INTRODUCTION

Emma Previato

"Every time someone invokes you, it is by a different name."

(The Muses, from Dialogues with Leu

co,

by C. Pavese)

When you try to tell your undergraduates what algebraic geometry is about,

you probably begin with the Greeks, as does Dieudonne [D], the cissoid being a

prime example of solution by geometry of an algebraic equation, the duplication

of the cube. But the progress of mathematical thought keeps revealing different

names of algebraic geometry, beyond what perhaps might have been the core dis-

cipline, namely Commutative Algebra: Complex Analysis, Differential Geometry,

Number Theory, Representation Theory, Model Theory, Skew Rings are some of the

names by which this "mother" who "existed before there were gods", like Pavese's

Mnemosyne, is now invoked. And the previous list does not even include the disci-

pline which is at the heart of this collection, Mathematical Physics.

The reciprocal influence of geometry and physics is especially crucial today:

it has truly changed the face of geometry, and this change has called to life anew

certain classical problems that had been shelved when the old techniques ran out

of computational power. Arguably the trend began in the late '60s: Mumford liked

the fact that "initial insight behind this and related discoveries was the work of

the first electronic computer!" [M], which allowed a group of applied mathemati-

cians in Princeton to simulate solutions of a non-linear wave equation, Korteweg-de

Vries (KdV for short), and discover a surprising stability property: for very large

time, the sum of two solutions ("solitons") was approximately a solution. Eventu-

ally this phenomenon, like Fagnano's duplication of the arc of the lemniscate [S,

§I.l], was interpreted as the Abelian sum on a Jacobian: algebraic geometry all

over again. Links between finite and infinite-dimensional, completely integrable

Hamiltonian systems ensued; the PDE was also interpreted by M. Sato as the

Plucker relations for an infinite-dimensional Grassmannian; representation theory,

W-algebras, Clifford algebras and symplectic reduction progressively fell into place.

N.J. Hitchin's generalization of the differential-geometric framework for the PDE

gave a new thrust to the theory of moduli spaces of vector bundles; geometric

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