Analytic and geometric function theory

Geometric and analytic function theory is concerned with understanding how complex-valued functions deform, distort, and map regions of the plane, with particular attention to functions that are analytic and univalent — meaning they preserve local geometry and never take the same value twice. A central preoccupation is conformal mapping, which allows one domain to be transformed into another while preserving angles, a property with consequences ranging from fluid dynamics to number theory. Researchers work to bound the coefficients in the power series representations of such functions, a line of inquiry famously anchored by the Bieberbach conjecture — proved by de Branges in 1985 — yet still generative of open problems involving subordination chains, harmonic mappings that allow mild angle distortion, and the interplay between hypergeometric functions and classical extremal problems. Active directions include sharpening estimates for newly defined subclasses of analytic functions and extending classical results from the conformal to the quasiconformal setting, where the rigidity of angle-preservation is deliberately relaxed.

Works

40,644

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252,150

Keywords

Geometric Function TheoryComplex AnalysisAnalytic FunctionsUnivalent FunctionsConformal MappingCoefficient Estimates