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The Theory of Gauge Fields in Four Dimensions
H. Blaine Lawson, Jr.
A co-publication of the AMS and CBMS.
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CBMS Regional Conference Series in Mathematics
1985; 101 pp; softcover
Number: 58
Reprint/Revision History:
reprinted 2001, reprinted 2005
ISBN-10: 0-8218-0708-0
ISBN-13: 978-0-8218-0708-8
List Price: US$28
Member Price: US$22.40
All Individuals: US$22.40
Order Code: CBMS/58
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Lawson's expository lectures, presented at a CBMS Regional Conference held in Santa Barbara in August 1983, provide an in-depth examination of the recent work of Simon Donaldson, and is of special interest to both geometric topologists and differential geometers. This work has excited particular interest, in light of Mike Freedman's recent profound results: the complete classification, in the simply connected case, of compact topological 4-manifolds. Arguing from deep results in gauge field theory, Donaldson has proved the nonexistence of differentiable structures on certain compact 4-manifolds. Together with Freedman's results, Donaldson's work implies the existence of exotic differentiable structures in \(\mathbb R^4\)-a wonderful example of the results of one mathematical discipline yielding startling consequences in another.

The lectures are aimed at mature mathematicians with some training in both geometry and topology, but they do not assume any expert knowledge. In addition to a close examination of Donaldson's arguments, Lawson also presents, as background material, the foundation work in gauge theory (Uhlenbeck, Taubes, Atiyah, Hitchin, Singer, et al.) which underlies Donaldson's work.

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Table of Contents

  • Introduction
  • The geometry of connections
  • The self-dual Yang-Mills equations
  • The moduli space
  • Fundamental results of K. Uhlenbeck
  • The Taubes existence theorem
  • Final arguments
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