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621

The Influence of in Geometry and 50 Years of at CINVESTAV

Ludmil Katzarkov Ernesto Lupercio Francisco J. Turrubiates Editors

American Mathematical Society The Influence of Solomon Lefschetz in Geometry and Topology 50 Years of Mathematics at CINVESTAV

Ludmil Katzarkov Ernesto Lupercio Francisco J. Turrubiates Editors

621

The Influence of Solomon Lefschetz in Geometry and Topology 50 Years of Mathematics at CINVESTAV

Ludmil Katzarkov Ernesto Lupercio Francisco J. Turrubiates Editors

American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Dennis DeTurck, Managing Editor Michael Loss Kailash Misra Martin J. Strauss

2010 Mathematics Subject Classification. Primary 14-06, 53-06, 55-06.

Library of Congress Cataloging-in-Publication Data The influence of Solomon Lefschetz in geometry and topology : 50 years of mathematics at CINVESTAV / Ludmil Katzarkov, Ernesto Lupercio, Francisco J. Turrubiates, editors. pages cm. – (Contemporary mathematics ; volume 621) Includes bibliographical references. ISBN 978-0-8218-9494-1 (alk. paper) 1. Lefschetz, Solomon, 1884–1972. 2. Centro de Investigaci´on y de Estudios Avanzados del I.P.N. 3. Institute for Geometry and Physics Miami-Cinvestav-Campinas. 4. Geometry, Al- gebraic. 5. . 6. Topology. I. Katzarkov, Ludmil, 1961– . II. Lupercio, Ernesto, 1970– . III. Turrubiates, Francisco J., 1970– . IV. American Mathematical Society.

QA565.I54 2014 516.3’5–dc23 2013049429

DOI: http://dx.doi.org/10.1090/conm/621

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c 2014 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the Government. Copyright of individual articles may revert to the public domain 28 years after publication. Contact the AMS for copyright status of individual articles. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 191817161514 Contents

Preface vii Solomon Lefschetz and Mexico 1 Recent Progress in Symplectic Flexibility 3 Equivariant Extensions of Differential Forms for Non-compact Lie Groups Hugo Garc´ıa-Compean,´ Pablo Paniagua, and Bernardo Uribe 19 From Classical Theta Functions to Topological Quantum Theory Razvan˘ Gelca and Alejandro Uribe 35 Toric Topology Samuel Gitler 69 Beilinson for Finite-dimensional Associative Algebras Dmitry Kaledin 77 Partial Monoids and Dold-Thom Functors Jacob Mostovoy 89 The Weak b-principle Rustam Sadykov 101 Orbit Configuration Spaces Miguel A. Xicotencatl´ 113 Dynamical Systems and Categories G. Dimitrov, F. Haiden, L. Katzarkov, and M. Kontsevich 133 The Nahm Pole Boundary Condition Rafe Mazzeo and 171

v

Preface

The purpose of this volume is to both celebrate half a century of Mathematics at the Center for Research and Advanced Studies in Mexico City (Cinvestav) and the opening of the Institute for Geometry and Physics Miami-Cinvestav-Campinas. Cinvestav is one of the most prestigious research centers in Latin America. It focuses most of its activities in exact sciences. It was founded in 1961, and one of the founding departments was the Mathematics Department under the direction of Jos´e Adem. At the time, there were two members: Adem, as a full professor, and Samuel Gitler, as an assistant professor (both working in the field of Algebraic Topology). This particular choice was influenced by the fact that both Adem and Gitler were recruited to study their Ph.D.s at Princeton by Solomon Lefschetz, who used to spend half a year in Mexico for many years lecturing at UNAM; he also began a Mexican school of Geometry and Topology. Solomon Lefschetz received the order of the Aguila´ Azteca from President L´opez Mateos in 1964 for his contributions to Mexican Mathematics. Nowadays, the Mathematics Department at Cinvestav organizes the Lefschetz Memorial Lecture Series biyearly: the conference is one of the foremost mathematical events in Latin America drawing some of the most distinguished in the world to deliver three talks at Cinvestav during the fall. The following is the list of all previous Lefschetz lecturers:

• Jack Hale • Michael Atiyah • • Shing S. Chern • • • • Ya G. Sinai • P. Maslov • Sergio Albeverio • • • Graeme Segal • Michael Hopkins • Edward Witten • Yasha Eliashberg

This volume contains the paper by Eliashberg corresponding to his 2012 Lef- schetz Lectures.

