English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

This masterly exposition of Raoul Bott, written in 1981, stands at several crossroads. The problem of closed geodesics on smooth Riemannian manifolds dates back to the late 19th century. Morse transformed the subject in the 1930s, and it is his basic critical point theory and its application to closed geodesics which Bott recalls in the first half of these lectures. This is finite dimensional Morse theory, and while it extends in this form to infinite dimensions via work of Palais and Smale, its more radical evolution to infinite dimensions in the hands of Floer and others lay in the future in 1981. The geodesic problem may be formulated on the infinite dimensional space of loops, but Bott approximates it by a finite space of polygons— piecewise geodesics—and so stays within the finite dimensional Morse theory. The first crossroads, then, is between finite and infinite dimensions. Another crossroads concerns the role of symmetry groups. Prior to 1980 Morse theory was profitably studied on the underlying manifold of a Lie group. The most prominent example is Bott’s great discovery in the 1950s that the homotopy groups of classical Lie groups are periodic. In the last of these lectures Lie groups play an entirely new role. Namely, Bott considers the Morse theory of a function f on a manifold M with a Lie group G acting on M leaving f invariant. This is equivariant Morse theory. This Morse theory with symmetry was developed just at the time these lectures were written. The equivariant theory provides a beautiful coda to many results in the closed geodesic problem, as developed by Bott’s student Hingston. Finally, these lectures lie at the beginning of a period in geometry of strong influence from physics, a current which is very much alive today. Indeed, a year or so before these lectures were written, Witten had already seen how supersymmetric gauge theory sheds light on Morse theory and he links it in a beautiful way with the de Rham complex. Bott’s account of Witten’s ideas is not in these lectures, but rather appeared in his next expository article on Morse theory eight years later [B].

Commentary on “Lectures on Morse theory, old and new”