Title: The Stingray
Abstract: The Earle slice S is an open domain in a 1-dimensional (complex) curve of the SL(2,C) character variety of a rank 2 free group, which is a model for the Teichmuller space of a once-punctured torus. It bears a remarkable resemblance to J, the Julia set of z -> z^2 + 1/4. Connected quadratic Julia sets are parameterized by the Mandelbrot set. This raises the question: is there a “Mandelbrot set” for which S is a “Julia set”? The answer is yes: the Stingray. We will define and describe this set, and some of its remarkable properties.

Bio:

Danny Calegari was born in Melbourne, Australia. He obtained a PhD in 2000 at UC Berkeley under the joint supervision of Andrew Casson and Bill Thurston. He was a postdoc at Harvard, and has held tenured positions at Caltech, the University of Cambridge, and (most recently) the University of Chicago, where he has been since 2012. He won a Clay Research Award in 2009 for his contributions to the solutions of the Marden Tameness Conjecture and the Ahlfors Measure Conjecture, and in 2022 he gave an invited lecture at the ICM. His research interests include geometry, dynamical systems, low-dimensional topology, and geometric group theory.