Curriculum Vitae for Marius Crainic
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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. -
Fundamental Algebraic Geometry
http://dx.doi.org/10.1090/surv/123 hematical Surveys and onographs olume 123 Fundamental Algebraic Geometry Grothendieck's FGA Explained Barbara Fantechi Lothar Gottsche Luc lllusie Steven L. Kleiman Nitin Nitsure AngeloVistoli American Mathematical Society U^VDED^ EDITORIAL COMMITTEE Jerry L. Bona Peter S. Landweber Michael G. Eastwood Michael P. Loss J. T. Stafford, Chair 2000 Mathematics Subject Classification. Primary 14-01, 14C20, 13D10, 14D15, 14K30, 18F10, 18D30. For additional information and updates on this book, visit www.ams.org/bookpages/surv-123 Library of Congress Cataloging-in-Publication Data Fundamental algebraic geometry : Grothendieck's FGA explained / Barbara Fantechi p. cm. — (Mathematical surveys and monographs, ISSN 0076-5376 ; v. 123) Includes bibliographical references and index. ISBN 0-8218-3541-6 (pbk. : acid-free paper) ISBN 0-8218-4245-5 (soft cover : acid-free paper) 1. Geometry, Algebraic. 2. Grothendieck groups. 3. Grothendieck categories. I Barbara, 1966- II. Mathematical surveys and monographs ; no. 123. QA564.F86 2005 516.3'5—dc22 2005053614 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. -
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. -
Campus Commits $30M to Altgeld/Illini Hall Renovation Project!
Department of Mathematics, Fall/Winter 2013 Campus commits $30M to Altgeld/Illini Hall renovation project! There has never been a greater need The proposed project will restore Altgeld Mathor demand for mathematics and Hall to its historicTimes splendor and bring its mathematicians. New science and facilities up to the standards worthy of a technology are creating unprecedented world-class University. It will transform opportunities to address serious Illini Hall from an aging anachronism into societal challenges and advance a vibrant center for collaborative learning the human condition. Mathematics and research. The proposed renovation provides uniquely effective tools for will move Illinois to the forefront of a solving problems and communicating revolution in new quantitative methods in about them; it makes interdisciplinary science and society, and it will benefit the research in the sciences possible. thousands of students who learn from our At Illinois, the number of majors in courses each year. Actuarial Science, Mathematics, and Specifically the renovation will Statistics is up 30% since 2003, and including majors and non-majors we • modernize 14,600 square feet of teach 30% more student-classes than classroom space and create 6,800 square we did then. feet of new classroom space, appropriate for the interactive learning which is The world-class scientific enterprise essential for 21st century instruction. at Illinois puts the Departments of Mathematics and Statistics • create 5,300 square feet of space for in an extraordinary position to collaborative instruction and research, an assume national leadership in the increase of almost 700%. mathematical sciences. And yet we • unlock 11,000 square feet of new space face a crisis: insufficient space that is Rendering of proposed collaborative space (top) for growth in these two departments. -
Curriculum Vitae - Rogier Bos
Curriculum Vitae - Rogier Bos Personal Full name: Rogier David Bos. Date and place of birth: August 7, 1978, Ter Aar, The Netherlands. Email: rogier bos [at] hotmail.com. Education 2010 M. Sc. Mathematics Education at University of Utrecht, 2007 Ph. D. degree in Mathematics at the Radboud University Nijmegen, thesis: Groupoids in geometric quantization, advisor: Prof. dr. N. P. Landsman. Manuscript committee: Prof. dr. Gert Heckman, Prof. dr. Ieke Moerdijk (Universiteit Utrecht), Prof. dr. Jean Renault (Universit´ed'Orl´eans), Prof. dr. Alan Weinstein (University of California), Prof. dr. Ping Xu (Penn State University). 2001 M. Sc. (Doctoraal) in Mathematics at the University of Utrecht, thesis: Operads in deformation quantization, advisor: Prof. dr. I. Moerdijk; minors in philosophy and physics. 