescu, J. Lepowsky, A. Milas Vertex-algebraic structure of the principal subspaces (1) of certain A1 -modules, II: higher level case , Journal of Pure and Applied Algebra, 212 (2008), pag. 1928– 1950 4. Dorin Cheptea, Kazuo Habiro, Gwenael Massuyeau: A functorial LMO invariant for Lagrangian cobordisms, Geometry & Topology 12:2 (2008), pag. 1091 – 1170 (MR 2403806) 5. Alexandru Constantinescu Hilbert Function and Betti Numbers of Algebras with Lefschetz Property of Order m, Communications in Algebra, 36 (2008), pag. 4704 – 4720 6. Bruno Benedetti, Alexandru Constantinescu, Matteo Varbaro Dimension, Depth and Zero-Divisors of the Algebra of Basic k-Covers of a Graph, Le Matematiche, Volume LXIII, Issue II, (2008) , pag. 117–156. 7. Alexandru Constantinescu, Le Dinh Nam The Standard Graded Property for Vertex Cover Algebras of Quasi-Trees, Le Matematiche, Volume LXIII, Issue II, (2008), pag. 173–183. 8. I.Aberbach, F. Enescu: Lower bounds for Hilbert-Kunz multiplicities in local rings of fixed dimension, Mich. Math. Journal vol. 57 (2008) special volume in honor of M. Hochster, pag. 1-16 9. -
The Duality Theorem on a Generalized Flag Variety
proceedings of the american mathematical society Volume 117, Number 3, March 1993 REMARKS ON LOCALIZATION AND STANDARD MODULES: THE DUALITY THEOREM ON A GENERALIZED FLAG VARIETY JEN-TSEH CHANG (Communicated by Jonathan M. Rosenberg) Abstract. An amplification of the duality theorem of Hecht, Milicic, Schmid, and Wolf is given in the setting of a generalized flag variety. As an application, we give a different proof of the reducibility theorem for principal series by Speh and Vogan. The purpose of this note is to sketch an amplification of the duality theo- rem of Hecht, Milicic, Schmid, and Wolf. The main theorem in [3] establishes a natural duality between Harish-Chandra modules constructed in two differ- ent ways: the ^-module construction on a flag variety (due to Beilinson and Bernstein) on the one hand and the cohomological induction from a Borel sub- algebra (due to Zuckerman) on the other hand. This also follows from a result of Bernstein, which gives a ^-module construction of Zuckerman's functors (for the construction, see [1]). In this note we introduce a similar Z)-module construction on a generalized flag variety. Following Hecht, Milicic, Schmid, and Wolf, this construction is then dual to the cohomological induction from a parabolic subalgebra (when containing a a-stable Levi factor). As an applica- tion of this result, we give a different proof of a result by Speh and Vogan [4], which asserts the irreducibility of certain standard Zuckerman modules. This introduction is followed by four sections. The first section recalls the notion of a generalized t.d.o. on a generalized flag variety introduced in [2], from which our construction stems. -
Rapport Annuel 2014-2015
RAPPORT ANNUEL 2014-2015 Présentation du rapport annuel 1 Programme thématique 2 Autres activités 12 Grandes Conférences et colloques 16 Les laboratoires du CRM 20 Les prix du CRM 30 Le CRM et la formation 34 Les partenariats du CRM 38 Les publications du CRM 40 Comités à la tête du CRM 41 Le CRM en chiffres 42 Luc Vinet Présentation En 2014-2015, contrairement à ce qui était le cas dans (en physique mathématique) à Charles Gale de l’Université les années récentes, le programme thématique du CRM a McGill et le prix CRM-SSC (en statistique) à Matías été consacré à un seul thème (très vaste !) : la théorie des Salibián-Barrera de l’Université de Colombie-Britannique. nombres. L’année thématique, intitulée « La théorie des Les Grandes conférences du CRM permirent au grand public nombres : de la statistique Arithmétique aux éléments Zêta », de s’initier à des sujets variés, présentés par des mathémati- a été organisée par les membres du CICMA, un laboratoire ciens chevronnés : Euler et les jets d’eau de Sans-Souci du CRM à la fine pointe de la recherche mondiale, auxquels il (par Yann Brenier), la mesure des émotions en temps réel faut ajouter Louigi Addario-Berry (du Groupe de probabilités (par Chris Danforth), le mécanisme d’Anticythère (par de Montréal). Je tiens à remercier les quatre organisateurs de James Evans) et l’optique et les solitons (par John Dudley). cette brillante année thématique : Henri Darmon de l’Univer- L’année 2014-2015 fut également importante du point de sité McGill, Chantal David de l’Université Concordia, Andrew vue de l’organisation et du financement du CRM. -
