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Commentary on “Lectures on Morse theory, old and new”. (English) Zbl 1277.57001

From the introduction: This masterly exposition of Raoul Bott, written in 1981 [Bull. Am. Math. Soc., New Ser. 7, 331–358 (1982; Zbl 0505.58001)], stands at several crossroads. …
We will comment briefly on these and other developments over the past 30 years. The subject is vast and we can only touch lightly on a few significant lines of progress. The remarks are organized in two sections: outgrowths of the work of Atiyah and Bott described in the last lecture, and then other recent work in Morse theory proper and its applications. We mention only some of the mathematicians who have contributed, and encourage the reader to be mindful of the legions of others not identified here. The bibliography consists of a few expository articles about this material.
This important article is reprinted in its entirety immediately following the commentary.

MSC:

57-03 History of manifolds and cell complexes
58-03 History of global analysis
57R70 Critical points and critical submanifolds in differential topology
57R56 Topological quantum field theories (aspects of differential topology)
57R57 Applications of global analysis to structures on manifolds
57R58 Floer homology
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
01A60 History of mathematics in the 20th century

Citations:

Zbl 0505.58001
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References:

[1] Raoul Bott, Morse theory indomitable, Inst. Hautes Études Sci. Publ. Math. 68 (1988), 99 – 114 (1989). · Zbl 0685.58009
[2] S. K. Donaldson, The Seiberg-Witten equations and 4-manifold topology, Bull. Amer. Math. Soc. (N.S.) 33 (1996), no. 1, 45 – 70. · Zbl 0872.57023
[3] Michael Hutchings, Taubes’s proof of the Weinstein conjecture in dimension three, Bull. Amer. Math. Soc. (N.S.) 47 (2010), no. 1, 73 – 125. · Zbl 1197.57023
[4] Paul S. Aspinwall, Tom Bridgeland, Alastair Craw, Michael R. Douglas, Mark Gross, Anton Kapustin, Gregory W. Moore, Graeme Segal, Balázs Szendrői, and P. M. H. Wilson, Dirichlet branes and mirror symmetry, Clay Mathematics Monographs, vol. 4, American Mathematical Society, Providence, RI; Clay Mathematics Institute, Cambridge, MA, 2009. · Zbl 1188.14026
[5] Edward Witten, Fivebranes and knots, preprint. · Zbl 1241.57041
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