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Jahresbericht der Deutschen Mathematiker-Vereinigung

, Volume 114, Issue 3, pp 119–130 | Cite as

A Short History of the Weinstein Conjecture

  • Federica Pasquotto
Survey Article

Abstract

A very natural and fundamental question (both from a historical and a mathematical point of view) is that of existence of periodic orbits of Hamiltonian flows on a fixed energy hypersurface. In 1978 Alan Weinstein conjectured that a geometric property of the hypersurface under consideration would provide a sufficient condition for the existence of such orbits. He called hypersurfaces with this property hypersurfaces of contact type. This article briefly describes the history of the Weinstein Conjecture, which has been one of the major driving forces behind the development of symplectic geometry at the end of the twentieth century, leading to some of the most fruitful interactions between analysis, geometry and topology. Weinstein’s Conjecture has been proved in a number of significant cases but remains, in its most general form, an extremely interesting and challenging open problem.

Keywords

Hamiltonian system Periodic orbit Symplectic form Contact form Reeb flow Almost complex structure 

Mathematics Subject Classification

37-02 53-02 37J45 53D42 

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Copyright information

© Deutsche Mathematiker-Vereinigung and Springer-Verlag 2012

Authors and Affiliations

  1. 1.Afdeling WiskundeVrije UniversiteitAmsterdamThe Netherlands

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