Localization of Cohomologically Induced Modules to Partial Flag Varieties
LOCALIZATION OF COHOMOLOGICALLY INDUCED MODULES TO PARTIAL FLAG VARIETIES by Sarah Kitchen A dissertation submitted to the faculty of The University of Utah in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Mathematics The University of Utah March 2010 Copyright c Sarah Kitchen 2010 All Rights Reserved THE UNIVERSITY OF UTAH GRADUATE SCHOOL SUPERVISORY COMMITTEE APPROVAL of a dissertation submitted by Sarah Kitchen This dissertation has been read by each member of the following supervisory committee and by majority vote has been found to be satisfactory. Chair: Dragan Miliˇci´c Aaron Bertram Peter Trapa Henryk Hecht Y.P. Lee THE UNIVERSITY OF UTAH GRADUATE SCHOOL FINAL READING APPROVAL To the Graduate Council of the University of Utah: I have read the dissertation of Sarah Kitchen in its final form and have found that (1) its format, citations, and bibliographic style are consistent and acceptable; (2) its illustrative materials including figures, tables, and charts are in place; and (3) the final manuscript is satisfactory to the Supervisory Committee and is ready for submission to The Graduate School. Date Dragan Miliˇci´c Chair, Supervisory Committee Approved for the Major Department Aaron Bertram Chair/Dean Approved for the Graduate Council Charles A. Wight Dean of The Graduate School ABSTRACT Abstract goes here. ?? CONTENTS ABSTRACT ...................................................... ii CHAPTERS 1. INTRODUCTION ............................................. 1 1.1 Cohomological Induction and Harish-Chandra Modules . 1 1.2 The Kazhdan-Lusztig Conjectures . 2 1.3 Localization of Harish-Chandra Modules . 3 1.4 The Beilinson-Ginzburg Equivariant Derived Category . 5 1.5 Equivariant Zuckerman Functors . -
Galois Cohomology and Homogeneous Varieties
GALOIS COHOMOLOGY IN DEGREE THREE AND HOMOGENEOUS VARIETIES∗ by Emmanuel Peyre Abstract. — The central result of this paper is the following generalization of a result of the author on products of Severi•Brauer varieties. Let G be a semi•simple linear algebraic group over a field k. Let V be a generalized flag variety under G. Then there exist finite extensions ki of k for 1 6 i 6 m, elements αi in Brki and a natural exact sequence m N (. α ) ki /k i k ∪ Ker H3(k,Q/Z(2)) H3(k(V ),Q/Z(2)) CH2(V ) 0. i∗ −−−−−−→ → → tors→ Mi=1 After giving a more explicit expression of the second morphism in a particular case, we apply this result to get classes in H3(Q,Q/Z), which are k•negligible for any field k of characteristic different from 2 which contains a fourth root of unity, for a group Q which is a central extension of an F2 vector space by another. Résumé. — Le résultat central de ce texte est la généralisation suivante d’un résultat de l’auteur sur les produits de variétés de Severi•Brauer. Soit G un groupe algébrique linéaire semi•simple sur un corps k. Soit V une variété de drapeaux généralisée sous G. Alors il existe des extensions finies ki de k pour 1 6 i 6 m, des éléments αi de Brki et une suite exacte naturelle m N (. α ) ki /k i k ∪ Ker H3(k,Q/Z(2)) H3(k(V ),Q/Z(2)) CH2(V ) 0. -
Henri Darmon
