CBMS Regional Conference Series in Mathematics 1985; 101 pp; softcover Number: 58 Reprint/Revision History: reprinted 2001, reprinted 2005 ISBN10: 0821807080 ISBN13: 9780821807088 List Price: US$30 Member Price: US$24 All Individuals: US$24 Order Code: CBMS/58
 Lawson's expository lectures, presented at a CBMS Regional Conference held in Santa Barbara in August 1983, provide an indepth 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 4manifolds. Arguing from deep results in gauge field theory, Donaldson has proved the nonexistence of differentiable structures on certain compact 4manifolds. 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. Readership Table of Contents  Introduction
 The geometry of connections
 The selfdual YangMills equations
 The moduli space
 Fundamental results of K. Uhlenbeck
 The Taubes existence theorem
 Final arguments
