Mathfields flourishing

[Notes on a public lecture delivered in March 1999  by Robert McPherson, of and at the Institute for Advanced Study, Princeton.]

The government is now demanding prediction and accountability from the things it funds. Reasonable enough.  A prime example of expert prescription/prediction and government funding was the Bahcall report on what's hot and what's not in astronomy.  The government followed his recommendations and heavily funded; results were in fact achieved.  But math has a bad track record of prediction.

            The worst was the COSRIMS report of 1969, which predicted a terrible shortage of math Ph.D.'s – so bad in fact, that they recommended a new degree, "Doctor of Arts" (basically an ABD).  But by 1971, math Ph.D.'s were driving cabs.

[A personal note:  
In 1971, I began in the math Ph.D. program at Berkeley.  But I was penniless, and there was no fellowship to be had.  So I switched to linguistics, where a fellowship in Arabic was available.
In retrospect, this was probably a good thing.  For I don't really much enjoy driving ...]

Examples of fields once out of favor, which found renewed life:

* Invariant theory

            The detailed calculations of Cayley were surpassed/subsumed by Hilbert (cf. "Hilbert's theorem" page in Lang's Algebra); Weyl suggested that this was all that need be said.

* Knot theory

            When Atiyah dominated topology, he dissed it

* Pathological spaces

            Milnor disliked these infinite-dimensional monstrosities, but one was used in Mike Freedman's proof of the 4-dim Poincaré conjecture

* Enumerative geometry

* Fermat's theorem

Hardy predicted that Number Theory would long have no applications.  Yet now it is so crucial for codes, there is a rumor that the DoD/NSA has made a breakthrough in factoring – the only case in which the cutting edge is not in the literature.  For one thing, they suddenly stopped funding factoring research.