Table Of ContentThe 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.
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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
