Clifford J. Earle: A Life in Mathematics and Music (1935–2017)
Linda Keen, Irwin Kra · Notices of the American Mathematical Society · 2017
Kleinian Groups, the most significant result on Kleinian groups in the twentieth century.It made it possible to develop the different structures of finitely generated Kleinian groups and to study various boundaries of moduli spaces.There was excitement in the air and rapid progress because so many people, including those from other fields, were attracted to it.Among these were Dennis Sullivan and William Thurston, who created a second revolution in the field.Sullivan gave a new proof of the Ahlfors Finiteness Theorem, realized that its proof also proves the No-wandering Domain Theorem in complex dynamics, and created a dictionary between Kleinian groups and complex dynamics.Thurston used Kleinian groups to develop his theory of hyperbolic three manifolds.A Riemann surface is a topological surface with a conformal structure.Teichmüller theory provides a way to distinguish distinct conformal structures that occur with a given topology.Cliff's thesis and a lot of his later work dealt with describing natural families of Riemann surfaces and providing moduli for the members that fit together as a pleasing whole "moduli space."A particular approach to this problem, developed by Earle and his co-authors, was via vector bundles over the moduli spaces.Among Earle's major contributions, we mention only three of the most influential.These were all joint work: first with James Eells in developing a differential geometric approach to Teichmüller theory [2]; second, his decades' long collaboration with Albert Marden on intrinsic coordinates for moduli spaces; and third, his work with Douady on conformally natural extensions of homeomorphisms of the circle.Over his long career Cliff collaborated with almost twenty others.It was a pleasure to write a paper with Earle.He was always generous in attributions to others.According to Marden, Cliff had the patience and fortitude to carefully study a situation before putting it in print.And then what he wrote, by himself or with others, was carefully presented.His exposition was clear and got to the key points in a very expeditious manner.Most co-au-