The Gelfand Mathematical Seminars, 1990–1992 PDF
Preview The Gelfand Mathematical Seminars, 1990–1992
The Gelfand Mathematical Seminars, 1990-1992 L. Corwin I. Gelfand ]. Lepowsky Editors Spririger Science+Business Media. LLC Israel M. Gelfand James Lepowsky Department of Mathematics Department of Mathematics Rutgers University Rutgers University New Brunswick, N.J. 08903 New Brunswick, N.J. 08903 Lawrence Corwin (deceased) Department of Mathematics Rutgers University New Brunswick, N.J. 08903 Library of Congress Cataloging In-Publication Data The Gelfand Mathematical Seminars, 1990-1992 ISSN 1068-7122 Printed on acid-free paper © Springer Science+Business Media New York 1993 Originally published by Birkbliuser Boston in 1993 Softcover reprint of the hardcover 1st edition 1993 Copyright is not claimed for works of U.S. Government employees. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopy ing, recording, or otherwise, without prior permission of the copyright owner. Permission to photocopy for internal or personal use of specific clients is granted by Springer Science+Business Media, LLC for libraries and other users registered with the Copyright Clearance Center (CCC), provided that the base fee of $6.00 per copy, plus $0.20 per page is paid directly to CCC, 21 Congress Street, Salem, MA 01970, U.S.A. Special requests should be addressed directly to Springer Science+Business Media, LLC. ISBN 978-1-4612-6717-1 ISBN 978-1-4612-0345-2 (eBook) DOI 10.1007/978-1-4612-0345-2 Typeset by the Authors in TEX. 9 8 7 6 5 432 1 This volume is dedicated to Lawrence Corwin ljanuary 20, 1943 -March 19,1992) and to Daniel Gorenstein ljanuary 1, 1923 -August 26, 1992) Contents Preface I.M. Gelfand and J. Lepowsky . . . . . . . . . . . . . . . vii Tribute to Daniel Gorenstein R. Lyons ..... . . . . . . . . . . . . . ix The Degeneracy of Two Spectral Sequences J.-L. Brylinski . . . . . . . . . . 1 Hopf Algebra Structures for the Heisenberg Algebra: I L. Corwin and I.M. Gelfand . . . . . .. ....... 11 Avalanches, Sandpiles and Thtte Decompositions A. Gabrielov . . . . . . . . . . . .. ......... 19 On the Dimension and Degree of the Projective Dual Variety: A q-Analog of the Katz-Kleiman Formula I. M. Gelfand and M. M. Kapranov . . . . . . . . . . . . . 27 Crofton Densities and Nonlocal Differentials I. M. Gelfand and M. M. Smirnov ... . ...... 35 On the Local Geometry of a Bihamiltonian Structure I. M. Gelfand and I. Zakharevich . . . . .. . ...... 51 The Classical Polylogarithms, Algebraic K -Theory and (F(n) A. B. Goncharov .......... . 113 A Brief History of the Sporadic Simple Groups D. Gorenstein ........... . 137 Vertex Operator Algebras and Operads Y.-Z. Huang and J. Lepowsky 145 Representations of the Quantized Function Algebras, 2-Categories and Zamolodchikov Tetrahedra Equation D. Kazhdan and Y. Soibelman ......... . 163 Formal (Non)-Commutative Symplectic Geometry M. Kontsevich . . . . . . . . . . . . . 173 Constructible Functions, Lagrangian Cycles and Computational Geometry P. Schapira. . . . . . . . . . . . . 189 Quantum Groups and Perverse Sheaves. An Example V. Schechtman . . . . . . . . . . . . . . . 203 Linearly Recursive Sequences, Witt Algebras and Quantum Groups E. J. Taft ............ . 217 Complexes of Connected Graphs V. A. Vassiliev . . . . . . 223 Preface This Seminar began in Moscow in November 1943 and has continued without interruption up to the present. We are happy that with this vol ume, Birkhiiuser has begun to publish papers of talks from the Seminar. It was, unfortunately, difficult to organize their publication before 1990. Since 1990, most of the talks have taken place at Rutgers University in New Brunswick, New Jersey. Parallel seminars were also held in Moscow, and during July, 1992, at IRES in Bures-sur-Yvette, France. Speakers were invited to submit papers in their own style, and to elaborate on what they discussed in the Seminar. We hope that readers will find the diversity of styles appealing, and recognize that to some extent this reflects the diversity of styles in a mathematical society. The principal aim was to have interesting talks, even if the topic was not especially popular at the time. The papers listed in the Table of Contents reflect some of the rich variety of ideas presented in the Seminar. Not all the speakers submit ted papers. Among the interesting talks that influenced the seminar in an important way, let us mention, for example, that of R. Langlands on per colation theory and those of J. Conway and J. McKay on sporadic groups. In addition, there were many extemporaneous talks as well as short discus sions. This volume is dedicated to the memory of our great friends Larry Corwin and Danny Gorenstein. The mathematical community has deeply felt their loss. The talented mathematician Larry Corwin, widely known for his research on harmonic analysis on p-adic and nilpotent Lie groups, had just begun a new page in his mathematical life. In this volume, we