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A Tribute to Henri Cartan
A Tribute to Henri Cartan This collection of articles paying tribute to the mathematician Henri Cartan was assembled and edited by Pierre Cartier, IHÉS, and Luc Illusie, Université Paris-Sud 11, in consultation with Jean-Pierre Serre, Collège de France. The collection begins with the present introductory article, which provides an overview of Cartan's work and a short contribution by Michael Atiyah. This overview is followed by three additional articles, each of which focuses on a particular aspect of Cartan's rich life. —Steven G. Krantz This happy marriage, which lasted until his death Jean-Pierre Serre (followed, a few months later, by that of his wife), Henri Cartan produced five children: Jean, Françoise, Étienne, 8 July 1904–13 August 2008 Mireille, and Suzanne. In September 1939, at the beginning of the Henri Cartan was, for many of the younger gen- war, he moved to Clermont-Ferrand, where the eration, the symbol of the resurgence of French University of Strasbourg had been evacuated. A mathematics after World War II. He died in 2008 year later he got a chair at the Sorbonne, where he at the age of 104 years. was given the task of teaching the students of the Personal Life ENS. This was a providential choice that allowed the “normaliens” (and many others) to benefit for Henri was the eldest son of the mathematician more than twenty-five years (1940–1965) from Élie Cartan (1869–1951), born in Dolomieu (Isère), his courses and seminars. In fact there was a two- and of his wife Marie-Louise Bianconi, of Corsican year interruption when he returned to Strasbourg origin. -
Lectures on Integrability of Lie Brackets
Geometry & Topology Monographs 17 (2011) 1–107 1 Lectures on integrability of Lie brackets MARIUS CRAINIC RUI LOJA FERNANDES These lectures discuss the problem of integrating infinitesimal geometric structures to global geometric structures. 22A22, 53D17, 58H05 Preface The subject of these lecture notes is the problem of integrating infinitesimal geometric structures to global geometric structures. We follow the categorical approach to differ- ential geometry, where the infinitesimal geometric structures are called Lie algebroids and the global geometric structures are called Lie groupoids. It is also one of our aims to convince you of the advantages of this approach. You may not be familiar with the language of Lie algebroids or Lie groupoids, but you have already seen several instances of the integrability problem. For example, the problem of integrating a vector field to a flow, the problem of integrating an involutive distribution to a foliation, the problem of integrating a Lie algebra to a Lie group or the problem of integrating a Lie algebra action to a Lie group action. In all these special cases, the integrability problem always has a solution. However, in general, this need not be the case and this new aspect of the problem, the obstructions to integrability, is one of the main topics of these notes. One such example, that you may have seen before, is the problem of geometric quantization of a presymplectic manifold, where the so-called prequantization condition appears. These notes are made up of five lectures. In Lecture 1, we introduce Lie groupoids and their infinitesimal versions. In Lecture 2, we introduce the abstract notion of a Lie algebroid and explain how many notions of differential geometry can be described in this language. -
Integrability of Jacobi and Poisson Structures Tome 57, No 4 (2007), P
R AN IE N R A U L E O S F D T E U L T I ’ I T N S ANNALES DE L’INSTITUT FOURIER Marius CRAINIC & Chenchang ZHU Integrability of Jacobi and Poisson structures Tome 57, no 4 (2007), p. 1181-1216. <http://aif.cedram.org/item?id=AIF_2007__57_4_1181_0> © Association des Annales de l’institut Fourier, 2007, tous droits réservés. L’accès aux articles de la revue « Annales de l’institut Fourier » (http://aif.cedram.org/), implique l’accord avec les conditions générales d’utilisation (http://aif.cedram.org/legal/). Toute re- production en tout ou partie cet article sous quelque forme que ce soit pour tout usage autre que l’utilisation à fin strictement per- sonnelle du copiste est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.cedram.org/ Ann. Inst. Fourier, Grenoble 57, 4 (2007) 1181-1216 INTEGRABILITY OF JACOBI AND POISSON STRUCTURES by Marius CRAINIC & Chenchang ZHU (*) Abstract. — We discuss the integrability of Jacobi manifolds by contact groupoids, and then look at what the Jacobi point of view brings new into Poisson geometry. In particular, using contact groupoids, we prove a Kostant-type theorem on the prequantization of symplectic groupoids, which answers a question posed by We- instein and Xu. The methods used are those of Crainic-Fernandes on A-paths and monodromy group(oid)s of algebroids. In particular, most of the results we obtain are valid also in the non-integrable case. -
