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Instantons and Four-Manifolds

  • Daniel S. Freed
  • Karen K. Uhlenbeck

Part of the Mathematical Sciences Research Institute Publications book series (MSRI, volume 1)

Table of contents

  1. Front Matter
    Pages i-x
  2. Daniel S. Freed, Karen K. Uhlenbeck
    Pages 1-12
  3. Daniel S. Freed, Karen K. Uhlenbeck
    Pages 13-16
  4. Daniel S. Freed, Karen K. Uhlenbeck
    Pages 17-30
  5. Daniel S. Freed, Karen K. Uhlenbeck
    Pages 31-50
  6. Daniel S. Freed, Karen K. Uhlenbeck
    Pages 51-73
  7. Daniel S. Freed, Karen K. Uhlenbeck
    Pages 74-87
  8. Daniel S. Freed, Karen K. Uhlenbeck
    Pages 88-98
  9. Daniel S. Freed, Karen K. Uhlenbeck
    Pages 99-118
  10. Daniel S. Freed, Karen K. Uhlenbeck
    Pages 119-140
  11. Daniel S. Freed, Karen K. Uhlenbeck
    Pages 141-161
  12. Daniel S. Freed, Karen K. Uhlenbeck
    Pages 162-187
  13. Daniel S. Freed, Karen K. Uhlenbeck
    Pages 188-195
  14. Back Matter
    Pages 196-232

About this book

Introduction

This book is the outcome of a seminar organized by Michael Freedman and Karen Uhlenbeck (the senior author) at the Mathematical Sciences Research Institute in Berkeley during its first few months of existence. Dan Freed (the junior author) was originally appointed as notetaker. The express purpose of the seminar was to go through a proof of Simon Donaldson's Theorem, which had been announced the previous spring. Donaldson proved the nonsmoothability of certain topological four-manifolds; a year earlier Freedman had constructed these manifolds as part of his solution to the four dimensional ; Poincare conjecture. The spectacular application of Donaldson's and Freedman's theorems to the existence of fake 1R4,s made headlines (insofar as mathematics ever makes headlines). Moreover, Donaldson proved his theorem in topology by studying the solution space of equations the Yang-Mills equations which come from ultra-modern physics. The philosophical implications are unavoidable: we mathematicians need physics! The seminar was initially very well attended. Unfortunately, we found after three months that we had covered most of the published material, but had made little real progress towards giving a complete, detailed proof. Mter joint work extending over three cities and 3000 miles, this book now provides such a proof. The seminar bogged down in the hard analysis (56 59), which also takes up most of Donaldson's paper (in less detail). As we proceeded it became clear to us that the techniques in partial differential equations used in the proof differ strikingly from the geometric and topological material.

Keywords

Manifold Manifolds Topology differential equation equation mathematics proof theorem

Authors and affiliations

  • Daniel S. Freed
    • 1
  • Karen K. Uhlenbeck
    • 2
  1. 1.Department of MathematicsUniversity of CaliforniaBerkeleyUSA
  2. 2.Department of MathematicsUniversity of ChicagoChicagoUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4684-0258-2
  • Copyright Information Springer-Verlag New York 1984
  • Publisher Name Springer, New York, NY
  • eBook Packages Springer Book Archive
  • Print ISBN 978-1-4684-0260-5
  • Online ISBN 978-1-4684-0258-2
  • Series Print ISSN 0940-4740
  • Buy this book on publisher's site