Jean Leray was a French mathematician, who worked on both partial differential equations and algebraic topology.
In mathematics, the uniformization theorem states that every simply connected Riemann surface is conformally equivalent to one of three Riemann surfaces: the open unit disk, the complex plane, or the Riemann sphere. The theorem is a generalization of the Riemann mapping theorem from simply connected open subsets of the plane to arbitrary simply connected Riemann surfaces.
In mathematics, in particular in the theory of modular forms, a Hecke operator, studied by Erich Hecke, is a certain kind of "averaging" operator that plays a significant role in the structure of vector spaces of modular forms and more general automorphic representations.
The Schauder fixed-point theorem is an extension of the Brouwer fixed-point theorem to topological vector spaces, which may be of infinite dimension. It asserts that if is a nonempty convex closed subset of a Hausdorff topological vector space and is a continuous mapping of into itself such that is contained in a compact subset of , then has a fixed point.
Juliusz Paweł Schauder was a Polish mathematician known for his work in functional analysis, partial differential equations and mathematical physics.
Hilbert's nineteenth problem is one of the 23 Hilbert problems, set out in a list compiled by David Hilbert in 1900. It asks whether the solutions of regular problems in the calculus of variations are always analytic. Informally, and perhaps less directly, since Hilbert's concept of a "regular variational problem" identifies this precisely as a variational problem whose Euler–Lagrange equation is an elliptic partial differential equation with analytic coefficients, Hilbert's nineteenth problem, despite its seemingly technical statement, simply asks whether, in this class of partial differential equations, any solution inherits the relatively simple and well understood property of being an analytic function from the equation it satisfies. Hilbert's nineteenth problem was solved independently in the late 1950s by Ennio De Giorgi and John Forbes Nash, Jr.
In mathematics, a mixed boundary condition for a partial differential equation defines a boundary value problem in which the solution of the given equation is required to satisfy different boundary conditions on disjoint parts of the boundary of the domain where the condition is stated. Precisely, in a mixed boundary value problem, the solution is required to satisfy a Dirichlet or a Neumann boundary condition in a mutually exclusive way on disjoint parts of the boundary.
Eberhard Frederich Ferdinand Hopf was a German mathematician and astronomer, one of the founding fathers of ergodic theory and a pioneer of bifurcation theory who also made significant contributions to the subjects of partial differential equations and integral equations, fluid dynamics, and differential geometry. The Hopf maximum principle is an early result of his (1927) that is one of the most important techniques in the theory of elliptic partial differential equations.
In mathematics, and specifically the field of partial differential equations (PDEs), a parametrix is an approximation to a fundamental solution of a PDE, and is essentially an approximate inverse to a differential operator.
Hans Maass was a German mathematician who introduced Maass wave forms and Koecher–Maass series and Maass–Selberg relations and who proved most of the Saito–Kurokawa conjecture. Maass was a student of Erich Hecke.
In mathematics, and more precisely, in functional Analysis and PDEs, the Schauder estimates are a collection of results due to Juliusz Schauder concerning the regularity of solutions to linear, uniformly elliptic partial differential equations. The estimates say that when the equation has appropriately smooth terms and appropriately smooth solutions, then the Hölder norm of the solution can be controlled in terms of the Hölder norms for the coefficient and source terms. Since these estimates assume by hypothesis the existence of a solution, they are called a priori estimates.
In the mathematical study of harmonic functions, the Perron method, also known as the method of subharmonic functions, is a technique introduced by Oskar Perron for the solution of the Dirichlet problem for Laplace's equation. The Perron method works by finding the largest subharmonic function with boundary values below the desired values; the "Perron solution" coincides with the actual solution of the Dirichlet problem if the problem is soluble.
In the mathematical theory of automorphic forms, a converse theorem gives sufficient conditions for a Dirichlet series to be the Mellin transform of a modular form. More generally a converse theorem states that a representation of an algebraic group over the adeles is automorphic whenever the L-functions of various twists of it are well-behaved.
In differential geometry, Bernstein's problem is as follows: if the graph of a function on Rn−1 is a minimal surface in Rn, does this imply that the function is linear? This is true for n at most 8, but false for n at least 9. The problem is named for Sergei Natanovich Bernstein who solved the case n = 3 in 1914.
In mathematics, the Valentiner group is the perfect triple cover of the alternating group on 6 points, and is a group of order 1080. It was found by Herman Valentiner in the form of an action of A6 on the complex projective plane, and was studied further by Wiman (1896).
In mathematics, modular units are certain units of rings of integers of fields of modular functions, introduced by Kubert and Lang. They are functions whose zeroes and poles are confined to the cusps.
Myron Mathisson was a theoretical physicist of Polish and Jewish descent. He is known for his work in general relativity, for developing a new method to analyze the properties of fundamental solutions of linear hyperbolic partial differential equations, and proved, in a special case, the Hadamard conjecture on the class of equations that satisfy the Huygens principle.
Edward Norman Dancer FAA is an Australian mathematician, specializing in nonlinear analysis.
Leifur Ásgeirsson was the first Icelandic mathematician to gain major international recognition.
Erich Hans Rothe was a German-born American mathematician, who did research in mathematical analysis, differential equations, integral equations, and mathematical physics. He is known for the Rothe method used for solving evolution equations.
