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Department of Mathematical Sciences

# Seminar Archives

## Pure Maths Colloquium: Matings of Quadratic Polynomials

Presented by "Adam Epstein (University of Warwick, Coventry) ",

19 November 2001 16:00 in CM221

" The moduli space of all quadratic rational maps up to M\"obius conjugacy is isomorphic to
${\Bbb C}^2$. It is possible, and also useful, to regard one of the coordinate axes as the moduli
with connected Julia set, thereby lies in this slice. Nearly twenty years ago, Douady conjectured
that the rational maps in the central portion of the moduli space of quadratic rational maps might
be understood as {\em matings} of pairs of quadratic polynomials. The proposed construction is
purely topological: one glues filled-in Julia sets back-to-back along complex-conjugate prime
ends to obtain a branched cover of the sphere. Work of Tan Lei and Mary Rees shows that under
favorable circumstances, the resulting branched cover is topologically conjugate to an essentially
unique quadratic rational map. According to Milnor, mating is an interesting operation because it
possesses none of the usual good properties: it is not injective, surjective, continuous, or even
everywhere defined. We will survey recent results concerning these issues - in particular, the
discontinuity of mating.

"