Abstract: Let p and q be two polynomials of degree d, with connected and locally connected filled in Julia sets K_p and K_q. Let γ_p, γ_q: R/Z --> J_p be the Caratheodory loop. We can contruct a topological space X_p,q by taking the disjoint union K_p U K_q and identifying γ_p(t) with γ_q(-t); the polynomial p on K_p and q on K_q then fit together to give a map f_p,q: X_p,q --> X_p,q, called the mating of p and q.

Sometimes, in fact often, X_p,q is homeomorphic to a sphere, and a homeomorphism can be chosen so that f_p,q is a rational function. This is especially exciting when K_p and K_q have empty interiors, in that case γ_p induces a map R/Z --> \overline C that is a Peano curve.

Matings have been studied extensively, by Douady and Hubbard, Tan Lei, Mary Rees, Shishikura, Milnor, Jiaqi Luo among others.

Matings appear to be a 1-dimensional phenomenon. Recently (with Sebastien Krief and Peter papadopol) we have found that a variant of matings naturally appears in Newton's method in two variables, as applied to the intersection of a pair of parallel lines with a conic.

I my lecture I will go over matings, illustrationg them with the computer, and will explain how they arise in Newton's method.