05 April 2008

Smale's problems

A lot of people refer to the Clay Mathematics Institute's seven "Millennium Prize Problems" as an analogue of Hilbert's problems for the 21st-century.

In 1998, Stephen Smale produced a list (of 18 problems) as well. (There is significant overlap with the Clay problems: both lists contain Riemann, P =? NP, Poincaré, and Navier-Stokes.) Two of them are Hilbert problems (the Riemann hypothesis and Hilbert's 16th problem). The list seems a bit biased, though, in that Smale made contributions to many of the problems mentioned; Smale acknowledges this as one of the criteria for forming his list, and the document isn't meant to stand alone; the essay was written in response to a query of Vladimir Arnold, and Smale was not the only person Arnold asked. One has to wonder if it would be possible for anyone to really be able to survey all of mathematics intelligently in the way I'm told Hilbert did. (I've read Hilbert's address but I don't know enough of the history to be able to assess whether it really covers all of mathematics at the time.)

In Smale's discussion of the Poincaré conjecture, after pointing out that a big part of the importance of the Poincaré conjecture is that it helped make manifolds respectable objects to study in their own right,he states:
I hold the conviction that there is a comparable phenomenon today in the notion of a "polynomial time algorithm". Algorithms are becoming worthy of analysis in their own right, not merely as a means to solve other problems. Thus I am suggesting that as the study of the set of solutions of an equation (e.g. a manifold) played such an important role in 20th century mathematics, the study of finding the solutions (e.g. an algorithm) may play an equally important role in the next century.

This introduces the discussion of P =? NP, although the reason people study algorithms is not to answer that question; but one often hears statements like Smale's statement on Poincaré's conjecture, or statements that Fermat's Last Theorem is more important for the development in number theory that it spurred than for the result itself.

1 comment:

Suresh said...

I may be a little dense here, but it seems to me that for people who study algorithms, it has ALWAYS been true that algorithms are worthy of analysis in their own right.

Also, the comparison to the Poincare conjecture and Fermat's theorem is not entirely fair to P =? NP, because at least in the latter case, there are direct and serious consequences of (for example) the question being resolved as P = NP, and there are profound philosophical implications of the other direction. I think one would be hard pressed to say this about either of the other two.