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Artin–Tits Group
In the mathematical area of group theory, Artin groups, also known as Artin–Tits groups or generalized braid groups, are a family of infinite discrete group (mathematics), groups defined by simple presentation of a group, presentations. They are closely related with Coxeter groups. Examples are free groups, free abelian groups, braid groups, and right-angled Artin–Tits groups, among others. The groups are named after Emil Artin, due to his early work on braid groups in the 1920s to 1940s, and Jacques Tits who developed the theory of a more general class of groups in the 1960s. Definition An Artin–Tits presentation is a group presentation of a group, presentation \langle S \mid R \rangle where S is a (usually finite) set of generators and R is a set of Artin–Tits relations, namely relations of the form stst\ldots = tsts\ldots for distinct s, t in S, where both sides have equal lengths, and there exists at most one relation for each pair of distinct generators ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field (mathematics), fields, and vector spaces, can all be seen as groups endowed with additional operation (mathematics), operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right. Various physical systems, such as crystals and the hydrogen atom, and Standard Model, three of the four known fundamental forces in the universe, may be modelled by symmetry groups. Thus group theory and the closely related representation theory have many important applications in physics, chemistry, and materials science. Group theory is also ce ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fundamental Group
In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of the topological space. The fundamental group is the first and simplest homotopy group. The fundamental group is a homotopy invariant—topological spaces that are homotopy equivalent (or the stronger case of homeomorphic) have isomorphic fundamental groups. The fundamental group of a topological space X is denoted by \pi_1(X). Intuition Start with a space (for example, a surface), and some point in it, and all the loops both starting and ending at this point— paths that start at this point, wander around and eventually return to the starting point. Two loops can be combined in an obvious way: travel along the first loop, then along the second. Two loops are considered equivalent if one can be deformed into the other without breakin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
HNN Extension
In mathematics, the HNN extension is an important construction of combinatorial group theory. Introduced in a 1949 paper ''Embedding Theorems for Groups'' by Graham Higman, Bernhard Neumann, and Hanna Neumann, it embeds a given group ''G'' into another group ''G' '', in such a way that two given isomorphic subgroups of ''G'' are conjugate (through a given isomorphism) in ''G' ''. Construction Let ''G'' be a group with presentation G = \langle S \mid R\rangle , and let \alpha\colon H \to K be an isomorphism between two subgroups of ''G''. Let ''t'' be a new symbol not in ''S'', and define :G*_ = \left \langle S,t \mid R, tht^=\alpha(h), \forall h\in H \right \rangle. The group G*_ is called the ''HNN extension of'' ''G'' ''relative to'' α. The original group G is called the ''base group'' for the construction, while the subgroups ''H'' and ''K'' are the ''associated subgroups''. The new generator ''t'' is called the ''stable letter''. Key properties Since the presentation for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complete Graph
In the mathematical field of graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge. A complete digraph is a directed graph in which every pair of distinct vertices is connected by a pair of unique edges (one in each direction). Graph theory itself is typically dated as beginning with Leonhard Euler's 1736 work on the Seven Bridges of Königsberg. However, drawings of complete graphs, with their vertices placed on the points of a regular polygon, had already appeared in the 13th century, in the work of Ramon Llull. Such a drawing is sometimes referred to as a mystic rose. Properties The complete graph on vertices is denoted by . Some sources claim that the letter in this notation stands for the German word , but the German name for a complete graph, , does not contain the letter , and other sources state that the notation honors the contributions of Kazimierz Kuratowski to graph theory. has edges (a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph (discrete Mathematics)
In discrete mathematics, and more specifically in graph theory, a graph is a structure amounting to a Set (mathematics), set of objects in which some pairs of the objects are in some sense "related". The objects correspond to mathematical abstractions called ''Vertex (graph theory), vertices'' (also called ''nodes'' or ''points'') and each of the related pairs of vertices is called an ''edge'' (also called ''link'' or ''line''). Typically, a graph is depicted in diagrammatic form as a set of dots or circles for the vertices, joined by lines or curves for the edges. Graphs are one of the objects of study in discrete mathematics. The edges may be directed or undirected. For example, if the vertices represent people at a party, and there is an edge between two people if they shake hands, then this graph is undirected because any person ''A'' can shake hands with a person ''B'' only if ''B'' also shakes hands with ''A''. In contrast, if an edge from a person ''A'' to a person ''B'' m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematische Annalen
''Mathematische Annalen'' (abbreviated as ''Math. Ann.'' or, formerly, ''Math. Annal.'') is a German mathematical research journal founded in 1868 by Alfred Clebsch and Carl Neumann. Subsequent managing editors were Felix Klein, David Hilbert, Otto Blumenthal, Erich Hecke, Heinrich Behnke, Hans Grauert, Heinz Bauer, Herbert Amann, Jean-Pierre Bourguignon, Wolfgang Lück, and Nigel Hitchin. Currently, the managing editor of Mathematische Annalen is Thomas Schick. Volumes 1–80 (1869–1919) were published by Teubner. Since 1920 (vol. 81), the journal has been published by Springer. In the late 1920s, under the editorship of Hilbert, the journal became embroiled in controversy over the participation of L. E. J. Brouwer on its editorial board, a spillover from the foundational Brouwer–Hilbert controversy. Between 1945 and 1947 the journal briefly ceased publication. References External links''Mathematische Annalen''homepage at Springer''Mathematische Annalen''archive (1869 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Biautomatic Group
