author={Christopher J. Bishop},
  journal={Proceedings of the International Congress of Mathematicians (ICM 2018)},
  • C. Bishop
  • Published 1 May 2019
  • Mathematics
  • Proceedings of the International Congress of Mathematicians (ICM 2018)
This is a brief survey of results related to planar harmonic measure, roughly from Makarov’s results of the 1980’s to recent applications involving 4-manifolds, dessins d’enfants and transcendental dynamics. It is non-chronological and rather selective, but I hope that it still illustrates various areas in analysis, topology and algebra that are influenced by harmonic measure, the computational questions that arise, the many open problems that remain, and how these questions bridge the gaps… 


Harmonic measure,L2-estimates and the Schwarzian derivative
We consider several results, each of which uses some type of “L2” estimate to provide information about harmonic measure on planar domains. The first gives an a.e. characterization of tangent points
Dessins d'Enfants on Riemann Surfaces
Historical and introductory background.- Graph embeddings.- Dessins and triangle groups.- Galois actions.- Quasiplatonic surfaces, and automorphisms.- Regular maps.- Regular embeddings of complete
Hausdorff dimension of harmonic measures in the plane
for all continuous u: aQ--*R, where /~ is the Perron solution of the Dirichlet problem with boundary values u. We must assume here that E has positive capacity, but not that 92 is regular for the
Tameness of hyperbolic 3-manifolds
We show that hyperbolic 3-manifolds with finitely generated fundamental group are tame, that is the ends are products. We actually work in slightly greater generality with pinched negatively curved
On the Hausdorff dimension of harmonic measure in higher dimension
Assume A=I~n\E a domain in F, n, where E is a compact set. Denote o~(A,A,x) the harmonic measure for A of A, evaluated at x ~ R d. According to 0ksendal 's theorem [O], o E= o(A, . ,x) is singular
Lectures on quasiconformal mappings
The Ahlfors Lectures: Acknowledgments Differentiable quasiconformal mappings The general definition Extremal geometric properties Boundary correspondence The mapping theorem Teichmuller spaces
Dynamic rays of bounded-type entire functions
We construct an entire function in the Eremenko-Lyubich class B whose Julia set has only bounded path-components. This answers a question of Eremenko from 1989 in the negative. On the other hand, we
A finiteness theorem for a dynamical class of entire functions
Abstract We define a class Σ of entire functions whose covering properties are similar to those of rational maps. The set Σ is closed under composition of functions, and we show that when regarded as
On the uniform rectifiability of AD-regular measures with bounded Riesz transform operator: the case of codimension 1
We prove that if μ is a d-dimensional Ahlfors-David regular measure in $${\mathbb{R}^{d+1}}$$Rd+1 , then the boundedness of the d-dimensional Riesz transform in L2(μ) implies that the non-BAUP