Celebratio Mathematica — Bott

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R. Bott: “Morse theory and the Yang–Mills equations,” pp. 269–275 in Differential geometrical methods in mathematical physics (3–7 September 1979, Aix-en-Provence, France and 10–14 September 1979, Salamanca, Spain). Edited by P. L. García, A. Pérez-Rendón, and J. M. Souriau. Lecture Notes in Mathematics 836. Springer (Berlin), 1980. MR 607701 Zbl 0486.58009 incollection

In the variational problems of mathematical physics one is usually interested only in the extremals which afford an absolute minimum to the functional being varied. The other, the unstable solutions, therefore tend to be forgotten. On the other hand, in the Morse theory the totality of the extrema satisfy relations which are forced by the topology of the space underlying the variational problem. There is thus a certain amount of feedback between the stable and unstable critical sets which enables one to derive information about the topology of the absolute minimum by studying the sometimes more accessible unstable critical sets. The Yang–Mills functional for a principal bundle \( P \) over a compact Riemann surface \( M \) gives a striking illustration of the principle, and the purpose of this talk is to sketch in the main steps of this application of the Morse theory for you.

I am reporting on joint work with M. Atiyah still very much in progress.

A. Pérez-Rendón	Related
P. L. García	Related
Jean-Marie Souriau	Related