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Loewner Energy
In complex analysis, the Loewner energy is an invariant of a domain in the complex plane, or equivalently an invariant of the boundary of the domain, a simple closed curve. According to the uniformization theorem, every domain has a conformal mapping to one of three uniform Riemann surfaces: an open unit disk, the complex plane, or the Riemann sphere. In a 1923 work on the Bieberbach conjecture, Charles Loewner showed that (a suitable normalization of) this uniform mapping can be described as the solution to the Loewner differential equation, which depends on a certain real-valued function, the ''driving function'', defined on the boundary of the domain. The Loewner energy was originally defined by Yilin Wang and (independently) by Peter Friz and Atul Shekhar as the Dirichlet energy of this driving function. In later work, Wang found an equivalent definition of the Loewner energy as the Dirichlet energy of the logarithmic derivative of the conformal mapping itself. This energy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 geometry, number theory, analytic combinatorics, and applied mathematics, as well as in physics, including the branches of hydrodynamics, thermodynamics, quantum mechanics, and twistor theory. By extension, use of complex analysis also has applications in engineering fields such as nuclear, aerospace, mechanical and electrical engineering. As a differentiable function of a complex variable is equal to the sum function given by its Taylor series (that is, it is analytic), complex analysis is particularly concerned with analytic functions of a complex variable, that is, '' holomorphic functions''. The concept can be extended to functions of several complex variables. Complex analysis is contrasted with real analysis, which dea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Charles Loewner
Charles Loewner (29 May 1893 – 8 January 1968) was an American mathematician. His name was Karel Löwner in Czech and Karl Löwner in German. Early life and career Karl Loewner was born into a Jewish family in Lany, about 30 km from Prague, where his father Sigmund Löwner was a store owner. Loewner received his Ph.D. from the University of Prague in 1917 under supervision of Georg Pick. One of his central mathematical contributions is the proof of the Bieberbach conjecture in the first highly nontrivial case of the third coefficient. The technique he introduced, the Loewner differential equation, has had far-reaching implications in geometric function theory; it was used in the final solution of the Bieberbach conjecture by Louis de Branges in 1985. Loewner worked at the University of Berlin, University of Prague, University of Louisville, Brown University, Syracuse University and eventually at Stanford University. His students include Lipman Bers, Roger Horn, A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quasicircle
In mathematics, a quasicircle is a Jordan curve in the complex plane that is the image of a circle under a quasiconformal mapping of the plane onto itself. Originally introduced independently by and , in the older literature (in German) they were referred to as quasiconformal curves, a terminology which also applied to arc (geometry), arcs. In complex analysis and geometric function theory, quasicircles play a fundamental role in the description of the universal Teichmüller space, through quasisymmetric map, quasisymmetric homeomorphisms of the circle. Quasicircles also play an important role in complex dynamical systems. Definitions A quasicircle is defined as the image of a circle under a quasiconformal mapping of the extended complex plane. It is called a ''K''-quasicircle if the quasiconformal mapping has dilatation ''K''. The definition of quasicircle generalizes the characterization of a Jordan curve as the image of a circle under a homeomorphism of the plane. In particul ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logarithmic Derivative
In mathematics, specifically in calculus and complex analysis, the logarithmic derivative of a function is defined by the formula \frac where is the derivative of . Intuitively, this is the infinitesimal relative change in ; that is, the infinitesimal absolute change in , namely scaled by the current value of . When is a function of a real variable , and takes real, strictly positive values, this is equal to the derivative of , or the natural logarithm of . This follows directly from the chain rule: \frac\ln f(x) = \frac \frac Basic properties Many properties of the real logarithm also apply to the logarithmic derivative, even when the function does ''not'' take values in the positive reals. For example, since the logarithm of a product is the sum of the logarithms of the factors, we have (\log uv)' = (\log u + \log v)' = (\log u)' + (\log v)' . So for positive-real-valued functions, the logarithmic derivative of a product is the sum of the logarithmic derivatives ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirichlet Energy
In mathematics, the Dirichlet energy is a measure of how ''variable'' a function is. More abstractly, it is a quadratic functional on the Sobolev space . The Dirichlet energy is intimately connected to Laplace's equation and is named after the German mathematician Peter Gustav Lejeune Dirichlet. Definition Given an open set and a differentiable function , the Dirichlet energy of the function is the real number :E = \frac 1 2 \int_\Omega \, \nabla u(x) \, ^2 \, dx, where denotes the gradient vector field of the function . Properties and applications Since it is the integral of a non-negative quantity, the Dirichlet energy is itself non-negative, i.e. for every function . Solving Laplace's equation -\Delta u(x) = 0 for all x \in \Omega, subject to appropriate boundary conditions, is equivalent to solving the variational problem of finding a function that satisfies the boundary conditions and has minimal Dirichlet energy. Such a solution is called ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Peter Friz
