### Shaun Bullett

Queen Mary, University of London

Thursday 09 November 2023, 4.00pm – 5.00pm

Lecture Theatre D (Mathematical Institute)

** Holomorphic correspondences mating rational maps with Kleinian groups **

Holomorphic correspondences on the Riemann sphere are multi-valued maps defined by polynomial relations P(z,w)=0. They generalise rational maps and finitely generated Kleinian groups. In 1994 Christopher Penrose and I discovered the first examples of matings between rational maps and Kleinian groups: these are holomorphic correspondences which exhibit the behaviour of an iterated quadratic polynomial on one part of the Riemann sphere and the behaviour of the modular group PSL(2,Z) on its complement. The examples lie in a 2-parameter family about which we made a number of conjectures. I will take the audience on an illustrated guided tour of this family (with a detour to visit Minkowski’s ‘question mark’ function), and report on how all the 1994 conjectures have been resolved in the last 5 years in joint work with Luna Lomonaco (IMPA, Brazil).

series: Pure Mathematics Colloquium

organiser: Scott Harper