Instantons and their relatives. (English) Zbl 1073.53505

Uhlenbeck, Karen

Browder, Felix E. (ed.), Mathematics into the twenty-first century. Proceedings of the AMS centennial symposium, Providence, RI, USA, August 8–12, 1988. Providence, RI: American Mathematical Society (ISBN 0-8218-0167-8). Am. Math. Soc. Centen. Publ. 2, 467-477 (1992).

This essay, based on an address to the American Mathematical Society at its centennial celebration, is much more than an incisive assessment of modern global differential geometry. Infused with wit and wisdom and perspicacity, it contains commentary on issues ranging from the nature of gauge theory to the modern world and the place of mathematics therein.
The first part of the essay is a beautifully clear description of gauge theory and of the nominal subjects of the essay, i.e., ”Instantons and their relatives”. There are two aspects that are particularly noteworthy here: the discussion of the analogy between Yang-Mills theory and the theory of minimal surfaces, and the inclusion of a useful bibliography of ”Early papers in gauge theory”.
Even those who find the first part too far removed from their own areas of expertise should be well rewarded by the second, more philisophical, part of this essay. In her discussion of current trends, and predictions for the future, the author provides an approach to her subject that all mathematicians should find thought-provoking and enlightening.
This essay deserves, and should be appreciated by, a wide audience.
For the entire collection see [Zbl 0921.00016].

Reviewer: Steven B. Bradlow (MR1184623)

MSC:

53C07	Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills)
32G13	Complex-analytic moduli problems
58D27	Moduli problems for differential geometric structures