Progress in Mathematics

By: R. Dijkgraaf, G. Van Der Geer, C. Faber

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Hardcover | 18 October 1995

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580 Pages

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The moduli space Mg of curves of fixed genus g - that is, the algebraic variety that parametrizes all curves of genus g - is one of the most intriguing objects of study in algebraic geometry these days. Its appeal results not only from its beautiful mathematical structure but also from recent developments in theoretical physics, in particular in conformal field theory.

Leading experts in the field explore in this volume both the structure of the moduli space of curves and its relationship with physics through quantum cohomology. Altogether, this is a lively volume that testifies to the ferment in the field and gives an excellent view of the state of the art for both mathematicians and theoretical physicists. It is a persuasive example of the famous Wignes comment, and its converse, on "the unreasonable effectiveness of mathematics in the natural science."

Witteen's conjecture in 1990 describing the intersection behavior of tautological classes in the cohomology of Mg arose directly from string theory. Shortly thereafter a stunning proof was provided by Kontsevich who, in this volume, describes his solution to the problem of counting rational curves on certain algebraic varieties and includes numerous suggestions for further development. The same problem is given an elegant treatment in a paper by Manin. There follows a number of contributions to the geometry, cohomology, and arithmetic of the moduli spaces of curves. In addition, several contributors address quantum cohomology and conformal field theory.

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Non-Fiction Mathematics Geometry Algebraic Geometry Science Physics Mathematical Physics Topology Algebra Applied Physics

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Published: 19th November 2011

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Participants
Contributors
Introduction
Distribution of rational points and Kodaira dimension of fiber products	p. 1
How many rational points can a curve have?	p. 13
Quantum cohomology of rational surfaces	p. 33
Quantum intersection rings	p. 81
Mirror symmetry and elliptic curves	p. 149
A generalized Jacobi theta function and quasimodular forms	p. 165
Boundary behaviour of Hurwitz schemes	p. 173
Operads and moduli spaces of genus 0 Riemann surfaces	p. 199
Resolution of diagonals and moduli spaces	p. 231
The Chow ring of the moduli space of curves of genus 5	p. 267
The cohomology of algebras over moduli spaces	p. 305
Enumeration of rational curves via torus actions	p. 335
Cellular decompositions of compactified moduli spaces of pointed curves	p. 369
Generating functions in algebraic geometry and sums over trees	p. 401
Holomorphicity and non-holomorphicity in N = 2 supersymmetric field theories	p. 419
An arithmetic problem in surface geometry	p. 427
An orbifold partition of [actual symbol not reproducible]	p. 467
Moduli of curves with non-abelian level structure	p. 483
Q-structure of conformal field theory with gauge symmetries	p. 511
On the cohomology of moduli spaces of rank two vector bundles over curves	p. 533
Table of Contents provided by Blackwell. All Rights Reserved.

Progress in Mathematics

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Hardcover

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You Can Find This Book In

Other Editions and Formats

Paperback

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