Henri Darmon Address: Dept of Math, McGill University, Burnside Hall, Montreal, PQ. E-mail: [email protected] Web Page: http://www.math.mcgill.ca/darmon Telephone: Work (514) 398-2263 Home: (514) 481-0174 Born: Oct. 22, 1965, in Paris, France. Citizenship: Canadian, French, and Swiss. Education: 1987. B.Sc. Mathematics and Computer Science, McGill University. 1991. Ph.D. Mathematics, Harvard University. Thesis: Refined class number formulas for derivatives of L-series. University Positions: 1991-1994. Princeton University, Instructor. 1994-1996. Princeton University, Assistant Professor. 1994-1997. McGill University, Assistant Professor. 1997-2000. McGill University, Associate Professor. 2000- . McGill University, Professor. 2005-2019. James McGill Professor, McGill University. Other positions: 1991-1994. Cercheur hors Qu´ebec, CICMA. 1994- . Chercheur Universitaire, CICMA. 1998- . Director, CICMA (Centre Interuniversitaire en Calcul Math´ematique Alg´ebrique). 1999- . Member, CRM (Centre de Recherches Math´ematiques). 2005-2014. External member, European network in Arithmetic Geometry. Visiting Positions: 1991. IHES, Paris. 1995. Universit´a di Pavia. 1996. Visiting member, MSRI, Berkeley. 1996. Visiting professor and guest lecturer, University of Barcelona. 1997. Visiting Professor, Universit´e Paris VI (Jussieu). 1997. Visitor, Institut Henri Poincar´e. 1998. Visiting Professor and NachDiplom lecturer, ETH, Zuric¨ h. 1999. Visiting professor, Universit`a di Pavia. 2001. Visiting professor, Universit`a di Padova. 2001. Korea Institute for Advanced Study. 2002. Visiting professor, RIMS and Saga University (Japan). 1 2003. Visiting Professor, Universit´e Paris VI, Paris. 2003. Visiting professor, Princeton University. 2004. Visiting Professor, Universit´e Paris VI, Paris. 2006. Visiting Professor, CRM, Barcelona, Spain. 2008. Visiting Professor, Universit´e Paris-Sud (Orsay). -
Visible Actions of Reductive Algebraic Groups on Complex Algebraic Varieties (簡約代数群の複素代数多様体への可視的作用について) 田中雄一郎
博士論文 (要約) 論文題目 Visible actions of reductive algebraic groups on complex algebraic varieties (簡約代数群の複素代数多様体への可視的作用について) 田中 雄一郎 1 2 Summary of Ph. D Thesis submitted to the University of Tokyo Yuichiro Tanaka We study visible actions on complex algebraic varieties, and the main result is a classifica- tion of visible actions on generalized flag varieties. Definition 1.1 (Kobayashi [Ko2]). We say a holomorphic action of a Lie group G on a complex manifold X is strongly visible if the following two conditions are satisfied: (1) There exists a real submanifold S (called a \slice") such that X0 := G · S is an open subset of X: (2) There exists an anti-holomorphic diffeomorphism σ of X0 such that σjS = idS; σ(G · x) = G · x for any x 2 X0: In the above setting, we say the action of G on X is S-visible. This terminology will be used also if S is just a subset of X. Definition 1.2 (Kobayashi [Ko2]). We say a holomorphic action of a Lie group G on a complex manifold X is previsible if the condition (1) of Definition 1.1 is satisfied for a totally real submanifold S of X. The notion of visible actions on complex manifolds was introduced by T. Kobayashi [Ko2] with the aim of uniform treatment of multiplicity-free representations of Lie groups. Definition 1.3. We say a unitary representation V of a locally compact group G is multiplicity-free if the ring EndG(V ) of intertwining operators on V is commutative. To prove the multiplicity-freeness property of representations of Lie groups in the frame- work of visible actions ([Ko1, Ko2, Ko5, Ko6]), we use Kobayashi's theory of the propaga- tion of the multiplicity-freeness property under the assumption of visible actions. -
Mathematics Calendar
