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Nikolai Georgievich Makarov
Nikolai Georgievich Makarov (russian: Николай Георгиевич Макаров; born January 1955) is a Russian mathematician. He is known for his work in complex analysis and its applications to dynamical systems, probability theory and mathematical physics. He is currently the Richard Merkin Distinguished Professor of Mathematics at Caltech, 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 with a bachelor's degree in 1982. He received his Ph.D. (Candidate of Science) from the Steklov Institute of Mathematics in 1986 under Nikolai Nikolski with thesis ''Metric properties of harmonic measure'' (title translated from Russian).. He was an academic at the Steklov Institute of Mathematics in Leningrad. Since 1991 he has been a professor at Caltech. In 1986 he was an Invited Speaker of the ICM in Berkeley, California. In 1986 he was awarded the Salem Prize fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Russia
Russia (, , ), or the Russian Federation, is a List of transcontinental countries, transcontinental country spanning Eastern Europe and North Asia, Northern Asia. It is the List of countries and dependencies by area, largest country in the world, with its internationally recognised territory covering , and encompassing one-eighth of Earth's inhabitable landmass. Russia extends across Time in Russia, eleven time zones and shares Borders of Russia, land boundaries with fourteen countries, more than List of countries and territories by land borders, any other country but China. It is the List of countries and dependencies by population, world's ninth-most populous country and List of European countries by population, Europe's most populous country, with a population of 146 million people. The country's capital and List of cities and towns in Russia by population, largest city is Moscow, the List of European cities by population within city limits, largest city entirely within E ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
International Congress Of Mathematicians
The International Congress of Mathematicians (ICM) is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union (IMU). The Fields Medals, the Nevanlinna Prize (to be renamed as the IMU Abacus Medal), the Carl Friedrich Gauss Prize, Gauss Prize, and the Chern Medal are awarded during the congress's opening ceremony. Each congress is memorialized by a printed set of Proceedings recording academic papers based on invited talks intended to be relevant to current topics of general interest. Being List of International Congresses of Mathematicians Plenary and Invited Speakers, invited to talk at the ICM has been called "the equivalent ... of an induction to a hall of fame". History Felix Klein and Georg Cantor are credited with putting forward the idea of an international congress of mathematicians in the 1890s.A. John Coleman"Mathematics without borders": a book review ''CMS Notes'', vol 31, no. 3, April 1999 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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-dimensional lattice models in statistical mechanics. Given a parameter ''κ'' and a domain in the complex plane ''U'', it gives a family of random curves in ''U'', with ''κ'' controlling how much the curve turns. There are two main variants of SLE, ''chordal SLE'' which gives a family of random curves from two fixed boundary points, and ''radial SLE'', which gives a family of random curves from a fixed boundary point to a fixed interior point. These curves are defined to satisfy conformal invariance and a domain Markov property. It was discovered by as a conjectured scaling limit of the planar uniform spanning tree (UST) and the planar loop-erased random walk (LERW) probabilistic processes, and developed by him together with Greg Lawler and Wendel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 elements are random variables. Many important properties of physical systems can be represented mathematically as matrix problems. For example, the thermal conductivity of a lattice can be computed from the dynamical matrix of the particle-particle interactions within the lattice. Applications Physics In nuclear physics, random matrices were introduced by Eugene Wigner to model the nuclei of heavy atoms. Wigner postulated that the spacings between the lines in the spectrum of a heavy atom nucleus should resemble the spacings between the eigenvalues of a random matrix, and should depend only on the symmetry class of the underlying evolution. In solid-state physics, random matrices model the behaviour of large disordered Hamiltonians in the mean-field approximation. In quantum chaos, the Bohigas–Giannoni–Schmit (BGS) conjecture asserts ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Universality (dynamical Systems)
In statistical mechanics, universality is the observation that there are properties for a large class of systems that are independent of the dynamical details of the system. Systems display universality in a scaling limit, when a large number of interacting parts come together. The modern meaning of the term was introduced by Leo Kadanoff in the 1960s, but a simpler version of the concept was already implicit in the van der Waals equation and in the earlier Landau theory of phase transitions, which did not incorporate scaling correctly. The term is slowly gaining a broader usage in several fields of mathematics, including combinatorics and probability theory, whenever the quantitative features of a structure (such as asymptotic behaviour) can be deduced from a few global parameters appearing in the definition, without requiring knowledge of the details of the system. The renormalization group provides an intuitively appealing, albeit mathematically non-rigorous, explanation of un ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 2006 for "his profound and seminal contributions to harmonic analysis and the theory of smooth dynamical systems." Life He was a student of