Nikolai Georgievich Makarov (; born January 1955) is a Russian-American mathematician. He is known for his work in

complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic ...

and its applications to

dynamical systems In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space, such as in a parametric curve. Examples include the mathematical models ...

probability theory Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...

and

mathematical physics Mathematical physics is the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the de ...

. He is currently the Richard Merkin Distinguished Professor of Mathematics at

Caltech The California Institute of Technology (branded as Caltech) is a private university, private research university in Pasadena, California, United States. The university is responsible for many modern scientific advancements and is among a small g ...

, where he has been teaching since 1991.

Career

Makarov belongs to the Leningrad school of geometric function theory. He graduated from the

Leningrad State University Saint Petersburg State University (SPBGU; ) is a public research university in Saint Petersburg, Russia, and one of the oldest and most prestigious universities in Russia. Founded in 1724 by a decree of Peter the Great, the university from the be ...

with a bachelor's degree in 1982. He received his Ph.D. (Candidate of Science) from the

Steklov Institute of Mathematics Steklov Institute of Mathematics or Steklov Mathematical Institute () is a premier research institute based in Moscow, specialized in mathematics, and a part of the Russian Academy of Sciences. The institute is named after Vladimir Andreevich Stek ...

in 1986 under Nikolai Nikolski with thesis ''Metric properties of harmonic measure'' (title translated from Russian). He was an academic at the

in Leningrad. Since 1991 he has been a professor at

. In 1986 he was an Invited Speaker of the ICM in

Berkeley, California Berkeley ( ) is a city on the eastern shore of San Francisco Bay in northern Alameda County, California, United States. It is named after the 18th-century Anglo-Irish bishop and philosopher George Berkeley. It borders the cities of Oakland, Cali ...

. 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 In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...

boundary using stochastic methods. In 2020, he was awarded the

Rolf Schock Prize The Rolf Schock Prizes were established and endowed by bequest of philosopher and artist Rolf Schock (1933–1986). The prizes were first awarded in Stockholm, Sweden, in 1993 and, since 2005, are awarded every three years. It is sometimes conside ...

, "for his significant contributions to

and its applications to mathematical physics". His doctoral students include the

Fields medal The Fields Medal is a prize awarded to two, three, or four mathematicians under 40 years of age at the International Congress of Mathematicians, International Congress of the International Mathematical Union (IMU), a meeting that takes place e ...

list Stanislav Smirnov, Alexei Poltoratski and .

Research

Makarov works in complex analysis and related fields (potential theory, harmonic analysis, spectral theory) as well as on various applications to

complex dynamics Complex dynamics, or holomorphic dynamics, is the study of dynamical systems obtained by Iterated function, iterating a complex analytic mapping. This article focuses on the case of algebraic dynamics, where a polynomial or rational function is it ...

random matrices In probability theory and mathematical physics, a random matrix is a matrix-valued random variable—that is, a matrix in which some or all of its entries are sampled randomly from a probability distribution. Random matrix theory (RMT) is the ...

and mathematical

conformal field theory A conformal field theory (CFT) is a quantum field theory that is invariant under conformal transformations. In two dimensions, there is an infinite-dimensional algebra of local conformal transformations, and conformal field theories can sometime ...

. Makarov's most well-known result concerns the theory of

harmonic measure In mathematics, especially potential theory, harmonic measure is a concept related to the theory of harmonic functions that arises from the solution of the classical Dirichlet problem. In probability theory, the harmonic measure of a subset of the ...

in the complex plane. Makarov's theorem states that: 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 In mathematics, Hausdorff dimension is a measure of ''roughness'', or more specifically, fractal dimension, that was introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point is zero, of a line ...

1. Makarov has also studied

diffusion-limited aggregation Diffusion-limited aggregation (DLA) is the process whereby particles undergoing a random walk due to Brownian motion cluster together to form aggregates of such particles. This theory, proposed by T.A. Witten Jr. and L.M. Sander in 1981, is ap ...

which describes crystal growth in two dimensions with

Lennart Carleson Lennart Axel Edvard Carleson (born 18 March 1928) is a Swedish mathematician, known as a leader in the field of harmonic analysis. One of his most noted accomplishments is his proof of Lusin's conjecture. He was awarded the Abel Prize in 200 ...

and Beurling-Malliavin theory with his former student Alexei Poltoratski. He has studied the thermodynamic formalism for iterations of the rational functions with another of his former students Stanislav Smirnov, Fields medallist. He has studied the universality laws and field convergence in normal

random matrix In probability theory and mathematical physics, a random matrix is a matrix-valued random variable—that is, a matrix in which some or all of its entries are sampled randomly from a probability distribution. Random matrix theory (RMT) is the ...

ensembles. His most recent research concerns the mathematical

and its relation to

Schramm–Loewner evolution In probability theory, the Schramm–Loewner evolution with parameter ''κ'', also known as stochastic Loewner evolution (SLE''κ''), is a family of random planar curves that have been proven to be the scaling limit of a variety of two-dimensiona ...

theory.

Selected publications

* English version: * * with S. Smirnov: * with L. Carleson: * with L. Carleson: * with I. Binder and S. Smirnov: * with Y. Ameur and H. Hedenmalm: * with N.-G. Kang: * with S.-Y. Lee:

References

External links

Nikolai G. Makarov, Mathematics Professor, caltech.edu

mathnet.ru
{{DEFAULTSORT:Makarov, Nikolai G 1955 births Living people 20th-century Russian mathematicians 21st-century Russian mathematicians Saint Petersburg State University alumni California Institute of Technology faculty