Mathematics Calendar The most comprehensive and up-to-date Mathematics Calendar information is available on e-MATH at http://www.ams.org/mathcal/. August 2005 Brown, Antonio Campillo, Max Karoubi, Jean-Louis Loday, Susan Montgomery, Adrian Ocneanu, Jose Antonio de la Pena,˜ Vladimir Mathematical Modeling of Infectious Diseases: Dynamics and Popov, Hans-Jurgen Schneider, Aron Simis, Frank Sottile, Richard Control, Institute for Mathematical Sciences, National University Stanley, Boris Tsygan, Mariusz Wodzicki. of Singapore, Singapore 118402. (May 2005, p. 568) Organizing and Scientific Committee: Walter Ferrer Santos (Co- Information: KP Chua, Administrative Officer, Institute for Mathe- ord.), Gerardo Gonzalez-Sprinberg, Alfredo Jones, Alvaro Rittatore, matical Sciences, National University of Singapore, 3 Prince George’s Andrea Solotar. Park, Singapore 118402; tel: (+65) 6874 1893; Facsimile: (+65) 6873 8292; http://www.ims.nus.edu.sg. Deadline: May 1st, 2005. Information:http://www.cmat.edu.uy/cmat/eventos/16cla/en; 1–5 14th USENIX Security Symposium, Baltimore, Maryland. (May Walter Ferrer: email: [email protected] 2005, p. 568) Description: The USENIX Security Symposium brings together 1–December 23 Pattern Formation in Large Domains, Isaac researchers, practitioners, system administrators, system pro- Newton Institute for Mathematical Sciences, Cambridge, England. grammers, and others interested in the latest advances in security (Aug. 2004, p. 834) of computer systems. Organizers: J.H.P. Dawes (Cambridge), M. Golubitsky (Houston), Information: Contact: Conference Department, email: conference@ P.C. Matthews (Nottingham), A.M. Rucklidge (Leeds). usenix.org; tel: 510-528-8649; http://www.usenix.org/sec05/. Information: http://www.newton.cam.ac.uk/programmes/PFD/; Isaac Newton Institute for Mathematical Sciences, 20 Clarkson 1–9 XVI Coloquio Latinoamericano de Algebra, Colonia, Uruguay. -
Overconvergent Eichler-Shimura Isomorphisms on Shimura Curves Over a Totally Real Field
Overconvergent Eichler-Shimura Isomorphisms on Shimura Curves over a Totally Real Field Shan Gao AThesis In the Department of Mathematics and Statistics Presented in Partial Fulfillment of the Requirements For the Degree of Doctor of Philosophy (Mathematics) at Concordia University Montreal, Quebec, Canada September 2016 c Shan Gao, 2016 CONCORDIA UNIVERSITY SCHOOL OF GRADUATE STUDIES This is to verify that the thesis prepared By: Shan Gao Entitled: Overconvergent Eichler-Shimura Isomorphisms on Shimura Curves over a Totally Real Field and submitted in partial fulfilment of requirements for the degree of Docotor of Philosophy (Mathematics) complies with the regulations of the University and meets the accepted standards with respect to originality and quality. Signed by the final examining committee: Dr. Marcie Frank Chair Dr. Henri Darmon External Examiner Dr. Pablo Bianucci External to Program Dr. Hershy Kisilevsky Examiner Dr. Christopher Cummins Examiner Dr. Adrian Iovita Thesis Supervisor Approved by Dr. Arusharka Sen Chair of Department or Graduate Program Director September 8, 2016 Dr. Andr´eRoy Dean of Faculty ABSTRACT Overconvergent Eichler-Shimura Isomorphisms on Shimura Curves over a Totally Real Field Shan Gao, Ph.D. Concordia University, 2016 In this work we construct overconvergent Eichler-Shimura isomorphisms on Shimura curves over a totally real field F . More precisely, for a prime p>2 and a wide open disk U in the weight space, we construct a Hecke-Galois-equivariant morphism from the space of families of overconvergent modular symbols over U to the space of families of overconvergent modular forms over U. In addition, for all but finitely many weights λ ∈ U, this morphism provides a description of the finite slope part of the space of overconvergent modular symbols of weight λ in terms of the finite slope part of the space of overconvergent modular forms of weight λ + 2. -
Linear Algebraic Groups
