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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