Preface

Le caractere propre des methodes de l'Analyse et de

la Geometrie modernes consiste dans l'emploi d'un

petit nombre de principes generaux, independants

de la situation respective des differentes parties ou

des valeurs relatives des differents symboles; et les

consequences sont d'autant plus etendues que les

principes eux-memes ont plus de generalite.

from G.

DARBOUX:

Principes de Geometrie Analy-

tique

This text is written for the graduate student who has previous training

in analysis and linear algebra, as for instance S. Lang's Analysis / a n d Lin-

ear Algebra. It is meant as an introduction to what is today an intensive

area of research linking several disciplines of mathematics and physics in

the sense of the Greek word avinrXeKStu (which means to interconnect, or

to interrelate in

English).1

The difficulty (but also the fascination) of the

area is the wide variety of mathematical machinery required. In order to

introduce this interrelation, this text includes extensive appendices which

include definitions and developments not usually covered in the basic train-

ing of students but which lay the groundwork for the specific constructions

I want to thank P. Slodowy for pointing out to me that the name symplectic group, which

eventually gave rise to the term symplectic geometry, was proposed by H. WEYL, [W], 1938, in

his book, The Classical Groups (see footnote on p. 165). The symplectic group was also called

the complex group or an Abelian linear group, this last to honor ABEL, who was the first to study

them.

XI