Analysis is the branch of mathematics dealing with limits and related theories, such as differentiation, integration, measure, sequences, series, and analytic functions.
In differential geometry, the Atiyah–Singer index theorem, proved by Michael Atiyah and Isadore Singer (1963), states that for an elliptic differential operator on a compact manifold, the analytical index is equal to the topological index. It includes many other theorems, such as the Chern–Gauss–Bonnet theorem and Riemann–Roch theorem, as special cases, and has applications to theoretical physics.
Elias Menachem Stein was an American mathematician who was a leading figure in the field of harmonic analysis. He was the Albert Baldwin Dod Professor of Mathematics, Emeritus, at Princeton University, where he was a faculty member from 1963 until his death in 2018.
Charles Louis Fefferman is an American mathematician at Princeton University, where he is currently the Herbert E. Jones, Jr. '43 University Professor of Mathematics. He has been awarded the Fields Medal in 1978 for his contributions to mathematical analysis.
In mathematical analysis a pseudo-differential operator is an extension of the concept of differential operator. Pseudo-differential operators are used extensively in the theory of partial differential equations and quantum field theory.
Walter Rudin was an Austrian-American mathematician and professor of Mathematics at the University of Wisconsin–Madison.
In mathematics, a varifold is, loosely speaking, a measure-theoretic generalization of the concept of a differentiable manifold, by replacing differentiability requirements with those provided by rectifiable sets, while maintaining the general algebraic structure usually seen in differential geometry. Varifolds generalize the idea of a rectifiable current, and are studied in geometric measure theory.
In mathematics, L2 cohomology is a cohomology theory for smooth non-compact manifolds M with Riemannian metric. It is defined in the same way as de Rham cohomology except that one uses square-integrable differential forms. The notion of square-integrability makes sense because the metric on M gives rise to a norm on differential forms and a volume form.
In mathematics, singular integrals are central to harmonic analysis and are intimately connected with the study of partial differential equations. Broadly speaking a singular integral is an integral operator
In mathematics, the Malgrange–Ehrenpreis theorem states that every non-zero linear differential operator with constant coefficients has a Green's function. It was first proved independently by Leon Ehrenpreis and Bernard Malgrange (1955–1956).
In mathematics, the Birkhoff–Grothendieck theorem classifies holomorphic vector bundles over the complex projective line. In particular every holomorphic vector bundle over is a direct sum of holomorphic line bundles. The theorem was proved by Alexander Grothendieck, and is more or less equivalent to Birkhoff factorization introduced by George David Birkhoff (1909).
Leon Melvyn Simon, born in 1945, is a Leroy P. Steele Prize and Bôcher Prize-winning mathematician, known for deep contributions to the fields of geometric analysis, geometric measure theory, and partial differential equations. He is currently Professor Emeritus in the Mathematics Department at Stanford University.
Robert Stephen Strichartz was an American mathematician who specialized in mathematical analysis.
Masatake Kuranishi was a Japanese mathematician who worked on several complex variables, partial differential equations, and differential geometry.
Karsten Grove is a Danish-American mathematician working in metric and differential geometry, differential topology and global analysis, mainly in topics related to global Riemannian geometry, Alexandrov geometry, isometric group actions and manifolds with positive or nonnegative sectional curvature.
The Stefan Bergman Prize is a mathematics award, funded by the estate of the widow of mathematician Stefan Bergman and supported by the American Mathematical Society. The award is granted for mathematical research in: "1) the theory of the kernel function and its applications in real and complex analysis; or 2) function-theoretic methods in the theory of partial differential equations of elliptic type with attention to Bergman's operator method."
Duong Hong Phong is an American mathematician of Vietnamese origin. He is a professor of mathematics at Columbia University. He is known for his research on complex analysis, partial differential equations, string theory and complex geometry.
Francis Michael Christ is an American mathematician and professor at University of California, Berkeley, specializing in harmonic analysis, partial differential equations, and several complex variables. He is known for the Christ–Kiselev maximal inequality.
Guy David is a French mathematician, specializing in analysis.
Ulrich Pinkall is a German mathematician, specializing in differential geometry and computer graphics.
