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A Short History of the Weinstein Conjecture

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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.

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Notes

  1. This result is sometimes referred to as the principle of the symplectic camel.

  2. According to the Oxford English Dictionary, the word symplectic was introduced by Weyl, who proposed to substitute the name ‘complex group’ by the corresponding Greek adjective ‘symplectic’. On a trip to Asia, Klaus Niederkrüger learned that the Chinese character for symplectic is one whose standard meaning is ‘hot, spicy’, so that, at least in China, symplectic geometry is ‘hot geometry’!

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Authors and Affiliations

  1. Afdeling Wiskunde, Vrije Universiteit, De Boelelaan 1081a, 1081 HV, Amsterdam, The Netherlands

    Federica Pasquotto

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  1. Federica Pasquotto
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Correspondence to Federica Pasquotto.

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This article is an extended and translated version of Een korte geschiedenis van het vermoeden van Weinstein [11], which originally appeared in the Nieuw Archief voor Wiskunde, the journal of the Royal Dutch Mathematical Society (KWG). Jahresbericht der DMV and the author are grateful to the KWG for their kind permission to reprint the article. The author would also like to thank Hansjörg Geiges and Kai Zehmisch for their careful reading of the translation and for many useful comments.

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Pasquotto, F. A Short History of the Weinstein Conjecture. Jahresber. Dtsch. Math. Ver. 114, 119–130 (2012). https://doi.org/10.1365/s13291-012-0051-1

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