Nuno Crato Paolo Paruolo Editors How Access to Microdata
Nuno Crato Paolo Paruolo Editors Data-Driven Policy Impact Evaluation How Access to Microdata is Transforming Policy Design Data-Driven Policy Impact Evaluation Nuno Crato • Paolo Paruolo Editors Data-Driven Policy Impact Evaluation How Access to Microdata is Transforming Policy Design Editors Nuno Crato Paolo Paruolo University of Lisbon Joint Research Centre Lisbon, Portugal Ispra, Italy ISBN 978-3-319-78460-1 ISBN 978-3-319-78461-8 (eBook) https://doi.org/10.1007/978-3-319-78461-8 Library of Congress Control Number: 2018954896 © The Editor(s) (if applicable) and The Author(s) 2019. This book is an open access publication. Open Access This book is licensed under the terms of the Creative Commons Attribution 4.0 Inter- national License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made. The images or other third party material in this book are included in the book’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the book’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. -
Higher Localized Analytic Indices and Strict Deformation Quantization3
Higher localized analytic indices and strict deformation quantization Paulo Carrillo Rouse Abstract. This paper is concerned with the localization of higher analytic indices for Lie groupoids. Let G be a Lie groupoid with Lie algebroid AG . Let ∞ τ be a (periodic) cyclic cocycle over the convolution algebra Cc (G ). We say that τ can be localized if there is a correspondence 0 ∗ Ind K (A G ) −→τ C satisfying Indτ (a) = hindDa, τi (Connes pairing). In this case, we call Indτ the higher localized index associated to τ. In[CR08a] we use the algebra of functions over the tangent groupoid introduced in [CR08b], which is in fact a strict deformation quantization of the Schwartz algebra S (AG ), to prove the following results: • Every bounded continuous cyclic cocycle can be localized. • If G is ´etale, every cyclic cocycle can be localized. We will recall this results with the difference that in this paper, a formula for higher localized indices will be given in terms of an asymptotic limit of a pairing at the level of the deformation algebra mentioned above. We will discuss how the higher index formulas of Connes-Moscovici, Gorokhovsky-Lott fit in this unifying setting. Contents 1. Introduction 1 2. Index theory for Lie groupoids 4 3. Indextheoryandstrictdeformationquantization 11 arXiv:0810.4480v1 [math.KT] 24 Oct 2008 4. Higher localized indices 15 References 20 1. Introduction This paper is concerned with the localization of higher analytic indices for Lie groupoids. In [CM90], Connes and Moscovici defined, for any smooth manifold M ¯ ev and every Alexander-Spanier class [¯ϕ] ∈ Hc (M), a localized index morphism 0 ∗ C (1) Indϕ : Kc (T M) −→ . -
Rostislav Grigorchuk Texas A&M University
CLAREMONT CENTER for MATHEMATICAL SCIENCES CCMSCOLLOQUIUM On Gap Conjecture for group growth and related topics by Rostislav Grigorchuk Texas A&M University Abstract: In this talk we will present a survey of results around the Gap Conjecture which was formulated by thep speaker in 1989. The conjecture states that if growth of a finitely generated group is less than e n, then it is polynomial. If proved, it will be a far reaching improvement of the Gromov's remarkable theorem describing groups of polynomial growth as virtually nilpotent groups. Old and new results around conjecture and its weaker and stronger forms will be formulated and discussed. About the speaker: Professor Grigorchuk received his undergraduate degree in 1975 and Ph.D. in 1978, both from Moscow State University, and a habilitation (Doctor of Science) degree in 1985 at the Steklov Institute of Mathematics in Moscow. During the 1980s and 1990s, Rostislav Grigorchuk held positions in Moscow, and in 2002 joined the faculty of Texas A&M University, where he is now a Distinguished Professor of Mathematics. Grigorchuk is especially famous for his celebrated construction of the first example of a finitely generated group of intermediate growth, thus answering an important problem posed by John Milnor in 1968. This group is now known as the Grigorchuk group, and is one of the important objects studied in geometric group theory. Grigorchuk has written over 100 research papers and monographs, had a number of Ph.D. students and postdocs, and serves on editorial boards of eight journals. He has given numerous talks and presentations, including an invited address at the 1990 International Congress of Mathematicians in Kyoto, an AMS Invited Address at the March 2004 meeting of the AMS in Athens, Ohio, and a plenary talk at the 2004 Winter Meeting of the Canadian Mathematical Society. -
Morita Equivalence and Continuous-Trace C*-Algebras, 1998 59 Paul Howard and Jean E