Arne Beurling and received his Ph.D. from Uppsala University in 1950. He did his post-doctoral work at Harvard University where he met and discussed Fourier series and their convergence with Antoni Zygmund and Raphaël Salem who were there in 1950 and 1951. He is a professor emeritus at Uppsala University, the Royal Institute of Technology in Stockholm, and the University of California, Los Angeles, and has served as director of the Mittag-Leffler Institute in Djursholm outside Stockholm 1968–1984. Between 1978 and 1982 he served as president of the International Mathematical Union. Carleson married Butte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 applicable to aggregation in any system where diffusion is the primary means of transport in the system. DLA can be observed in many systems such as electrodeposition, Hele-Shaw flow, mineral deposits, and dielectric breakdown. The clusters formed in DLA processes are referred to as Brownian trees. These clusters are an example of a fractal. In 2D these fractals exhibit a dimension of approximately 1.71 for free particles that are unrestricted by a lattice, however computer simulation of DLA on a lattice will change the fractal dimension slightly for a DLA in the same embedding dimension. Some variations are also observed depending on the geometry of the growth, whether it be from a single point radially outward or from a plane or line ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hausdorff Dimension
In mathematics, Hausdorff dimension is a measure of ''roughness'', or more specifically, fractal dimension, that was first introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point is zero, of a line segment is 1, of a square is 2, and of a cube is 3. That is, for sets of points that define a smooth shape or a shape that has a small number of corners—the shapes of traditional geometry and science—the Hausdorff dimension is an integer agreeing with the usual sense of dimension, also known as the topological dimension. However, formulas have also been developed that allow calculation of the dimension of other less simple objects, where, solely on the basis of their properties of scaling and self-similarity, one is led to the conclusion that particular objects—including fractals—have non-integer Hausdorff dimensions. Because of the significant technical advances made by Abram Samoilovitch Besicovitch allowing computation of di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 boundary of a bounded domain in Euclidean space R^n, n\geq 2 is the probability that a Brownian motion started inside a domain hits that subset of the boundary. More generally, harmonic measure of an Itō diffusion ''X'' describes the distribution of ''X'' as it hits the boundary of ''D''. In the complex plane, harmonic measure can be used to estimate the modulus of an analytic function inside a domain ''D'' given bounds on the modulus on the boundary of the domain; a special case of this principle is Hadamard's three-circle theorem. On simply connected planar domains, there is a close connection between harmonic measure and the theory of conformal maps. The term ''harmonic measure'' was introduced by Rolf Nevanlinna in 1928 for planar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 sometimes be exactly solved or classified. Conformal field theory has important applications to condensed matter physics, statistical mechanics, quantum statistical mechanics, and string theory. Statistical and condensed matter systems are indeed often conformally invariant at their thermodynamic or quantum critical points. Scale invariance vs conformal invariance In quantum field theory, scale invariance is a common and natural symmetry, because any fixed point of the renormalization group is by definition scale invariant. Conformal symmetry is stronger than scale invariance, and one needs additional assumptions to argue that it should appear in nature. The basic idea behind its plausibility is that ''local'' scale invariant theories have their ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 elements are random variables. Many important properties of physical systems can be represented mathematically as matrix problems. For example, the thermal conductivity of a lattice can be computed from the dynamical matrix of the particle-particle interactions within the lattice. Applications Physics In nuclear physics, random matrices were introduced by Eugene Wigner to model the nuclei of heavy atoms. Wigner postulated that the spacings between the lines in the spectrum of a heavy atom nucleus should resemble the spacings between the eigenvalues of a random matrix, and should depend only on the symmetry class of the underlying evolution. In solid-state physics, random matrices model the behaviour of large disordered Hamiltonians in the mean-field approximation. In quantum chaos, the Bohigas–Giannoni–Schmit (BGS) conjecture asserts ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Dynamics
Complex dynamics is the study of dynamical systems defined by Iterated function, iteration of functions on complex number spaces. Complex analytic dynamics is the study of the dynamics of specifically analytic functions. Techniques *General **Montel's theorem **Poincaré metric **Schwarz lemma **Riemann mapping theorem **Carathéodory's theorem (conformal mapping) **Böttcher's equation *Combinatorics, Combinatorial ** Hubbard trees ** Spider algorithm ** Tuning **Lamination (topology), Laminations **Cantor function, Devil's Staircase algorithm (Cantor function) **Orbit portraits **Jean-Christophe Yoccoz, Yoccoz puzzles Parts * Holomorphic dynamics (dynamics of holomorphic functions) ** in one complex variable ** in several complex variables * Conformal dynamics unites holomorphic dynamics in one complex variable with differentiable dynamics in one real variable. See also *Arithmetic dynamics *Chaos theory *Complex analysis *Complex quadratic polynomial *Fatou set *Infinite co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
