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INTRODUCTION

quantization [H] and conformal blocks instigated the use of infinite-dimensional

algebraic geometry [BL]. A deep, slowly unfolding analogy of operations on bun-

dles with number theory was represented at the Lowell meeting (see Gaitsgory's

abstract) but there was no time to include it in this book. This multiple-front

development shows no sign of abating, as random matrices and Brauer classes as-

sociated to orbit problems, for example, come to the fore. In parallel, Yang-Mills

theory should be mentioned; it overlaps with the KdV theory but the two have not

yet been 'unified'.

A second main area of growth for geometry is connected with quantum field

theory. The most robust geometric application is intersection theory on moduli

spaces; new invariants (Donaldson or Gromov-Witten); new algebraic structures

(Frobenius manifolds, Hopf algebras, quantum cohomology) on old spaces. But you

could argue that this harks back to integrable systems, for the connections exist

(KdV, Toda, Virasoro and monodromy equations) though they are still mysterious.

There is a third way, however, in which geometry was revitalized by physics;

so this spring when I had the privilege of organizing a special session of the AMS

meeting in Lowell, MA, I decided to learn "with a little help from my friends" (Ringo

Starr) how old problems of enumerative geometry were solved by new ideas of mirror

symmetry. The two Lowell days of cutting-edge talks, representing such diverse

and strong-growing areas, energized us enough that we decided to put together this

book. Ravi Vakil suggested the title. The timeline for proceedings is rigid, but the

work came in as planned, to give both a snapshot of what is known, and a set of

previously unpublished results. Each article was refereed by an expert.

Some of the Lowell participants were unable to produce an article by the end

of the summer, so to give an idea of what we learned I compiled the abstracts as

an Appendix, in the order they were delivered. Some of the present authors were

unable to be physically present in Lowell but were so good as to contribute their

written work.

In the first chapter you will find results of enumerative nature, from counting

automorphisms, a contribution by Sandor Kovacs, to issues of reality conditions in

counting curves, a contribution by Frank Sottile, who found an alternative approach

to quantum cohomology techniques in the literature of systems theory; Alexander

Suciu provides a survey of numerical invariants of line arrangements in the com-

plex plane, as well as new examples and conjectures. In chapter 2, I grouped work

that has to do with deformation theory: Dan Abramovich and Aaron Bertram

prove by deformation a coincidence of Gromov-Witten invariants for certain ratio-

nal surfaces; Dan Abramovich and Frans Oort give results on the Hurwitz stack,

parametrizing simply branched covers of the projective line, in mixed character-

istic; Lucia Caporaso investigates numerical properties of tangent hyperplanes to

canonical curves, in a way that is compatible with degenerations; Eduardo Cat-

tani and Javier Fernandez establish a correspondence between quantum potentials

and polarized variations of Hodge structures in a given asymptotic sense; Herbert

Clemens shows by deformation theory that certain families of rational curves in

projective space are stably rational; Ravi Vakil introduces certain graph invariants

to reproduce results on the degeneration of a family of maps from nodal curves

to a surface. Chapter 3 is loosely titled mirror symmetry and Gromov-Witten

invariants, although these phenomena were already present in the previous chap-

ters. David Cox, Sheldon Katz and Yuan-Pin Lee discuss a conjecture of Andreas

Gathmann for a certain virtual fundamental class and prove it in some cases; Tyler