1997 Propedeuse in Mathematics at the University of Utrecht. 1997 Propedeuse in Physics at the University of Utrecht. 1996 Gymnasium at Erasmus College Zoetermeer. General Working experience 2010-now Co-author dutch mathematics teaching series \Wagen- ingse Methode". 2010-now Lecturer at Master Mathematics Education at Hogeschool Utrecht. 2009-now Mathematics and general science teacher at the Chris- telijk Gymnasium Utrecht. 2007-2008 Post-doctoral researcher at the Center for Mathematical Analysis, Geometry, and Dynamical Systems, Instituto Superior T´ecnicoin Lisbon (Portugal). 2002-2007 Ph.D.-student mathematical physics at the University of Amsterdam (Radboud University Nijmegen from 2004); funded by FOM; promotor: Prof. dr. N. P. Landsman. 2002 march-august Mathematics teacher at the highschool Minkema College in Woerden. 1 1999-2001 Teacher of mathematics exercise classes at the Univer- sity of Utrecht. University Teaching Experience 2007 Mathematics deficiency repair course for highschool students (in preparation of studying mathematics or physics), Radboud Univer- sity Nijmegen. -
An Outline of Algebraic Set Theory
An Outline of Algebraic Set Theory Steve Awodey Dedicated to Saunders Mac Lane, 1909–2005 Abstract This survey article is intended to introduce the reader to the field of Algebraic Set Theory, in which models of set theory of a new and fascinating kind are determined algebraically. The method is quite robust, admitting adjustment in several respects to model different theories including classical, intuitionistic, bounded, and predicative ones. Under this scheme some familiar set theoretic properties are related to algebraic ones, like freeness, while others result from logical constraints, like definability. The overall theory is complete in two important respects: conventional elementary set theory axiomatizes the class of algebraic models, and the axioms provided for the abstract algebraic framework itself are also complete with respect to a range of natural models consisting of “ideals” of sets, suitably defined. Some previous results involving realizability, forcing, and sheaf models are subsumed, and the prospects for further such unification seem bright. 1 Contents 1 Introduction 3 2 The category of classes 10 2.1 Smallmaps ............................ 12 2.2 Powerclasses............................ 14 2.3 UniversesandInfinity . 15 2.4 Classcategories .......................... 16 2.5 Thetoposofsets ......................... 17 3 Algebraic models of set theory 18 3.1 ThesettheoryBIST ....................... 18 3.2 Algebraic soundness of BIST . 20 3.3 Algebraic completeness of BIST . 21 4 Classes as ideals of sets 23 4.1 Smallmapsandideals . .. .. 24 4.2 Powerclasses and universes . 26 4.3 Conservativity........................... 29 5 Ideal models 29 5.1 Freealgebras ........................... 29 5.2 Collection ............................. 30 5.3 Idealcompleteness . .. .. 32 6 Variations 33 References 36 2 1 Introduction Algebraic set theory (AST) is a new approach to the construction of models of set theory, invented by Andr´eJoyal and Ieke Moerdijk and first presented in [16]. -
Integrability of Lie Brackets
Annals of Mathematics, 157 (2003), 575–620 Integrability of Lie brackets By Marius Crainic and Rui Loja Fernandes* Abstract In this paper we present the solution to a longstanding problem of dif- ferential geometry: Lie’s third theorem for Lie algebroids. We show that the integrability problem is controlled by two computable obstructions. As ap- plications we derive, explain and improve the known integrability results, we establish integrability by local Lie groupoids, we clarify the smoothness of the Poisson sigma-model for Poisson manifolds, and we describe other geometrical applications. Contents 0. Introduction 1. A-paths and homotopy 1.1. A-paths 1.2. A-paths and connections 1.3. Homotopy of A-paths 1.4. Representations and A-paths 2. The Weinstein groupoid 2.1. The groupoid G(A) 2.2. Homomorphisms 2.3. The exponential map 3. Monodromy 3.1. Monodromy groups 3.2. A second-order monodromy map 3.3. Computing the monodromy 3.4. Measuring the monodromy ∗ The first author was supported in part by NWO and a Miller Research Fellowship. The second author was supported in part by FCT through program POCTI and grant POCTI/1999/MAT/33081. Key words and phrases. Lie algebroid, Lie groupoid. 576 MARIUS CRAINIC AND RUI LOJA FERNANDES 4. Obstructions to integrability 4.1. The main theorem 4.2. The Weinstein groupoid as a leaf space 4.3. Proof of the main theorem 5. Examples and applications 5.1. Local integrability 5.2. Integrability criteria 5.3. Tranversally parallelizable foliations Appendix A. Flows A.1. Flows and infinitesimal flows A.2. -
