Caroline Series

Caroline Series is a British mathematician whose research connects hyperbolic geometry, Kleinian groups, and dynamical systems.

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Hyperbolic geometry: Geometry in which the parallel postulate fails and space has constant negative curvature. It is a central setting for Series’s work on geometric structures and group actions.
Geometric group theory: The study of groups through their actions on geometric spaces. Kleinian groups provide a concrete meeting point for group theory and hyperbolic geometry.
William Thurston: An American mathematician who transformed the study of three-dimensional manifolds and hyperbolic geometry. Thurston’s geometric ideas helped establish the setting in which Series’s work developed.
Complex dynamics: The study of iteration of holomorphic and meromorphic functions. It shares tools and questions with the dynamics of Möbius group actions.
Kleinian group: A discrete group of Möbius transformations acting on the Riemann sphere. Series studies these groups and the geometric structures their actions produce.
Geometrization theorem: Thurston’s theorem describing the geometric decomposition of compact three-dimensional manifolds. Kleinian groups and hyperbolic structures are key tools in the study of three-manifold geometry.
Benson Farb: An American mathematician known for research in geometric group theory and topology. Farb coauthored work with Series connecting geometry, groups, and dynamics.
Low-dimensional topology: The study of topological spaces and manifolds in dimensions two, three, and four. Hyperbolic structures and Kleinian groups are powerful tools for understanding three-dimensional manifolds.
Möbius transformation: A fractional linear map of the extended complex plane, represented by a two-by-two matrix. These transformations form the ambient group from which Kleinian groups are drawn.
Symbolic dynamics: The study of dynamical systems through sequences of symbols and shift maps. Symbolic descriptions can encode complicated actions arising in hyperbolic and Kleinian dynamics.