Title:  Singularities in algebraic geometry: how many times does a polynomial vanish at a point?

Abstract:  We all learn early on how to count the number of times a given number appears as a root of a polynomial in one variable. But for polynomials in several variables, the analogous question is much more interesting. The most naive generalization leads to the multiplicity of a singular point on an algebraic curve or hypersurface, and I will review this beautiful chapter of classical algebraic geometry. In recent years a more subtle invariant, defined via considerations of integrability, has come into prominence. I will conclude by discussing how this new invariant governs many analytic, arithmetic and geometric properties of a polynomial.