http://dx.doi.org/10.1090/surv/060 Selected Titles in This Series 60 Iain Raeburn and Dana P. Williams, Morita equivalence and continuous-trace C*-algebras, 1998 59 Paul Howard and Jean E. Rubin, Consequences of the axiom of choice, 1998 58 Pavel I. Etingof, Igor B. Frenkel, and Alexander A. Kirillov, Jr., Lectures on representation theory and Knizhnik-Zamolodchikov equations, 1998 57 Marc Levine, Mixed motives, 1998 56 Leonid I. Korogodski and Yan S. Soibelman, Algebras of functions on quantum groups: Part I, 1998 55 J. Scott Carter and Masahico Saito, Knotted surfaces and their diagrams, 1998 54 Casper Goffman, Togo Nishiura, and Daniel Waterman, Homeomorphisms in analysis, 1997 53 Andreas Kriegl and Peter W. Michor, The convenient setting of global analysis, 1997 52 V. A. Kozlov, V. G. Maz'ya> and J. Rossmann, Elliptic boundary value problems in domains with point singularities, 1997 51 Jan Maly and William P. Ziemer, Fine regularity of solutions of elliptic partial differential equations, 1997 50 Jon Aaronson, An introduction to infinite ergodic theory, 1997 49 R. E. Showalter, Monotone operators in Banach space and nonlinear partial differential equations, 1997 48 Paul-Jean Cahen and Jean-Luc Chabert, Integer-valued polynomials, 1997 47 A. D. Elmendorf, I. Kriz, M. A. Mandell, and J. P. May (with an appendix by M. Cole), Rings, modules, and algebras in stable homotopy theory, 1997 46 Stephen Lipscomb, Symmetric inverse semigroups, 1996 45 George M. Bergman and Adam O. Hausknecht, Cogroups and co-rings in categories of associative rings, 1996 44 J. Amoros, M. Burger, K. Corlette, D. -
Arxiv:1510.00545V4 [Math.GR] 29 Jun 2016 7
SCHREIER GRAPHS OF GRIGORCHUK'S GROUP AND A SUBSHIFT ASSOCIATED TO A NON-PRIMITIVE SUBSTITUTION ROSTISLAV GRIGORCHUK, DANIEL LENZ, AND TATIANA NAGNIBEDA Abstract. There is a recently discovered connection between the spectral theory of Schr¨o- dinger operators whose potentials exhibit aperiodic order and that of Laplacians associated with actions of groups on regular rooted trees, as Grigorchuk's group of intermediate growth. We give an overview of corresponding results, such as different spectral types in the isotropic and anisotropic cases, including Cantor spectrum of Lebesgue measure zero and absence of eigenvalues. Moreover, we discuss the relevant background as well as the combinatorial and dynamical tools that allow one to establish the afore-mentioned connection. The main such tool is the subshift associated to a substitution over a finite alphabet that defines the group algebraically via a recursive presentation by generators and relators. Contents Introduction 2 1. Subshifts and aperiodic order in one dimension 3 2. Schr¨odingeroperators with aperiodic order 5 2.1. Constancy of the spectrum and the integrated density of states (IDS) 5 2.2. The spectrum as a set and the absolute continuity of spectral measures 9 2.3. Aperiodic order and discrete random Schr¨odingeroperators 10 3. The substitution τ, its finite words Subτ and its subshift (Ωτ ;T ) 11 3.1. The substitution τ and its subshift: basic features 11 3.2. The main ingredient for our further analysis: n-partition and n-decomposition 14 3.3. The maximal equicontinuous factor of the dynamical system (Ωτ ;T ) 15 3.4. Powers and the index (critical exponent) of Subτ 19 3.5. -
Arxiv:1003.6025V1 [Math.HO] 31 Mar 2010 Karl Stein (1913-2000)
Karl Stein (1913-2000) Alan Huckleberry Karl Stein was born on the first of January 1913 in Hamm in Westfalen, grew up there, received his Abitur in 1932 and immediately thereafter be- gan his studies in M¨unster. Just four years later, under the guidance of Heinrich Behnke, he passed his Staatsexam, received his promotion and became Behnke’s assistant. Throughout his life, complex analysis, primarily in higher dimensions (“mehrere Ver¨anderliche”), was the leitmotif of Stein’s mathematics. As a fresh Ph.D. in M¨unster in 1936, under the leadership of the master Behnke, he had already been exposed to the fascinating developments in this area. The brilliant young Peter Thullen was proving fundamental theorems, Henri Cartan had visited M¨unster, and Behnke and Thullen had just writ- ten the book on the subject. It must have been clear to Stein that this was the way to go. Indeed it was! The amazing phenomenon of analytic continuation in higher dimensions had already been exemplified more than 20 years be- fore in the works of Hartogs and E. E. Levi. Thullen’s recent work had gone much further. In the opposite direction, Cartan and Thullen had arXiv:1003.6025v1 [math.HO] 31 Mar 2010 proved their characterization of domains in Cn which admit a holomor- phic function which can not be continued any further. Behnke himself was also an active participant in mathematics research, always bringing new ideas to M¨unster. This was indeed an exciting time for the young researcher, Karl Stein. Even though the pest of the Third Reich was already invading academia, Behnke kept things going for as long as possible. -
