Preface
The articles in this collection mainly grew out of the talks given at a Conference
held at UCLA in January 2008, which honored V. S. Varadarajan on his 70th
birthday. The main theme of the Conference was symmetry in mathematics and
physics. More precisely, the talks at the conference were dedicated to the interplay
between geometry, group theory, and fundamental physics. In addition to the
speakers there were a number of doctoral and post doctoral fellows including several
students of Varadarajan who had worked under him on these topics throughout his
career.
Varadarajan’s work over the past 50 years represents a broad spectrum of math-
ematics but its main emphasis has been on symmetry in mathematics and math-
ematical physics, broadly interpreted. Some of his significant achievements are:
development of the infinitesimal method in the theory of infinite dimensional repre-
sentations of real semi simple Lie algebras; Fourier transform theory in the complex
domain on Riemannian symmetric spaces; theory of local moduli for ordinary mero-
morphic linear differential equations at an irregular singularity; theory of unitary
representations of super Lie groups and the classification of super particles; and
more recently, studies on the physics associated to non-archimedean space-time.
The relevance of the representation theory of Lie groups and Lie algebras to
the physics of elementary particles and fields has been known for a very long time,
going back to the famous 1939 paper of E. P. Wigner on the representations of the
Poincare group. Since then this link between representation theory and physics has
deepened enormously, and includes quantum field theory and conformal field theory.
Then something marvelous happened. In the 1970’s the physicists created a new
extension of geometry where the underlying manifolds acquired anti-commuting co-
ordinates in addition to the usual commuting ones, reflecting the Fermionic struc-
ture of matter. This introduced supergeometry and super Lie groups into the mix
and made the connection between geometry and physics much richer. Together
with his students, he has made many important contributions to this area.
It thus seemed appropriate to have a conference at UCLA devoted to some
of these themes. The Conference turned out to be very exciting and stimulating
because of the contributions of the participants who came from the United States
and abroad. Most of the articles in this volume are thus naturally concerned with
the above-mentioned themes: representations of finite and infinite dimensional Lie
groups and Lie algebras, super Lie groups and supergeometry, which are at the
interface of mathematics and fundamental particle physics, and supersymmetry.
The discussions on supergeometry and supersymmetry are especially relevant at
this time since some of the experiments at the Large Hadron Collider at CERN
may help determine whether supersymmetry is a feature of the world of elementary
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