Chern Characters Via Connections up to Homotopy
Chern characters via connections up to homotopy ∗ Marius Crainic Department of Mathematics, Utrecht University, The Netherlands 1 Introduction: The aim of this note is to point out that Chern characters can be com- puted using curvatures of “connections up to homotopy”, and to present an application to the vanishing theorem for Lie algebroids. Classically, Chern characters are computed with the help of a connection and its cur- vature. However, one often has to relax the notion of connection so that one gains more freedom in representing these characteristic classes by differential forms. A well known ex- ample is Quillen’s notion of super-connection [8]. Here we remark that one can weaken the notion of (super-)connections even further, to what we call “up to homotopy”. Our interest on this type of connections comes from the theory of characteristic classes of Lie algebroids [2, 5] (hence, in particular of Poisson manifolds [6]). From our point of view, the intrinsic characteristic classes are secondary classes which arise from a vanishing result: the Chern classes of the adjoint representation vanish (compare to Bott’s approach to char- acteristic classes for foliations). We have sketched a proof of this in [2] for a particular class of Lie algebroids (the so called regular ones). The problem is that the adjoint representation is a representation up to homotopy only [4]. For the general setting, we have to show that Chern classes can be computed using connections up to homotopy. Since we believe that this result might be of larger interest, we have chosen to present it at the level vector bundles over manifolds. -
Lectures on Poisson Geometry
Lectures on Poisson Geometry Marius Crainic Rui Loja Fernandes Ioan M˘arcut, Mathematical Institute, Utrecht University E-mail address: [email protected] Department of Mathematics, University of Illinois at Urbana- Champaign E-mail address: [email protected] Institute for Mathematics, AstrophysicsMarch 11, and 2021 Particle Physics, Radboud University Nijmegen E-mail address: [email protected] Draft Draft March 11, 2021 To our families Draft March 11, 2021 Draft March 11, 2021 Contents Preface xiii List of notations and symbols xvii List of conventions xix Part 1. Basic Concepts Chapter 1. Poisson BracketsMarch 11, 2021 3 §1.1. Poisson brackets 3 §1.2. Orbits 5 §1.3. Poisson and Hamiltonian diffeomorphisms 6 §1.4. Examples 8 §1.5. PoissonDraft actions and quotients 14 Problems 19 Chapter 2. Poisson Bivectors 23 §2.1. The point of view of bivectors 23 §2.2. A slight twist: π] 28 §2.3. Poisson maps and bivector fields 30 §2.4. Examples 31 Problems 40 Chapter 3. Local Structure of Poisson Manifolds 43 §3.1. The Weinstein Splitting Theorem 43 §3.2. Regular points 46 vii viii Contents §3.3. Singular points 47 §3.4. The isotropy Lie algebra 50 §3.5. Linearization of Poisson structures 53 Problems 55 Notes and references for Part 1 57 Part 2. Poisson Geometry around Leaves Chapter 4. Symplectic Leaves and the Symplectic Foliation 61 §4.1. The symplectic foliation 61 §4.2. Regular Poisson structures 67 §4.3. More examples of symplectic foliations 71 §4.4. The coupling construction 74 Problems 79 Chapter 5. Poisson Transversals 83 §5.1. -
