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Double Groupoid
In mathematics, especially in higher-dimensional algebra and homotopy theory, a double groupoid generalises the notion of groupoid and of category to a higher dimension. Definition A double groupoid D is a higher-dimensional groupoid involving a relationship for both 'horizontal' and 'vertical' groupoid structures. (A double groupoid can also be considered as a generalization of certain higher-dimensional groups.) The geometry of squares and their compositions leads to a common representation of a ''double groupoid'' in the following diagram: where M is a set of 'points', H and V are, respectively, 'horizontal' and 'vertical' groupoids, and S is a set of 'squares' with two compositions. The composition laws for a double groupoid D make it also describable as a groupoid internal to the category of groupoids. Given two groupoids H and V over a set M, there is a double groupoid \Box(H,V) with H,V as horizontal and vertical edge groupoids, and squares given by quadruples :: \be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Homomorphism
In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" and () meaning "form" or "shape". However, the word was apparently introduced to mathematics due to a (mis)translation of German meaning "similar" to meaning "same". The term "homomorphism" appeared as early as 1892, when it was attributed to the German mathematician Felix Klein (1849–1925). Homomorphisms of vector spaces are also called linear maps, and their study is the subject of linear algebra. The concept of homomorphism has been generalized, under the name of morphism, to many other structures that either do not have an underlying set, or are not algebraic. This generalization is the starting point of category theory. A homomorphism may also be an isomorphis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Orbit Space
In mathematics, a group action of a group G on a set S is a group homomorphism from G to some group (under function composition) of functions from S to itself. It is said that G acts on S. Many sets of transformations form a group under function composition; for example, the rotations around a point in the plane. It is often useful to consider the group as an abstract group, and to say that one has a group action of the abstract group that consists of performing the transformations of the group of transformations. The reason for distinguishing the group from the transformations is that, generally, a group of transformations of a structure acts also on various related structures; for example, the above rotation group also acts on triangles by transforming triangles into triangles. If a group acts on a structure, it will usually also act on objects built from that structure. For example, the group of Euclidean isometries acts on Euclidean space and also on the figures drawn ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Graph Of Groups
In geometric group theory, a graph of groups is an object consisting of a collection of groups indexed by the vertices and edges of a graph, together with a family of monomorphisms of the edge groups into the vertex groups. There is a unique group, called the fundamental group, canonically associated to each finite connected graph of groups. It admits an orientation-preserving action on a tree: the original graph of groups can be recovered from the quotient graph and the stabilizer subgroups. This theory, commonly referred to as Bass–Serre theory, is due to the work of Hyman Bass and Jean-Pierre Serre. Definition A graph of groups over a graph is an assignment to each vertex of of a group and to each edge of of a group as well as monomorphisms and mapping into the groups assigned to the vertices at its ends. Fundamental group Let be a spanning tree for and define the fundamental group to be the group generated by the vertex groups and elements for each edge of w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Ian Stewart (mathematician)
Ian Nicholas Stewart (born 24 September 1945) is a British mathematician and a popular-science and science-fiction writer. He is Emeritus Professor of Mathematics at the University of Warwick, England. Education and early life Stewart was born in 1945 in Folkestone, England. While in the sixth form at Harvey Grammar School in Folkestone he came to the attention of the mathematics teacher. The teacher had Stewart sit mock A-level examinations without any preparation along with the upper-sixth students; Stewart was placed first in the examination. He was awarded a scholarship to study at the University of Cambridge as an undergraduate student of Churchill College, Cambridge, where he studied the Mathematical Tripos and obtained a first-class Bachelor of Arts degree in mathematics in 1966. Stewart then went to the University of Warwick where his PhD on Lie algebras was supervised by Brian Hartley and completed in 1969. Career and research After his PhD, Stewart was offere ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Marty Golubitsky
Martin Aaron Golubitsky is an American Distinguished professor of mathematics at Ohio State University and the former director of the Mathematical Biosciences Institute. Biography Education Marty Golubitsky was born on April 5, 1945, in Philadelphia, Pennsylvania. He graduated with bachelor's degree in 1966 from the University of Pennsylvania and the same year got his master's there as well. He obtained his Ph.D. from Massachusetts Institute of Technology in 1970 where his advisor was Victor Guillemin. Career Full-time From September 1974 to December 1976 he was an assistant professor at the Queens College and from January of next year to August 1979 served as an associate professor there. Starting from the same month of 1979 he relocated himself to the Arizona State University where he became a professor and served there till August 1983. In September of the same year he held the same position at the University of Houston where he remained till November 2008. From then until 20 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Alan Weinstein
Alan David Weinstein (born 17 June 1943) is a professor of mathematics at the University of California, Berkeley, working in the field of differential geometry, and especially in Poisson manifold, Poisson geometry. Early life and education Weinstein was born in New York City. After attending Roslyn High School, Weinstein obtained a bachelor's degree at the Massachusetts Institute of Technology in 1964. His teachers included, among others, James Munkres, Gian-Carlo Rota, Irving Segal, and, for the first senior course of differential geometry, Sigurður Helgason (mathematician), Sigurður Helgason. He received a PhD at University of California, Berkeley in 1967 under the direction of Shiing-Shen Chern. His dissertation was entitled "''The Cut locus (Riemannian manifold), cut locus and conjugate locus of a Riemannian manifold''". Career Weinstein worked then at MIT on 1967 (as Moore Instructor, Moore instructor) and at University of Bonn, Bonn University in 1968/69. In 1969 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Ana Cannas Da Silva
Ana M. L. G. Cannas da Silva (born 1968) is a Portuguese mathematician specializing in symplectic geometry and geometric topology. She works in Switzerland as a professor in mathematics at ETH Zurich. Early life and education Cannas was born in Lisbon. After studying at St. John de Britto College, she earned a licenciatura in mathematics in 1990 from the Instituto Superior Técnico in the University of Lisbon. She then went to the Massachusetts Institute of Technology for graduate studies, earning a master's degree in 1994 and completing her Ph.D. in 1996. Her dissertation, ''Multiplicity Formulas for Orbifolds'', was supervised by Victor Guillemin. Career After a temporary position as Morrey Assistant Professor at the University of California, Berkeley, Cannas returned to the Instituto Superior Técnico as a faculty member in 1997. She took a second position as a senior lecturer and research scholar in mathematics at Princeton University in 2006, keeping at the same time her posi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