vii viii PREFACE

The celebration for the 50th anniversary of the Mathematics Department at Cinvestav took place in 2012 with three main events: TQFT, Langlands and Mirror Symmetry: the Opening Conference of the Institute for Geometry and Physics Miami-Cinvestav-Campinas (IGP-MCC), the 50th year anniversary of Cinvestav, (all together celebrating 50 years of Topology, Geometry and Physics at Cinvestav), and finishing with the International Conference on Symplectic Geometry 2012. This volume contains contributions of the participants from all these events. The Institute for Geometry and Physics Miami-Cinvestav-Campinas (IGP- MCC) is a joint virtual venture to stimulate a North-South collaboration in Geome- try and Physics. The directors are Elisabeth Gasparim, Maxim Kontsevich, Ernesto Lupercio, and Dennis Sullivan, and its founding coincided with the celebrations for the 50th anniversary of Cinvestav. The events were centered in the fields of Geometry (Algebraic and Symplectic) and Algebraic Topology, which are some of the strongest fields at Cinvestav and, as it turned out to be, they reflect quite nicely that they are still of current interest in the Mathematics Department 50 years after Lefschetz. This volume is thus a witness to his enormous influence in Mexican Mathematics. Let us describe very briefly the contents of this volume. In the opening paper, Eliashberg surveys the relation between flexibility and rigidity in Symplectic Ge- ometry and Topology, concentrating primarily on the former. This is an important survey of very recent developments in the area by the leading expert. The second paper by Garc´ıa-Compe´an, Paniagua, and Uribe generalizes a result of Witten on the equivalence of absence of anomalies in gauged WZW actions on compact Lie groups to the existence of equivariant extension of the WZW term in the case in which the gauge group is the special linear group. The third contribution by Gelca and Uribe provides a rigorous construction of Abelian Chern-Simons theory using only the classical theory of theta functions and recovering the Murakami-Ohtsuki-Okada formula for invariants of 3-dimensional . The volume also contains a survey paper by Samuel Gitler (a founding member of the Mathematics Department at Cinvestav 50 years ago) dealing with recent developments on the field of Toric Topology by him and his collaborators. In his paper Kaledin proposes a non-commutative version of Beilinson’s famous conjecture predicting that the regulator map induces an isomorphism between the higher Algebraic K-groups of an Arithmetic scheme and its so-called Beilinson’s co- homology. This non-commutative version of the conjecture predicts the existence of a certain exact triangle of complexes associated to smooth and proper dg-Algebras defined over Z, for which each term is related to either (negative or periodic) cyclic homology or Algebraic K-theory. The main result of this work is the statement that the non-commutative version of Beilinson’s conjecture holds for finite dimensional Algebras. This is a work in the field of non-commutative Algebraic Geometry. The paper by Mostovoy generalizes the classical Dold-Thom result in Algebraic Topology from ordinary homology to any generalized homology theory. Sadykov’s contribution deals with his b-principle. The h-principle is a general observation that differential Geometry problems can often be reduced to problems in (unstable) theory. Similarly, the b-principle is a general observation that differential Geometry problems can often be reduced to problems in stable PREFACE ix ; in his paper, Sadykov surveys some recent developments in the b-principle. Xicot´encatl contributes to the volume with a survey on orbit configuration spaces. Some of the results deal with the existence of certain natural fibrations extending those of Fadell and Neuwirth, Lie algebras associated to the lower cen- tral series of fundamental groups of these spaces, cohomology rings, the Longhoni- Salvatore result of the failure of configuration spaces to preserve homotopy equiv- alences of closed manifolds, and structures of loop-spaces. The volume continues with a paper written by Dimitrov, Haiden, Katzarkov, and Kontsevich. This is a paper densely packed with deep results. One of the main objects of study in this contribution is the entropy of automorphisms of non- commutative Algebraic varieties as exemplified by saturated A∞ categories. The paper is organized around two major results. The first is an explicit identification between the categorical entropy of an endofunctor of the wrapped Fukaya category of a punctured Riemann surface and the classical entropy of a pseudo-Anosov map in the corresponding mapping class. With their second result, the authors give a complete answer to the problem of detecting the density of phases of stable objects in categories of representations of quivers. Finally, Rafe Mazzeo and Edward Witten conclude this volume with a study of the Nahm pole boundary condition in an ambitious program to generalize the classical Chern-Simons theory of knot invariants. As the reader can notice, many active fields within Geometry and Topology have been represented as we have celebrated our first 50 years of Mathematics at Cinvestav.

