| Date: | Tue, 15 Apr 2008 |
| From: | Leonid Chindelevitch |
| Subject: | SPAMS Thursday, 5 PM, in 4-231 |
Hello everyone,
This week's SPAMS talk will be presented by Ben Mares. You will find out how to
climb a flat field... and see cute pictures like the one attached. See you then!
Leonid, on behalf of the SPAMS organizers.
Title: How to ascend a flat trail
A paradox of mathematical analysis asserts that it's possible for a path to
ascend flat terrain. More precisely, there is a continuously differentiable
function f : R^2 -> R and a path p: [0,1] -> R^2 such that f'(p(t))=0 for all
t, but f(p(0)) < f(p(1)).
Such a function seems to violate both the chain rule and Sard's theorem (that
the set of critical values has measure zero).
The only existence theorem currently in the literature is due to Whitney, based
on his famous extension theorem. Fear not, constructivists! I will explicitly
construct such a function and use a computer to visualize its graph.
Time permitting, I might also discuss the unrelated topic of how to define the
set of smooth functions algebraically.