|
Joseph H. Sampson
Joseph Harold Sampson Jr. (1926 – 2003) was an American mathematician known for his work in mathematical analysis, geometry and topology, especially his work about harmonic maps in collaboration with James Eells. He obtained his Ph.D. in mathematics from Princeton University in 1951 under the supervision of Salomon Bochner. Mathematical work In 1964, Sampson and James Eells introduced harmonic maps, which are mappings between Riemannian manifolds which solve a geometrically-defined system of partial differential equations. They can also be defined via the calculus of variations. Generalizing Bochner's work on harmonic functions, Eells and Sampson derived the Bochner identity, and used it to prove the triviality of harmonic maps under certain curvature conditions. Eells and Sampson established the existence of harmonic maps whenever the domain manifold is closed manifold, closed and the target has nonpositive sectional curvature. Their proof analyzed the harmonic map, harmonic ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
United States
The United States of America (U.S.A. or USA), commonly known as the United States (U.S. or US) or America, is a country primarily located in North America. It consists of 50 states, a federal district, five major unincorporated territories, nine Minor Outlying Islands, and 326 Indian reservations. The United States is also in free association with three Pacific Island sovereign states: the Federated States of Micronesia, the Marshall Islands, and the Republic of Palau. It is the world's third-largest country by both land and total area. It shares land borders with Canada to its north and with Mexico to its south and has maritime borders with the Bahamas, Cuba, Russia, and other nations. With a population of over 333 million, it is the most populous country in the Americas and the third most populous in the world. The national capital of the United States is Washington, D.C. and its most populous city and principal financial center is New York City. Paleo-Americ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homotopy Class
In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deformation being called a homotopy (, ; , ) between the two functions. A notable use of homotopy is the definition of homotopy groups and cohomotopy groups, important invariants in algebraic topology. In practice, there are technical difficulties in using homotopies with certain spaces. Algebraic topologists work with compactly generated spaces, CW complexes, or spectra. Formal definition Formally, a homotopy between two continuous functions ''f'' and ''g'' from a topological space ''X'' to a topological space ''Y'' is defined to be a continuous function H: X \times ,1\to Y from the product of the space ''X'' with the unit interval , 1to ''Y'' such that H(x,0) = f(x) and H(x,1) = g(x) for all x \in X. If we think of the second p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Mathematicians
American(s) may refer to: * American, something of, from, or related to the United States of America, commonly known as the "United States" or "America" ** Americans, citizens and nationals of the United States of America ** American ancestry, people who self-identify their ancestry as "American" ** American English, the set of varieties of the English language native to the United States ** Native Americans in the United States, indigenous peoples of the United States * American, something of, from, or related to the Americas, also known as "America" ** Indigenous peoples of the Americas * American (word), for analysis and history of the meanings in various contexts Organizations * American Airlines, U.S.-based airline headquartered in Fort Worth, Texas * American Athletic Conference, an American college athletic conference * American Recordings (record label), a record label previously known as Def American * American University, in Washington, D.C. Sports teams Soccer * B ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Annales Scientifiques De L'École Normale Supérieure
''Annales Scientifiques de l'École Normale Supérieure'' is a French scientific journal of mathematics published by the Société Mathématique de France. It was established in 1864 by the French chemist Louis Pasteur and published articles in mathematics, physics, chemistry, biology, and geology. In 1900, it became a purely mathematical journal. It is published with help of the Centre national de la recherche scientifique. Its web site is hosted by the mathematics department of the École Normale Supérieure École may refer to: * an elementary school in the French educational stages normally followed by secondary education establishments (collège and lycée) * École (river), a tributary of the Seine flowing in région Île-de-France * École, Savoi .... External links * Archive(1864–2013) Mathematics journals Publications established in 1864 Multilingual journals Multidisciplinary scientific journals Société Mathématique de France academic journals {{mat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Journal Of Mathematics
The ''American Journal of Mathematics'' is a bimonthly mathematics journal published by the Johns Hopkins University Press. History The ''American Journal of Mathematics'' is the oldest continuously published mathematical journal in the United States, established in 1878 at the Johns Hopkins University by James Joseph Sylvester, an English-born mathematician who also served as the journal's editor-in-chief from its inception through early 1884. Initially W. E. Story was associate editor in charge; he was replaced by Thomas Craig in 1880. For volume 7 Simon Newcomb became chief editor with Craig managing until 1894. Then with volume 16 it was "Edited by Thomas Craig with the Co-operation of Simon Newcomb" until 1898. Other notable mathematicians who have served as editors or editorial associates of the journal include Frank Morley, Oscar Zariski, Lars Ahlfors, Hermann Weyl, Wei-Liang Chow, S. S. Chern, André Weil, Harish-Chandra, Jean Dieudonné, Henri Cartan, Stephen S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shing-Tung Yau