Peter K. Friz (born 1974 in Klagenfurt) is a mathematician working in the fields of partial differential equations, quantitative finance, and stochastic analysis. Education and career He studied at the Vienna University of Technology, Ecole Centrale Paris, University of Cambridge and Courant Institute of Mathematical Sciences (New York University), and obtained his PhD in 2004 under the supervision of S. R. Srinivasa Varadhan. He worked as a quantitative associate at Merrill Lynch, then held academic positions at the University of Cambridge, and the Radon Institute. Since 2009, he is full professor at Technische Universität Berlin (TU Berlin; also known as Berlin Institute of Technology and Technical University of Berlin, although officially the name should not be translated) is a public university, public research university located in Berlin, Germany. It was the first ..., and associated with the Weierstrass Institute for Applied Analysis and Stochastics in Berlin. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Yilin Wang
Yilin Wang (; born 1991) is a mathematician whose research has involved complex analysis and probability theory, including Teichmüller theory, the Schramm–Loewner evolution, and Loewner energy. Originally from China, and educated in France and Switzerland, she is a junior professor at the Institut des Hautes Études Scientifiques in France and has accepted a position at ETH Zurich in Switzerland starting in July 2025. Education and career Wang was born in Shanghai in 1991 and attended Shanghai Foreign Language School where she elected to learn French as a foreign language. In Wang's third year of high school, the French Ministry of Education held a recruiting campaign in China for students who excelled in mathematics. While Wang did not score particularly well on the recruitment exam, she was the only student to answer her questionnaire in French rather than English.This caught the attention of the interviewers and she was eventually accepted into the programme. After cl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Loewner Differential Equation
In mathematics, the Loewner differential equation, or Loewner equation, is an ordinary differential equation discovered by Charles Loewner in 1923 in complex analysis and geometric function theory. Originally introduced for studying slit mappings (conformal mappings of the open disk onto the complex plane with a curve joining 0 to ∞ removed), Loewner's method was later developed in 1943 by the Russian mathematician Pavel Parfenevich Kufarev (1909–1968). Any family of domains in the complex plane that expands continuously in the sense of Carathéodory to the whole plane leads to a one parameter family of conformal mappings, called a Loewner chain, as well as a two parameter family of holomorphic univalent self-mappings of the unit disk, called a Loewner semigroup. This semigroup corresponds to a time dependent holomorphic vector field on the disk given by a one parameter family of holomorphic functions on the disk with positive real part. The Loewner semigroup generalizes the no ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bieberbach Conjecture
In complex analysis, de Branges's theorem, or the Bieberbach conjecture, is a theorem that gives a necessary condition on a holomorphic function in order for it to map the open unit disk of the complex plane injectively to the complex plane. It was posed by and finally proven by . The statement concerns the Taylor coefficients a_n of a univalent function, i.e. a one-to-one holomorphic function that maps the unit disk into the complex plane, normalized as is always possible so that a_0=0 and a_1=1. That is, we consider a function defined on the open unit disk which is holomorphic and injective ('' univalent'') with Taylor series of the form :f(z)=z+\sum_ a_n z^n. Such functions are called ''schlicht''. The theorem then states that : , a_n, \leq n \quad \textn\geq 2. The Koebe function (see below) is a function for which a_n=n for all n, and it is schlicht, so we cannot find a stricter limit on the absolute value of the nth coefficient. Schlicht functions The normalization ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Invariant (mathematics)
In mathematics, an invariant is a property of a mathematical object (or a class of mathematical objects) which remains unchanged after operations or transformations of a certain type are applied to the objects. The particular class of objects and type of transformations are usually indicated by the context in which the term is used. For example, the area of a triangle is an invariant with respect to isometries of the Euclidean plane. The phrases "invariant under" and "invariant to" a transformation are both used. More generally, an invariant with respect to an equivalence relation is a property that is constant on each equivalence class. Invariants are used in diverse areas of mathematics such as geometry, topology, algebra and discrete mathematics. Some important classes of transformations are defined by an invariant they leave unchanged. For example, conformal maps are defined as transformations of the plane that preserve angles. The discovery of invariants is an import ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemann Sphere
In mathematics, the Riemann sphere, named after Bernhard Riemann, is a Mathematical model, model of the extended complex plane (also called the closed complex plane): the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers plus a value \infty for infinity. With the Riemann model, the point \infty is near to very large numbers, just as the point 0 is near to very small numbers. The extended complex numbers are useful in complex analysis because they allow for division by zero in some circumstances, in a way that makes expressions such as 1/0=\infty well-behaved. For example, any rational function on the complex plane can be extended to a holomorphic function on the Riemann sphere, with the Pole (complex analysis), poles of the rational function mapping to infinity. More generally, any meromorphic function can be thought of as a holomorphic function whose codomain is the Riemann sphere. In geometr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemann Surfaces
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed versions of the complex plane: locally near every point they look like patches of the complex plane, but the global topology can be quite different. For example, they can look like a sphere or a torus or several sheets glued together. Examples of Riemann surfaces include graphs of multivalued functions such as √''z'' or log(''z''), e.g. the subset of pairs with . Every Riemann surface is a surface: a two-dimensional real manifold, but it contains more structure (specifically a complex structure). Conversely, a two-dimensional real manifold can be turned into a Riemann surface (usually in several inequivalent ways) if and only if it is orientable and metrizable. Given this, the sphere and torus admit complex structures but the Möbius ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
