Nikolai Georgievich Makarov

Nikolai Georgievich Makarov, (Николай Георгиевич Макаров, born January 1955), is a Russian mathematician, specializing in harmonic analysis.

Makarov belongs to the Leningrad school of geometric function theory. He studied at the Leningrad State University with undergraduate degree in 1982 and with Ph.D. (Candidate of Science) in 1986 under Nikolai Nikolski with thesis Metric properties of harmonic measure (title translated from Russian).[1] In 1986 he was an Invited Speaker of the ICM in Berkeley, California.[2] In 1986 he was awarded the Salem Prize for solving difficult problems involving the boundary behavior of the conformal mapping of a disk onto a domain with a Jordan curve boundary using stochastic methods. He was an academic at the Steklov Institute of Mathematics in Leningrad. Since the 1990s he has been a professor at Caltech.

His doctoral students include the Fields medallist Stanislav Smirnov and Dapeng Zhan. With Zhan, Makarov published research on the stochastic properties of iterated polynomial maps (theory of Julia sets).

Makarov's theorem

Let Ω be a simply connected domain in the complex plane. Suppose that ∂Ω (the boundary of Ω) is a Jordan curve. Then the harmonic measure on ∂Ω has Hausdorff dimension 1.[3][4]

Selected publications
  • Probability methods in the theory of conformal mappings, Algebra i Analiz, 1:1 (1989), pp. 3–59; English version: Leningrad Mathematical Journal, 1990, 1:1, 1–56
  • Fine structure of harmonic measure, St. Petersburg Math. J. 10 (1999), 217–268
  • with S. Smirnov: On thermodynamics of rational maps, I. Negative spectrum, Comm. Math. Phys. 211 (2000), 705–743 doi:10.1007/s002200050833
  • with S. Smirnov: On thermodynamics of rational maps, II. Non-recurrent maps, J. London Math. Soc. 67 (2003), 417-–32 doi:10.1112/S0024610702003964
  • with Lennart Carleson: Aggregation in the plane and Loewner's equation, Comm. Math. Phys. 216 (2001), 583–607 doi:10.1007/s002200000340
  • with Lennart Carleson: Laplacian path models, J. Analyse Math. 87 (2002), 103–150 doi:10.1007/BF02868471
  • with I. Binder and S. Smirnov: Harmonic measure and polynomial Julia sets, Duke Math. J. 117 (2003), 343–365 doi:10.1215/S0012-7094-03-11725-1
  • with Seung-Yeop Lee: Topology of quadrature domains, Journal of the American Mathematical Society 29, no. 2 (2016): 333–369 arXiv.org preprint
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References
  1. Nikolai G. Makarov at the Mathematics Genealogy Project
  2. Makarov, N. G. (1987). "Metric properties of harmonic measure". In: Proceedings of the International Congress of Mathematicians, Berkeley, 1986. Amer. Math. Soc. pp. 766–776.
  3. Makarov, On the distortion of boundary sets under conformal mappings, Proc. London Math. Soc. Ser. 3, 52, vol. 1985, pp. 369–384 doi:10.1112/plms/s3-51.2.369
  4. Ivrii, Oleg (9 August 2017). "On Makarov's principle in conformal mapping". International Mathematical Research Notes. arXiv:1604.05619. doi:10.1093/imrn/rnx129. arXiv.org preprint, 2016
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