The Weil Algebra and the Van Est Isomorphism Tome 61, No 3 (2011), 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 Camilo ARIAS ABAD & Marius CRAINIC The Weil algebra and the Van Est isomorphism Tome 61, no 3 (2011), p. 927-970. <http://aif.cedram.org/item?id=AIF_2011__61_3_927_0> © Association des Annales de l’institut Fourier, 2011, 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 61, 3 (2011) 927-970 THE WEIL ALGEBRA AND THE VAN EST ISOMORPHISM by Camilo ARIAS ABAD & Marius CRAINIC (*) Abstract. — This paper belongs to a series of papers devoted to the study of the cohomology of classifying spaces. Generalizing the Weil algebra of a Lie algebra and Kalkman’s BRST model, here we introduce the Weil algebra W (A) associated to any Lie algebroid A. We then show that this Weil algebra is related to the Bott-Shulman complex (computing the cohomology of the classifying space) via a Van Est map and we prove a Van Est isomorphism theorem. -
43567137.Pdf
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Radboud Repository PDF hosted at the Radboud Repository of the Radboud University Nijmegen The following full text is an author's version which may differ from the publisher's version. For additional information about this publication click this link. http://hdl.handle.net/2066/129021 Please be advised that this information was generated on 2017-12-05 and may be subject to change. Foliation groupoids and their cyclic homology∗ Marius Crainic and Ieke Moerdijk Department of Mathematics, Utrecht University, The Netherlands Introduction The purpose of this paper is to prove two theorems which concern the position of ´etale groupoids among general smooth (or “Lie”) groupoids. Our motivation comes from the non-commutative geometry and algebraic topology concerning leaf spaces of foliations. Here, one is concerned with invariants of the holonomy groupoid of a foliation [4, 34], such as the cohomology of its classifying space [14], the cyclic homology of its smooth convolution algebra [2, 7], or the K-theory of the C∗-convolution algebras. Many results here depend on the fact that such a holonomy groupoid can be “reduced” to what is called a complete transversal of the foliation, giving rise to an equivalent ´etale groupoid. For ´etale groupoids (sometimes called r-discrete groupoids in the literature [30, 33]), the cyclic homology, sheaf theory and classifying spaces are each well understood, as is the relation between these. Our first theorem provides a criterion for determining whether a given Lie groupoid is equivalent to an ´etale one. -
Normal Forms for Poisson Maps and Symplectic Groupoids Around Poisson Transversals
Lett Math Phys DOI 10.1007/s11005-017-1007-2 Normal forms for Poisson maps and symplectic groupoids around Poisson transversals 1 2 Pedro Frejlich · Ioan M˘arcut, Received: 31 January 2017 / Revised: 19 September 2017 / Accepted: 19 September 2017 © The Author(s) 2017. This article is an open access publication Abstract Poisson transversals are submanifolds in a Poisson manifold which intersect all symplectic leaves transversally and symplectically. In this communication, we prove a normal form theorem for Poisson maps around Poisson transversals. A Poisson map pulls a Poisson transversal back to a Poisson transversal, and our first main result states that simultaneous normal forms exist around such transversals, for which the Poisson map becomes transversally linear, and intertwines the normal form data of the transversals. Our second result concerns symplectic integrations. We prove that a neighborhood of a Poisson transversal is integrable exactly when the Poisson transversal itself is integrable, and in that case we prove a normal form theorem for the symplectic groupoid around its restriction to the Poisson transversal, which puts all structure maps in normal form. We conclude by illustrating our results with examples arising from Lie algebras. Keywords Symplectic manifolds, general · Poisson manifolds · Poisson groupoids and algebroids · Lie algebras and Lie superalgebras Mathematics Subject Classification 53D05 · 53D17 · 17Bxx B Ioan M˘arcut, [email protected] Pedro Frejlich [email protected] 1 Universidade Federal do Rio Grande do Sul, Campus Litoral Norte Rodovia RS 030, 11.700 - Km 92, Emboaba, Tramandaí, RS CEP 95590-000, Brazil 2 IMAPP, Radboud University Nijmegen, 6500 GL Nijmegen, The Netherlands 123 P.