see only the first steps of a promising and exciting program which he was initiating. Of course, we will now forever miss him. We were most fortunate to be able to hear Danny Gorenstein's talk on the sporadic groups. The classification of the finite simple groups is one of the proudest achievements in the mathematics of this century. At the time that the seminar began in the 1940s, when the work of Richard Brauer on characters and representations was being studied in Moscow, it seemed inconceivable that this problem could ever be solved within any reasonable time. Ideas and constructions which were developed during the process of solving this problem have already begun to influence many subjects in mathematics. One of the aims of the Seminar was to begin to learn more about the sporadic groups and their many connections with other topics; we were indeed lucky to hear a lecture by one of the leading pioneers in the classification program. Listening to a talk is not the same as just reading a paper. It is great to be able to feel the spirit of a person who has kept alive this tradition from the time of Brauer until today. Acknowledgments. We owe a great deal to Ann Kostant and Valera Dolotin for their invaluable assistance and editorial help with this volume. Much of our successful effort to expedite publication is due to their com petence and persevering efficiency. I. Gelfand and J. Lepowsky Rutgers University Tribute to Daniel Gorenstein Daniel Gorenstein died on August 26, 1992, at his summer home in Chilmark, Massachusetts. It was a hard and sudden loss to the mathematics community, especially because his final illness was brief, and with characteristic spirit and purpose he was at work until the very end. For Danny (as he was known to all) every morning of the year, virtually without exception, began around dawn with several hours of ferociously intense work on mathematics. For the last thirty years he spent those mornings plotting global strategy for the classification of the finite simple groups, and writing papers many of which were, in his own word, "blockbusters." Though there were other giants in the field, what made Danny unique was his combination of vision, focus, and leadership - both mathematical and personal. He was fascinated with the structure of a "generic" finite simple group. In the 60's, in a fruitful collaboration with John H. Walter, and through his study of Brauer, Feit, Suzuki, and Thompson, his ideas developed and focussed to the point that in the early 70's he proposed - with his considerable expository skill - a specific framework for the classification. This was daring, at a time when one might wake up in the morning to find that someone had discovered yet another sporadic simple group, and when many of the problem busting theorems which later settled the classification were simply dreams; Aschbacher had only just come on the scene. Through the 70's, as the classification accelerated toward its con clusion, he made Rutgers a kind of clearing-house for the problem. He took on the role of "coach" of the classification "team," and a playing coach in his prime, at that - all the while turning out big theorems. He found out everything that was going on and shared his knowledge and insights freely. As people came through his of fice, he probed their mathematics - he loved to schmooze - and by gently persuading them, or becoming impatient, or overwhelming them, or by leaving well enough alone, as he felt the need, he tried to keep them on a track that would be most valuable to solving THE problem. By doing this he played a vital role in the careers of a num ber of young mathematicians. His single-mindedness of purpose and strong opinions and tastes could easily provoke disagreement. But his ideas were engaging and effective, and his purpose was so true that he won the deep loyalty of almost everyone who worked with him. He was never conflicted or afraid about reaching the goal, and no person relished more than he the moment when the classification was finished. In the 80's, he recognized that the classification proof, tens of thousands of pages by a hundred people, in chronological or der, needed to be rewritten for the sake of clarity of logic as well as detail, and to allow earlier results the benefit of later insights. So he began a "revision" project, with several collaborators, which is still unfinished but continuing. The same driving energy, pragmatic optimism, and sure leader ship marked his efforts as chairman of the Mathematics Department at Rutgers, as chairman of a number of prestigious University Com mittees, and as Founding Director of DIMACS from 1988 until his death. To his family and friends he was a devoted, lively, and wise companion, someone who got into our lives because he cared. And so there are many of us who have felt his loss deeply. Richard Lyons Department of Mathematics Rutgers University
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