Curtis McMullen

fl. 1958, Berkeley, California, United States

Differential equations and dynamical systemsTopologyAnalysisFields Medal 1998

American mathematician who unified complex dynamics, hyperbolic 3-manifold geometry and renormalization theory, and proved Kra's theta conjecture on Riemann surfaces.

Curtis Tracy McMullen was born in Berkeley, California, studied at Williams College and Cambridge University, and completed his PhD at Harvard in 1985 under Dennis Sullivan. His dissertation already introduced his lifelong program: analyzing rational maps, Kleinian groups and the geometry of the moduli space of Riemann surfaces as manifestations of a single dynamical picture.

Among his early results was the proof of Kra's theta conjecture and the solution of the 'problem of the simplicity of the spectrum' type questions for dynamics on the moduli space; he also proved that the Julia sets of certain families of rational maps have positive area only in specific cases. His most influential body of work concerned renormalization: he showed that infinitely renormalizable quadratic polynomials are rigid (carrying no invariant line fields on their Julia sets) and that successive renormalizations converge to universal limits, giving a conceptual explanation of the universality first observed numerically by Feigenbaum. In 'Dynamics on K3 surfaces' and later work he developed relations between dynamics, automorphisms of surfaces and Salem numbers.

McMullen held positions at MIT, Princeton and Berkeley before returning to Harvard in 1998 as a professor. He received the Fields Medal at the 1998 ICM in Berlin. He is also known for expository texts such as 'Complex Dynamics and Renormalization' and 'Renormalization and 3-Manifolds which Fiber over the Circle', and for the McMullen rigidity and universality results that frame modern holomorphic dynamics.

Workplaces: Harvard University, University of California, Berkeley, Princeton University

United States

Award-winning work

Fields Medal 1998McMullen was awarded the medal for his work on the dynamics of one complex variable, hyperbolic 3-manifolds and renormalization. He proved Kra's 'theta conjecture' on the geometry of the moduli space of Riemann surfaces, showing that every Riemann surface admits a canonical hyperbolic metric whose shape is determined by the theta function. He made fundamental advances on the renormalization theory of quadratic polynomials — proving rigidity results and the convergence of renormalizations toward Feigenbaum-type universality — and on the topology and geometry of hyperbolic 3-manifolds, including sharp bounds on the number of cusps and the proof that the iterates of a rational map with an attracting fixed point have tame geometry. His work tied together complex dynamics, Kleinian groups and Teichmüller theory.

Contributions, linked to the library

Library topicFunctions of a complex variableHis work on iteration of rational maps in one complex variable founded much of modern complex dynamics.Library topicNon-Euclidean geometryHis theorems describe the geometry of hyperbolic 3-manifolds and the hyperbolic metric on moduli spaces.