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Configurations de Particules et Espaces de Modules

  • J. C. Hurtubise (a1)

Résumé

Cet article de survol est le résumé de la conférence Coxeter-James de l'auteur, prononcée à la réunion d'hiver 1993 de la Société Mathématique du Canada.

La théorie de Morse décrit les liens entre la topologie d'une variété et la topologie des points critiques d'une fonction sur cette variété. La fonctionnelle d'énergie pour les applications d'une surface dans une variété, dont les points critiques seront des applications harmoniques et parfois holomorphes, et la fonctionnelle de Yang-Mills pour des connections sur une variété de dimension quatre sont deux cas en dimension infinie pour lesquels la théorie de Morse ne tient pas. Néanmois, dans les deux cas, on peut récupérer une quantité étonnante d'information, pourvu qu'on stabilise par rapport à un degré ou une charge qui sont des données du problème. Les preuves recyclent des résultats de la théorie de l'homotopie des années '70, et les combinent à des idées de géométrie complexe pour donner de jolis modèles des espaces en cause en termes de "particules". Nous espérons donner un survol général et accessible des idées utilisées.

Abstract

This survey is the written summary of the author's Coxeter-James lecture, delivered at the 1993 Winter Meeting of the Canadian Mathematical Society.

Morse theory relates the topology of the critical set of a function on a manifold to the topology of the whole manifold. The energy functional for maps of surfaces into a manifold, whose critical points are harmonic and occasionally holomorphic maps, and the Yang-Mills functional for connections on a four-manifold are two infinite dimensional cases where Morse theory fails. Nevertheles, in both cases a surprising amount can be said, providing one stabilises with respect to a natural charge or degree. The proofs borrow from the homotopy theory of the 1970's and combine it with some input from complex geometry to give some nice "particle" models of the spaces involved. This paper gives a fairly general and, it is hoped, accessible survey of the ideas involved.

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References

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Configurations de Particules et Espaces de Modules

  • J. C. Hurtubise (a1)

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