Abstract: 

An intriguing dictionary exists between two branches of conformal dynamics: the theory of Kleinian groups and dynamics of rational maps. After surveying various aspects of this dictionary, we will explain some novel connections between Kleinian (reflection) groups, (anti-)holomorphic rational maps, Schwarz reflection maps, and algebraic correspondences. The last class of objects often combine the actions of Kleinian groups and rational maps in a common dynamical plane. We will present a general framework for constructing such combinations, and talk about its parameter space implications. We will also mention applications of our theory to problems in complex analysis and Teichmüller theory.

Bio-Sketch: 

Sabyasachi Mukherjee is an associate professor at the Tata Institute of Fundamental Research (TIFR), Mumbai. Before joining TIFR, he was a Milnor lecturer at the Institute for Mathematical Sciences at Stony Brook University. He completed his PhD in Mathematics at Jacobs University Bremen in 2015, following earlier education at Université Paris 13 and the University of Calcutta.

His research lies at the crossroads of conformal dynamics, complex analysis, and Teichmuller theory, with particular emphasis on parameter spaces of rational maps, antiholomorphic dynamics, combination theorems, holomorphic and antiholomorphic correspondences, and the Sullivan dictionary linking rational maps and Kleinian groups. He is an invited speaker for ICM 2026.