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Bulletin of the American Mathematical Society

The Bulletin publishes expository articles on contemporary mathematical research, written in a way that gives insight to mathematicians who may not be experts in the particular topic. The Bulletin also publishes reviews of selected books in mathematics and short articles in the Mathematical Perspectives section, both by invitation only.

ISSN 1088-9485 (online) ISSN 0273-0979 (print)

The 2020 MCQ for Bulletin of the American Mathematical Society is 0.84.

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Taubes’s proof of the Weinstein conjecture in dimension three
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by Michael Hutchings PDF
Bull. Amer. Math. Soc. 47 (2010), 73-125 Request permission

Abstract:

Does every smooth vector field on a closed three-manifold, for example the three-sphere, have a closed orbit? The answer is no, according to counterexamples by K. Kuperberg and others. On the other hand, there is a special class of vector fields, called Reeb vector fields, which are associated to contact forms. The three-dimensional case of the Weinstein conjecture asserts that every Reeb vector field on a closed oriented three-manifold has a closed orbit. This conjecture was recently proved by Taubes using Seiberg-Witten theory. We give an introduction to the Weinstein conjecture, the main ideas in Taubes’s proof, and the bigger picture into which it fits.
References
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Additional Information
  • Michael Hutchings
  • Affiliation: Mathematics Department, 970 Evans Hall, University of California, Berkeley, California 94720
  • Email: hutching@math.berkeley.edu
  • Received by editor(s): June 11, 2009
  • Received by editor(s) in revised form: August 26, 2009
  • Published electronically: October 29, 2009
  • Additional Notes: Partially supported by NSF grant DMS-0806037
  • © Copyright 2009 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Bull. Amer. Math. Soc. 47 (2010), 73-125
  • MSC (2010): Primary 57R17, 57R57, 53D40
  • DOI: https://doi.org/10.1090/S0273-0979-09-01282-8
  • MathSciNet review: 2566446