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Gauge Theory and the Topology of Four-Manifolds
Edited by: Robert Friedman and John W. Morgan, Columbia University, New York, NY
A co-publication of the AMS and IAS/Park City Mathematics Institute.
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IAS/Park City Mathematics Series
1998; 221 pp; hardcover
Volume: 4
ISBN-10: 0-8218-0591-6
ISBN-13: 978-0-8218-0591-6
List Price: US$47
Member Price: US$37.60
Order Code: PCMS/4
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See also:

Low Dimensional Topology - Tomasz S Mrowka and Peter S Ozsvath

The lectures in this volume provide a perspective on how 4-manifold theory was studied before the discovery of modern-day Seiberg-Witten theory. One reason the progress using the Seiberg-Witten invariants was so spectacular was that those studying \(SU(2)\)-gauge theory had more than ten years' experience with the subject. The tools had been honed, the correct questions formulated, and the basic strategies well understood. The knowledge immediately bore fruit in the technically simpler environment of the Seiberg-Witten theory.

Gauge theory long predates Donaldson's applications of the subject to 4-manifold topology, where the central concern was the geometry of the moduli space. One reason for the interest in this study is the connection between the gauge theory moduli spaces of a Kähler manifold and the algebro-geometric moduli space of stable holomorphic bundles over the manifold. The extra geometric richness of the \(SU(2)\)-moduli spaces may one day be important for purposes beyond the algebraic invariants that have been studied to date. It is for this reason that the results presented in this volume will be essential.

Titles in this series are co-published with the Institute for Advanced Study/Park City Mathematics Institute. Members of the Mathematical Association of America (MAA) and the National Council of Teachers of Mathematics (NCTM) receive a 20% discount from list price.

Readership

Graduate students and research mathematicians working in algebraic geometry.

Table of Contents

Geometric invariant theory and the moduli of bundles
  • D. Gieseker -- Geometric invariant theory
  • D. Gieseker -- The numerical criterion
  • D. Gieseker -- The moduli of stable bundles
  • D. Gieseker -- References
Anti-self-dual connections and stable vector bundles
  • J. Li -- Hermitian bundles, Hermitian connections and their curvatures
  • J. Li -- Hermitian-Einstein connections and stable vector bundles
  • J. Li -- The existence of Hermitian-Einstein metrics
  • J. Li -- References
An introduction to gauge theory
  • J. W. Morgan -- The context of Gauge theory
  • J. W. Morgan -- Principal bundles and connections
  • J. W. Morgan -- Curvature and characteristic classes
  • J. W. Morgan -- The space of connections
  • J. W. Morgan -- The ASD equations and the moduli space
  • J. W. Morgan -- Compactness and gluing theorems
  • J. W. Morgan -- The Donaldson polynomial invariants
  • J. W. Morgan -- The connected sum theorem
  • J. W. Morgan -- References
Computing Donaldson invariants
  • R. J. Stern -- Overview
  • R. J. Stern -- -2 spheres and the blowup formula
  • R. J. Stern -- Simple-type criteria and elliptic surfaces
  • R. J. Stern -- Elementary rational blowdowns
  • R. J. Stern -- Taut configurations and Horikowa surfaces
  • R. J. Stern -- References
Donaldson-Floer theory
  • C. Taubes and J. A. Bryan -- Introduction
  • C. Taubes and J. A. Bryan -- Quantization
  • C. Taubes and J. A. Bryan -- Simplicial decomposition of \(\Cal{M}^0_X\)
  • C. Taubes and J. A. Bryan -- Half-infinite dimensional spaces
  • C. Taubes and J. A. Bryan -- References
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