
Vaughan McDonald,
Tue, February 19, 6pm  7pm, Science Center 507.
How to Not Integrate: Impossibilities Theorems for Integrals.
Abstract: When we first learn calculus, we learn that taking a derivative is
much easier than calculating an integral. Whereas we have closed expressions
for any reasonable derivative, the opposite is not so clear for the integral.
Integrals such as the integral of e^{x2}
(The Gaussian integral from probability) are never presented to us in more
"elementary" forms such as exponentials, logarithms, and polynomials.
In fact, it cannot be, as proved by Liouville in the 19th century (for a
suitable definition of elementary)! In this talk, we will highlight how
Liouville proved such a result. Amazingly, such a result is directly
analogous to the fact that certain quintic polynomials do not have
roots expressed by radicals, a beautiful result from Galois theory.
After discussing Liouville's proof, we will pursue this analogy by
talking about a Galois theory for linear differential equations that
mirrors the normal theory for algebraic extensions. We will then
hint at how this theory can also lead to impossibility results for integrals.
