Analysis, Geometry and Probability: Essays in honour of K. R. Parthasarathy

425 Pages·1996·29.3 MB·English

by  Rajendra Bhatia (eds.)

t 0
TEXTS AND READINGS
IN MATHEMATICS
Analysis, Geometry and Probability
Texts and Readings in Mathematics
Advisory Editor
C. S. Seshadri, sPIe Scienee Foundation, Madras.
Managing Editor
Rajendra Bhatia, Indian Statistical Institute, New Delhi.
Editors
V. S. Borkar, Indian Institute of Scienee. Bangalore.
R. L. Karandikar, Indian Statistical Institute. New Delhi.
C. Musili, University of Hyderabad. Hyderabad.
K. H. Paranjape, Tata Institute of Fundamental Research, Bombay.
V. S. Sunder, Indian StatisticaI Institute, Bangalore.
Already Published Volumes
R. B. Bapat: Linear Algebra and Linear Models
R. Bhatia: Fourier Series
C. Musili: Representations of Finite Groups
H. Helson: Linear Algebra ( Second Edition)
D. Sarason: Notes on Complex Function Theory
M.G. Nadkami: Basic Ergodie Theory
H. Helson: Harmonie Analysis ( Second Edition)
K. Chandrasekharan: A Course on Integration Theory
K. Chandrasekharan: A Course on Topologieal Groups
Analysis, Geometry and Probability
Essays in honour of K. R . Parthasarathy
Edited by
Rajendra Bbatia
with the assistance uf
Abhay G. Bhatt
~OUrß HINDUSTA N
BOOKAGENCY
Published by Hindustan Book Agency (lndia)
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Copyright © 1996 by Hindustan Book Agency ( India)
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ISBN 978-81-85931-12-8 ISBN 978-93-80250-87-8 (eBook)
DOI 10.1007/978-93-80250-87-8
Contents
Prefa.ee vii
1 Covariant Poisson Fields In Fock Space
David Applebaum . . . . . . . . . 1
2 Remarks On The Moduli Space 0/ Principal Bundles
V. Balaji and C. S. Seshadri .......... . 16
3 Scattering 0/ Products 0/ Random Matrices
S. G. Dani . . . . . .......... . 33
4 The Representation Theory 0/ P-adic GL(n) And
Deligne-Langlands Parnmeters
J. E. HodgiIlß and R. J. Plymen . . . . . . . 54
5 The Structure 0/ Classical Noise
A. S. Holevo . . . . . . 73
. 6 Thinking Noncommutatively
R. L. Hudson .. 87
7 Fermat's Last Theorem
V. Kumar Murty . . . . . . . . . . . . . . . . 125
8 Axiomatic And Coordinate Geometry
Kapil Paranjape . . . . . . . . . . . . . . . . . . . . . . . . . 140
9 Counting Finite Groups
M. Ram Murty . . . . . 161
10 From Infinitely Divisible Representations To Cohomological
Rigidity
Klaus Schmidt . . . . . . . . . . . . . . . . . . . . . . . . . . 173
11 Riemann Sur/aces
R. R. Simha and V. Srinivas . . . . . . . . . . . 198
12 Representations 0/ Q-Commutation Relations
Kalyan B. Sinha. . . . . . . . . . . . . . . . . . . . . 274
13 Some Qualitative Uncertainty Prineiples In Harmonie
Analysis
A. Sitaram . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
14 Notes On The Imprimitivity Theorem
V. S. Sunder . . . . . . . . . . . . . . . . . . . . . . . . 299
15 Homogenization And Related Developments
M. Vanninathan . . . . . . . . . . .. ............ 322
16 Quantum Kinematios And Projective Unitary
Representations 0/ Abelian Groups
V. S. Varadarajan . . . . . . . . . . . . . . . . . . . . . . . . 362
17 Nongradient Models In Hydrodynamic Scaling
S. R. S. Varadhan . . . . . . . . . . . . . . . . . . . . . . . . 397
Preface
This book is a collection of articles dedicated to K.R. Parthasarathy on
bis sixtieth birthday.
K.R.P., as he is commonly called, is an wlUsual, complex and many
faceted person. He is a mathematician of rare distinction and has made
important contributions to different areas of mathematics: probability, rep
resentations of groups and quantum stochastic calculus. He has written
elementary and advanced books that have been widely influential. He has
served on many committees, adding to their staid and self-important pro
ceedings some colour by his irreverent remarks. He is engaging and enter
taining in conversation, acerbic but not malicious, exasperating at times,