Shing-Tung Yau (; ; born April 4, 1949) is a Chinese-American mathematician and the William Caspar Graustein Professor of Mathematics at Harvard University. In April 2022, Yau announced retirement from Harvard to become Chair Professor of mathematics at Tsinghua University. Yau was born in Shantou, China, moved to Hong Kong at a young age, and to the United States in 1969. He was awarded the Fields Medal in 1982, in recognition of his contributions to partial differential equations, the Calabi conjecture, the positive energy theorem, and the Monge–Ampère equation. Yau is considered one of the major contributors to the development of modern differential geometry and geometric analysis. The impact of Yau's work can be seen in the mathematical and physical fields of differential geometry, partial differential equations, convex geometry, algebraic geometry, enumerative geometry, mirror symmetry, general relativity, and string theory, while his work has also touched upon applied ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diffeomorphism
In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable. Definition Given two manifolds M and N, a differentiable map f \colon M \rightarrow N is called a diffeomorphism if it is a bijection and its inverse f^ \colon N \rightarrow M is differentiable as well. If these functions are r times continuously differentiable, f is called a C^r-diffeomorphism. Two manifolds M and N are diffeomorphic (usually denoted M \simeq N) if there is a diffeomorphism f from M to N. They are C^r-diffeomorphic if there is an r times continuously differentiable bijective map between them whose inverse is also r times continuously differentiable. Diffeomorphisms of subsets of manifolds Given a subset X of a manifold M and a subset Y of a manifold N, a function f:X\to Y is said to be smooth if for all p in X there is a neighbor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemann Surface
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. The main interest in Riemann surfaces is that holomorphic functions may be defined between them. Riemann surfaces are nowadays considered the natural setting for studying the global behavior of these functions, especially multi-valued functions such as the square root and other algebraic functions, or the logarithm. Every Riemann surface is a two-dimensional real analytic manifold (i.e., a surface), but it contains more structure (specifically a complex structure) which is needed for the unambiguous definitio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Degree Of A Continuous Mapping
In topology, the degree of a continuous mapping between two compact oriented manifolds of the same dimension is a number that represents the number of times that the domain manifold wraps around the range manifold under the mapping. The degree is always an integer, but may be positive or negative depending on the orientations. The degree of a map was first defined by Brouwer, who showed that the degree is homotopy invariant (invariant among homotopies), and used it to prove the Brouwer fixed point theorem. In modern mathematics, the degree of a map plays an important role in topology and geometry. In physics, the degree of a continuous map (for instance a map from space to some order parameter set) is one example of a topological quantum number. Definitions of the degree From ''S''''n'' to ''S''''n'' The simplest and most important case is the degree of a continuous map from the n-sphere S^n to itself (in the case n=1, this is called the winding number): Let f\colon S^n\t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maximum Principle
In the mathematical fields of partial differential equations and geometric analysis, the maximum principle is any of a collection of results and techniques of fundamental importance in the study of elliptic and parabolic differential equations. In the simplest case, consider a function of two variables such that :\frac+\frac=0. The weak maximum principle, in this setting, says that for any open precompact subset of the domain of , the maximum of on the closure of is achieved on the boundary of . The strong maximum principle says that, unless is a constant function, the maximum cannot also be achieved anywhere on itself. Such statements give a striking qualitative picture of solutions of the given differential equation. Such a qualitative picture can be extended to many kinds of differential equations. In many situations, one can also use such maximum principles to draw precise quantitative conclusions about solutions of differential equations, such as control over the size ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Karen Uhlenbeck
Karen Keskulla Uhlenbeck (born August 24, 1942) is an American mathematician and one of the founders of modern geometric analysis. She is a professor emeritus of mathematics at the University of Texas at Austin, where she held the Sid W. Richardson Foundation Regents Chair. She is currently a distinguished visiting professor at the Institute for Advanced Study and a visiting senior research scholar at Princeton University. Uhlenbeck was elected to the American Philosophical Society in 2007. She won the 2019 Abel Prize for "her pioneering achievements in geometric partial differential equations, gauge theory, and integrable systems, and for the fundamental impact of her work on analysis, geometry and mathematical physics." She is the first, and so far only, woman to win the prize since its inception in 2003. She donated half of the prize money to organizations which promote more engagement of women in research mathematics. Life and career Uhlenbeck was born in Cleveland, Ohio, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Richard Schoen
Richard Melvin Schoen (born October 23, 1950) is an American mathematician known for his work in differential geometry and geometric analysis. He is best known for the resolution of the Yamabe problem in 1984. Career Born in Celina, Ohio, and a 1968 graduate of Fort Recovery High School, he received his B.S. from the University of Dayton in mathematics. He then received his PhD in 1977 from Stanford University. After faculty positions at the Courant Institute, NYU, University of California, Berkeley, and University of California, San Diego, he was Professor at Stanford University from 1987–2014, as Bass Professor of Humanities and Sciences since 1992. He is currently Distinguished Professor and Excellence in Teaching Chair at the University of California, Irvine. His surname is pronounced "Shane." Schoen received an NSF Graduate Research Fellowship in 1972 and a Sloan Research Fellowship in 1979. Schoen is a 1983 MacArthur Fellow. He has been invited to speak at the Interna ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
.jpg "Click for more on -> American Journal Of Mathematics")