Appendix A Linear Algebraic Groups A.1 Linear Algebraic Groups Let us fix an algebraically closed field K. We work in the category of affine varieties over K. The purpose of this section is to give a brief exposition on the basic facts of algebraic groups. Definition A.1.1 A linear algebraic group is an affine algebraic variety G together with a unit element e 2 G and morphisms m W G G ! G and i W G ! G satisfying the group axioms (a) m.; e/ D m.e;/D for all 2 G; (b) m.; i.// D m.i./; / D e for all 2 G; (c) m.; m.; // D m.m.; /; / (associativity) for all ; ; 2 G. Often we will just write and 1 instead of m.; / and i./, respectively. Example A.1.2 The group GLn, the set of invertible nn matrices over K, is a linear algebraic group. Any Zariski closed subgroup G GLn is a linear algebraic group. In fact, every linear algebraic group is isomorphic to a Zariski closed subgroup of GLn (see Borel [1, Proposition I.1.10]). This justifies calling it a linear algebraic group. G The axioms for a linear algebraic group can also be stated in terms of the coordinate ring. The maps m W G G ! G and i W G ! G correspond to ring homomorphisms m W KŒG ! KŒG ˝ KŒG and i W KŒG ! KŒG. The unit element e 2 G corresponds to a ring homomorphism W KŒG ! K defined by f 7! f .e/. The axioms for a linear algebraic group translate to (a) .id ˝/ ı m D . -
RATIONAL POINTS on GENERALIZED FLAG VARIETIES and UNIPOTENT CONJUGACY in FINITE GROUPS of LIE TYPE 1. Introduction Let Gln(Q)
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 361, Number 1, January 2009, Pages 177–206 S 0002-9947(08)04442-5 Article electronically published on July 30, 2008 RATIONAL POINTS ON GENERALIZED FLAG VARIETIES AND UNIPOTENT CONJUGACY IN FINITE GROUPS OF LIE TYPE SIMON M. GOODWIN AND GERHARD ROHRLE¨ Dedicated to Professor J. A. Green on the occasion of his 80th birthday Abstract. Let G be a connected reductive algebraic group defined over the finite field Fq,whereq is a power of a good prime for G.WewriteF for F the Frobenius morphism of G corresponding to the Fq-structure, so that G is a finite group of Lie type. Let P be an F -stable parabolic subgroup of G and let U be the unipotent radical of P . In this paper, we prove that the number of U F -conjugacy classes in GF is given by a polynomial in q, under the assumption that the centre of G is connected. This answers a question of J. Alperin (2006). In order to prove the result mentioned above, we consider, for unipotent u ∈ F P0 G , the variety u of G-conjugates of P whose unipotent radical contains u. F P0 We prove that the number of q-rational points of u is given by a polynomial in q with integer coefficients. Moreover, in case G is split over Fq and u is split (in the sense of T. Shoji, 1987), the coefficients of this polynomial are given by P0 the Betti numbers of u. We also prove the analogous results for the variety Pu consisting of conjugates of P that contain u. -
Multiple Flag Varieties and Tensor Product Decompositions 3.1
Multiple flag varieties EVGENY SMIRNOV Abstract. This is a survey of results on multiple flag varieties, i.e. varieties of the form G=P1 × · · · × G=Pr. We provide a classi- fication of multiple flag varieties of complexity 0 and 1 and results on the combinatorics and geometry of B-orbits and their closures in double cominuscule flag varieties. We also discuss questions of finiteness for the number of G-orbits and existence of an open G-orbits on a multiple flag variety. 1. Introduction 1.1. Multiple flag varieties. Grassmann varieties and flag varieties first appeared at the end of the 19th century in the papers of Her- mann G¨unter Grassmann, Julius Pl¨ucker, Hermann Schubert and other mathematicians. These varieties are homogeneous spaces of the group G = GL(V ), with a parabolic subgroup P as the stabilizer of a point. One can also consider homogeneous spaces G=P not only for GL(V ), but also for other connected reductive group. In this survey we consider multiple flag varieties: these are direct products of several flag varieties, i.e. varieties of the form G=P1 × · · · × G=Pr (for r = 2 they are called double flag varieties). Each multiple flag variety can be viewed as a G-variety for the diagonal action of the group G. Among these, spherical multiple flag varieties are of particular inter- est. They are given by the following property: the algebra of regular functions on them as a G-module is decomposed into the sum of ir- reducible G-modules with multiplicity zero or one.