FOM: Re: Measures of semi algebraic sets
Lou van den Dries
vddries at math.uiuc.edu
Mon Dec 15 15:28:57 EST 1997
The "variantion of parameters" result (for volumes of sets) also holds
for sets definable in R_an expanded by the power functions x^r.
But nothing like that is known for sets definable in R_an,exp.
(To my knowledge). In general, one would like to prove that
the integral of a function f(x,y) with respect to y
(where x=(x_1,...,x_m) and y=(y_1,...,y_n), and the function is
definable in some o-minimal expansion of the real field)
is definable (as a function of x) in some larger o-minimal expansion.
(I consider that as an important open problem.)
Well, P=NP is certainly highly interesting as a mathematical problem,
and so is the abc-conjecture, and scores of other big problems,
like the Lang conjectures. I am not inclined to compare them as to
general intellectual interest. Who, 200 years ago, would have
predicted the central role of elliptic curves in so much that's
going on? Or zeta functions? (Well, Euler had some inkling, presumably.)
If these things are not of general intellectual interest, so much the
worse for "general intellectual interests". Of course, this is not
to say I do not highly value good expositions on these things, and
improvement in this direction is very desirable. But I don't see
any reason for mathematicians to aim for approval by, for example,
the big media, if this would subvert the intellectual standards
we should be keeping up. -Lou van den Dries-
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