hut not offensive. He has an austerity and simplicity that come with a rieh
inner life.
Above all, K.R.P. is an extraordinarily perfect teacher. His clear, metic
ulously organised and sonorously delivered lectures are a1\'1ays instructive
and inspiring. This collection oi essays is a tribute to this aspect of his lire.
I invited a few mathematicians to contribute expository articles, not
necessarily related to their own research work, that will explain to advanced
students some interesting ideas and themes. The model I had in mind was
a general seminar talk by K.R.P. The essays collected here were received in
response to this invitation. They are on different topics and are at different
levels. But a1l of them are instructional and educative. I am grateful to
all the authors for giving generously their time and effort. Before going
into print each article was read by another mathematician, whose comments
sometimes led to changes in the original article. To a11 these co11eagues, I
am thankful for their help. Dr. Abhay Bhatt has helped me in editing, and
has done a11 the tedmical work in compiling the papers.
On turning 60, K.R.P. was retired from the services of the Indian Star
tistical Institute, where he has worked with dedication and distinction for
the better part of bis life. K.R.P., of course, remains physically and men
taUy younger than almost all his colleagues. It is, therefore, safe to predict
what retirement would mean to him. He will continue doing what he haB
done every day for several years. Early in the morning, he will sit down,
unmindful of the world around rum, a leaky pen in hand, an old irtkpot by
bis side, and start writing symbols in his neat hand on faded paper. Those
who enter bis room will continue to wonder whether he is trying to prove
bis next lemma, or thinking of his next lecture, which might turn out to be
the most exhilarating part of their day.
Whatever it might be, an of US, :who have known K.R,P" and cherish
this sight, wish him tbe very best in the years to come.
July 1996 Rajendra Bhatia
Covariant Poisson Fields In Fock Space
DAVID ApPLEBAUM
Department of Mathematics, Statistics and Operational Research
Burton Street, Nottingham, NG! 4BU, England
Dedicated to K.R. Parthasarathy on his 60th birthday
1 Introduction
As I type these words, I have in front of me a collection of notes from a
seminar in Quantum Probability run by Luigi Acca.rdi in Roma in 1984-5
where I was at the time based as a research fellow. In those days I was
rather meticulous in ordering my jottings and on turning to page 131 in
my file I find twenty pages entitled "Representations of Current Groups" by
K.R. Parthasarathy. I remember being fascinated by these lectures (I think
there were two) in which Partha gave an account of how the Levy-Khinchine
formula arises. naturally from the construction of continuous tensor products
of unitary group representations (see e.g. [St], [Ar], [PS]) and then revealed
the beautiful Fock space decomposition of infinitely divisible random vari
ables as sums of creation, conservation and armihilation operators. This i8
in fact one of the many perspectives through which the historiau cau see
the "birth" of quantum probability - as a student of Robin Hudson, I had
seen the subject evolve from a different point of view through the search for
I
quantum centrallimit theorems and non-commuting analogue8 of Brownian
motion. The marriage of these two viewpoints turned out to be a wonderful
thing, leading as it did to the creation of quantum stochastic calculus.
In 1990, I began to make a systematic study of infinite divisibility and
I was fortunate to obtain a copy of the beautiful essay [P2] which I believe
grew out ofthe 1985 Roma lectures I had attended earlier. The account givell
therein of the Fock space structure of Levy processes (stochastic processes
with stationary and independent illcrements) was the essential starting point