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111 | 2012
Annuaire du Collège de France 2010-2011
Résumé des cours et travaux 111e année
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A Geometer Explores the Universe

Un géomètre explore l’univers
Michel Atiyah
p. 875-876

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1In this lecture I put forward two new mathematical approaches to fundamental physics. The first is joint work with G. Moore of Rutgers University and will appear in Surveys in Differential Geometry, volume XII, entitled “A Shifted View of Fundamental Physics”.

2In this we show how the Dirac operator enables one to introduce retarded and advanced differential equations in a relativistically invariant way. There are two surprising consequencies, one in atomic physics where we find that masses of stable fermions are quantized and the second in cosmology where we seem to encounter the cosmological constant.

3The second idea can be seen as a refinement of the Skyrme model of baryons. I propose using 4-dimensional Riemannian manifolds to model matter. Time is ignored and the 4 “spatial” dimensions are those of the Kaluza-Klein 5-dimensional extension of Minkowski space, but with the roles of electricity and magnetism reversed.

4The manifolds used are oriented and conformally self-dual, precisely those which have Penrose twistor spaces, and can be treated by complex analysis. Electrically neutral matter (neutrons and neutrinos) is modelled by compact manifolds, while charged matter (protons and electrons) is modelled by complete non-compact manifolds.

5More precisely the models with their metrics are:


standard 4-sphere


standard complex projective plane


Taub-NUT manifold


Atiyah-Hitchin manifold

6All of these have SO(3) symmetry.

7Note that the simply-connected Atiyah-Hitchin manifold (which arose as a moduli space of monopoles) is topologically the complement of the real projective plane in the complex projective plane, with its SO(3) symmetry.

8The charged models have asymptotic ends in the background Kaluza-Klein 4-space, while the neutral models intersect this background in a surface (2-sphere or real projective plane).

9Many aspects including the dynamical evolution and spectral properties remain to be investigated.

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Cours et travaux du Collège de France. Annuaire 111e année, Collège de France, Paris, avril 2012, p. 875-876. ISBN 978-2-7226-0156-7

Référence électronique

Michel Atiyah, « A Geometer Explores the Universe », L’annuaire du Collège de France [En ligne], 111 | 2012, mis en ligne le 22 novembre 2013, consulté le 17 février 2018. URL :

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Michel Atiyah

Professeur honoraire à l’Université d’Édinburg (Écosse). Trinity College, Cambridge & University of Edinburgh

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Droits d’auteur

Collège de France

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