In mathematics, an automatic group is a finitely generated group equipped with several finite-state automata. These automata represent the Cayley graph of the group. That is, they can tell if a given word representation of a group element is in a "canonical form" and can tell if two elements given in canonical words differ by a generator. More precisely, let ''G'' be a group and ''A'' be a finite set of generators. Then an ''automatic structure'' of ''G'' with respect to ''A'' is a set of finite-state automata: * the ''word-acceptor'', which accepts for every element of ''G'' at least one word in A^\ast representing it; *''multipliers'', one for each a \in A \cup \, which accept a pair (''w''1, ''w''2), for words ''w''''i'' accepted by the word-acceptor, precisely when w_1 a = w_2 in ''G''. The property of being automatic does not depend on the set of generators. Properties Automatic groups have word problem solvable in quadratic time. More strongly, a given word can actua ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperplane Arrangement
In geometry and combinatorics, an arrangement of hyperplanes is an arrangement of a finite set ''A'' of hyperplanes in a linear, affine, or projective space ''S''. Questions about a hyperplane arrangement ''A'' generally concern geometrical, topological, or other properties of the complement, ''M''(''A''), which is the set that remains when the hyperplanes are removed from the whole space. One may ask how these properties are related to the arrangement and its intersection semilattice. The intersection semilattice of ''A'', written ''L''(''A''), is the set of all subspaces that are obtained by intersecting some of the hyperplanes; among these subspaces are ''S'' itself, all the individual hyperplanes, all intersections of pairs of hyperplanes, etc. (excluding, in the affine case, the empty set). These intersection subspaces of ''A'' are also called the flats of ''A''. The intersection semilattice ''L''(''A'') is partially ordered by ''reverse inclusion''. If the whole space ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kyoji Saito
Kyōji Saitō (齋藤 恭司, Saitō Kyōji; born 25 December 1944) is a Japanese mathematician, specializing in algebraic geometry and complex analytic geometry. Education and career Saito received in 1971 his promotion Ph.D. from the University of Göttingen under Egbert Brieskorn, with thesis ''Quasihomogene isolierte Singularitäten von Hyperflächen'' (Quasihomogeneous isolated singularities of hypersurfaces). Saito is a professor at the Research Institute for Mathematical Sciences (RIMS) of Kyoto University. Saito's research deals with the interplay among Lie algebras, reflection groups (Coxeter groups), braid groups, and singularities of hypersurfaces. From the 1980s, he did research on underlying symmetries of period integrals in complex hypersurfaces. Saito introduced higher-dimensional generalizations of elliptic integrals. These generalizations are integrals of "primitive forms", first considered in the study of the unfolding of isolated singularities of complex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Egbert Brieskorn
Egbert Valentin Brieskorn (7 July 1936, in Rostock – 11 July 2013, in Bonn) was a German mathematician who introduced Brieskorn spheres and the Brieskorn–Grothendieck resolution. Education Brieskorn was born in 1936 as the son of a mill construction engineer in East Prussia. He grew up in Freudenberg (Siegerland) and studied mathematics and physics at the Ludwig-Maximilians-Universität München and the Rheinische Friedrich-Wilhelms-Universität Bonn. In 1963 he received his doctorate at Bonn under Friedrich Hirzebruch with thesis ''Zur differentialtopologischen und analytischen Klassifizierung gewisser algebraischer Mannigfaltigkeiten'', followed by his habilitation in 1968. Career From 1969 until 1973 he was professor ordinarius at Georg-August-Universität Göttingen and from 1973 to 1975 at the Sonderforschungsbereich Theoretische Mathematik in Bonn (since 1980 called the Max-Planck-Institut für Mathematik). From 1975 until his retirement as professor emeritus in 2001 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pierre Deligne
Pierre René, Viscount Deligne (; born 3 October 1944) is a Belgian mathematician. He is best known for work on the Weil conjectures, leading to a complete proof in 1973. He is the winner of the 2013 Abel Prize, 2008 Wolf Prize, 1988 Crafoord Prize, and 1978 Fields Medal. Early life and education Deligne was born in Etterbeek, attended school at Athénée Adolphe Max and studied at the Université libre de Bruxelles (ULB), writing a dissertation titled ''Théorème de Lefschetz et critères de dégénérescence de suites spectrales'' (Theorem of Lefschetz and criteria of degeneration of spectral sequences). He completed his doctorate at the University of Paris-Sud in Orsay 1972 under the supervision of Alexander Grothendieck, with a thesis titled ''Théorie de Hodge''. Career Starting in 1972, Deligne worked with Grothendieck at the Institut des Hautes Études Scientifiques (IHÉS) near Paris, initially on the generalization within scheme theory of Zariski's main theorem. In 196 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Cohomology
In mathematics (more specifically, in homological algebra), group cohomology is a set of mathematical tools used to study groups using cohomology theory, a technique from algebraic topology. Analogous to group representations, group cohomology looks at the group actions of a group ''G'' in an associated ''G''-module ''M'' to elucidate the properties of the group. By treating the ''G''-module as a kind of topological space with elements of G^n representing ''n''-simplices, topological properties of the space may be computed, such as the set of cohomology groups H^n(G,M). The cohomology groups in turn provide insight into the structure of the group ''G'' and ''G''-module ''M'' themselves. Group cohomology plays a role in the investigation of fixed points of a group action in a module or space and the quotient module or space with respect to a group action. Group cohomology is used in the fields of abstract algebra, homological algebra, algebraic topology and algebraic number theory, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