Interview with Henri Cartan, Volume 46, Number 7
fea-cartan.qxp 6/8/99 4:50 PM Page 782 Interview with Henri Cartan The following interview was conducted on March 19–20, 1999, in Paris by Notices senior writer and deputy editor Allyn Jackson. Biographical Sketch in 1975. Cartan is a member of the Académie des Henri Cartan is one of Sciences of Paris and of twelve other academies in the first-rank mathe- Europe, the United States, and Japan. He has re- maticians of the twen- ceived honorary doc- tieth century. He has torates from several had a lasting impact universities, and also through his research, the Wolf Prize in which spans a wide va- Mathematics in 1980. riety of areas, as well Early Years as through his teach- ing, his students, and Notices: Let’s start at the famed Séminaire the beginning of your Cartan. He is one of the life. What are your founding members of earliest memories of Bourbaki. His book Ho- mathematical inter- mological Algebra, writ- est? ten with Samuel Eilen- Cartan: I have al- berg and first published ways been interested Henri Cartan’s father, Élie Photograph by Sophie Caretta. in 1956, is still in print in mathematics. But Cartan. Henri Cartan, 1996. and remains a standard I don’t think it was reference. because my father The son of Élie Cartan, who is considered to be was a mathematician. I had no doubt that I could the founder of modern differential geometry, Henri become a mathematician. I had many teachers, Cartan was born on July 8, 1904, in Nancy, France. good or not so good. -
Contemporary Mathematics 526
CONTEMPORARY MATHEMATICS 526 Nonlinear Partial Differential Equations and Hyperbolic Wave Phenomena The 2008–2009 Research Program on Nonlinear Partial Differential Equations Centre for Advanced Study at the Norwegian Academy of Science and Letters Oslo, Norway Helge Holden Kenneth H. Karlsen Editors American Mathematical Society http://dx.doi.org/10.1090/conm/526 Nonlinear Partial Differential Equations and Hyperbolic Wave Phenomena CONTEMPORARY MATHEMATICS 526 Nonlinear Partial Differential Equations and Hyperbolic Wave Phenomena The 2008–2009 Research Program on Nonlinear Partial Differential Equations Centre for Advanced Study at the Norwegian Academy of Science and Letters Oslo, Norway Helge Holden Kenneth H. Karlsen Editors American Mathematical Society Providence, Rhode Island Editorial Board Dennis DeTurck, managing editor George Andrews Abel Klein Martin J. Strauss 2000 Mathematics Subject Classification. Primary 35L65, 35Q30, 35G25, 35J70, 35K65, 35B65, 39A14, 35B25, 35L05, 65M08. Library of Congress Cataloging-in-Publication Data Det Norske Videnskaps-Akademi. Research Program on Nonlinear Partial Differential Equations (2008–2009 : Centre for Advanced Study at the Norwegian Academy of Science and Letters) Nonlinear partial differential equations and hyperbolic wave phenomena : 2008–2009 Research Program on Nonlinear Partial Differential Equations, Centre for Advanced Study at the Norwegian Academy of Science and Letters, Oslo, Norway / [edited by] Helge Holden, Kenneth H. Karlsen. p. cm. — (Contemporary mathematics ; v. 526) Includes -
Attachment Theory – What Do You Really Need to Know?
Issue 3 February 2019 EVIDENCE-INFORMED EDUCATION – MYTH-BUSTING – BUILDING A COMMUNITY Nick Rose Attachment theory – what do you really need to know? A LIFE LESS Jodie Lomax ORDINARY Leading evidence in your school PAUL KIRSCHNER Nuno Crato Everything starts INTERVIEW with a C – curriculum Postgraduate Certificate in Educational Assessment and Examinations Take your understanding of assessment to the next level The Postgraduate Certificate in Educational Assessment and “It was a very valuable Examinations is a practice-based qualification designed to directly impact your work as you learn to apply various research learning experience that methodologies to your professional context. I will be returning to It is taught through a mix of online learning and Saturday Day repeatedly over the next Schools in Cambridge and led by experts from Cambridge 12 months, re-reading, Assessment and the University of Cambridge Faculty of Education. re-thinking and You will be awarded a Postgraduate Certificate in Educational adapting practice.” Studies (PCES) worth 60 credits at Master’s level (Level 7) at the end of the course. Find out more: www.cambridgeassessment.org.uk/events/pgca/ 1240_PGCA_advert_210x297mm.indd 1 21/09/2018 12:08 CONTENTS Message from the EDitor Everything starts with the curriculum 4 27 Nuno Crato This issue’s contributors 5 Cognitive load theory in the classroom 33 Tom Needham 6 To sleep, perchance to learn Joe Kirby How I became the leader of evidence in my 37 school Attachment theory: what do teachers need to Jodie Lomax CONTENTS9