A Topological Proof of the Abel-Ruffini Theorem - 
The roots of general quadratic, cubic, or quartic polynomials can be expressed in terms of their coefficients using the usual four field operations as well as square roots and cube roots.  The Abel-Ruffini theorem says that the general quintic polynomial does not have a ``radical'' expression of this form, no matter how many roots are allowed.   Loosely speaking, there is no ``quintic formula.''

A topological proof of this, due to Arnold, has been gradually refined over the years, to the point where recent versions give a visual expression of the ideas that provides a compelling reason for why the quintic polynomial has the smallest degree for  which a solution in radicals is impossible.  (Even a physicist might be happy with this proof, and in fact much of the recent work on this has been done by physicists.)

The goal of this talk is to explain all of this.