Ernesto Lupercio Ludmil Katzarkov Francisco Turrubiates

Selected Published Titles in This Series

621 Ludmil Katzarkov, Ernesto Lupercio, and Francisco J. Turrubiates, Editors, The Influence of Solomon Lefschetz in Geometry and Topology, 2014 614 James W. Cogdell, Freydoon Shahidi, and David Soudry, Editors, Automorphic Forms and Related Geometry, 2014 612 Patricio Cifuentes, Jos´eGarc´ıa-Cuerva, Gustavo Garrig´os, Eugenio Hern´andez, Jos´eMar´ıa Martell, Javier Parcet, Keith M. Rogers, Alberto Ruiz, Fernando Soria, and Ana Vargas, Editors, Harmonic Analysis and Partial Differential Equations, 2014 611 Robert Fitzgerald Morse, Daniela Nikolova-Popova, and Sarah Witherspoon, Editors, , Combinatorics, and Computing, 2014 610 Pavel Etingof, Mikhail Khovanov, and Alistair Savage, Editors, Perspectives in Representation Theory, 2014 609 Dinh Van Huynh, S. K. Jain, Sergio R. L´opez-Permouth, S. Tariq Rizvi, andCosminS.Roman,Editors, Ring Theory and Its Applications, 2014 608 Robert S. Doran, Greg Friedman, and Scott Nollet, Editors, Hodge Theory, Complex Geometry, and Representation Theory, 2014 607 Kiyoshi Igusa, Alex Martsinkovsky, and Gordana Todorov, Editors, Expository Lectures on Representation Theory, 2014 606 Chantal David, Matilde Lal´ın, and Michelle Manes, Editors, Women in Numbers 2, 2013 605 Omid Amini, Matthew Baker, and Xander Faber, Editors, Tropical and Non-Archimedean Geometry, 2013 604 Jos´eLuisMonta˜na and Luis M. Pardo, Editors, Recent Advances in Real Complexity and Computation, 2013 603 Azita Mayeli, Alex Iosevich, Palle E. T. Jorgensen, and Gestur Olafsson,´ Editors, Commutative and Noncommutative Harmonic Analysis and Applications, 2013 602 Vyjayanthi Chari, Jacob Greenstein, Kailash C. Misra, K. N. Raghavan, and Sankaran Viswanath, Editors, Recent Developments in Algebraic and Combinatorial Aspects of Representation Theory, 2013 601 David Carf`ı, Michel L. Lapidus, Erin P. J. Pearse, and Machiel van Frankenhuijsen, Editors, Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics II, 2013 600 David Carf`ı, Michel L. Lapidus, Erin P. J. Pearse, and Machiel van Frankenhuijsen, Editors, Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics I, 2013 599 Mohammad Ghomi, Junfang Li, John McCuan, Vladimir Oliker, Fernando Schwartz, and Gilbert Weinstein, Editors, Geometric Analysis, Mathematical Relativity, and Nonlinear Partial Differential Equations, 2013 598 Eric Todd Quinto, Fulton Gonzalez, and Jens Gerlach Christensen, Editors, Geometric Analysis and Integral Geometry, 2013 597 Craig D. Hodgson, William H. Jaco, Martin G. Scharlemann, and Stephan Tillmann, Editors, Geometry and Topology Down Under, 2013 596 Khodr Shamseddine, Editor, Advances in Ultrametric Analysis, 2013 595 James B. Serrin, Enzo L. Mitidieri, and Vicent¸iu D. R˘adulescu, Editors, Recent Trends in Nonlinear Partial Differential Equations II, 2013 594 James B. Serrin, Enzo L. Mitidieri, and Vicent¸iu D. R˘adulescu, Editors, Recent Trends in Nonlinear Partial Differential Equations I, 2013 593 Anton Dzhamay, Kenichi Maruno, and Virgil U. Pierce, Editors, Algebraic and Geometric Aspects of Integrable Systems and Random Matrices, 2013

For a complete list of titles in this series, visit the AMS Bookstore at www.ams.org/bookstore/conmseries/. CONM 621 0Yaso ahmtc tCINVESTAV at Mathematics of Years 50

The influence of Solomon Lefschetz (1884–1972) in geometry and topology 40 years after his death has been very profound. Lefschetz’s influence in Mexican mathematics has been even greater. In this volume, celebrating 50 years of mathematics at Cinvestav-Mexico,´ many of the fields of geometry and topology are represented by some of the leaders of their respective fields. This volume opens with Michael Atiyah reminiscing about his encounters with Lef- schetz and Mexico.´ Topics covered in this volume include symplectic flexibility, Chern– Simons theory and the theory of classical theta functions, toric topology, the Beilinson con- jecture for finite-dimensional associative algebras, partial monoids and Dold–Thom func- tors, the weak b-principle, orbit configuration spaces, equivariant extensions of differential forms for noncompact Lie groups, dynamical systems and categories, and the Nahm pole boundary condition. • azro ta. Editors al., et Katzarkov

ISBN 978-0-8218-9494-1

9 780821 894941

CONM/621 AMS