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Mnëv's Universality Theorem
In mathematics, Mnëv's universality theorem is a result in the intersection of combinatorics and algebraic geometry used to represent algebraic (or semialgebraic) varieties as realization spaces of oriented matroids. Informally it can also be understood as the statement that point configurations of a fixed combinatorics can show arbitrarily complicated behavior. The precise statement is as follows: : Let V be a semialgebraic variety in ^n defined over the integers. Then V is stably equivalent to the realization space of some oriented matroid. The theorem was discovered by Nikolai Mnëv in his 1986 Ph.D. thesis. Oriented matroids For the purposes of this article, an ''oriented matroid'' of a finite subset S\subset ^n is the list of partitions of S induced by hyperplanes in ^n (each oriented hyperplane partitions S into the points on the "positive side" of the hyperplane, the points on the "negative side" of the hyperplane, and the points that lie on the hyperplane). In part ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Kempe's Universality Theorem
In algebraic geometry, Kempe's universality theorem states that any bounded subset of an algebraic curve may be traced out by the motion of one of the joints in a suitably chosen linkage. It is named for British mathematician Alfred B. Kempe, who in 1876 published his article ''On a General Method of describing Plane Curves of the nth degree by Linkwork,'' which showed that for any arbitrary algebraic plane curve, a linkage can be constructed that draws the curve. However, Kempe's proof was flawed and the first complete proof was provided in 2002 based on his ideas. This theorem has been popularized by describing it as saying, "One can design a linkage which will sign your name!" Kempe recognized that his results demonstrate the existence of a drawing linkage but it would not be practical. He states It is hardly necessary to add, that this method would not be practically useful on account of the complexity of the linkwork employed, a necessary consequence of the perfect gene ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Oriented Matroids
In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, Surface (topology), surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "anticlockwise". A space is orientable if such a consistent definition exists. In this case, there are two possible definitions, and a choice between them is an orientation of the space. Real vector spaces, Euclidean spaces, and spheres are orientable. A space is non-orientable if "clockwise" is changed into "counterclockwise" after running through some loop (topology), loops in it, and coming back to the starting point. This means that a geometric shape, such as , that moves continuously along such a loop is changed into its own mirror image . A Möbius strip is an example of a non-orientable space. Various equivalent formulations of orientability can be given, depending on the desired application and level of generality. Formulations applicable to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Algebraic Geometry
In mathematics, real algebraic geometry is the sub-branch of algebraic geometry studying real algebraic sets, i.e. real-number solutions to algebraic equations with real-number coefficients, and mappings between them (in particular real polynomial mappings). Semialgebraic geometry is the study of semialgebraic sets, i.e. real-number solutions to algebraic inequalities with-real number coefficients, and mappings between them. The most natural mappings between semialgebraic sets are semialgebraic mappings, i.e., mappings whose graphs are semialgebraic sets. Terminology Nowadays the words 'semialgebraic geometry' and 'real algebraic geometry' are used as synonyms, because real algebraic sets cannot be studied seriously without the use of semialgebraic sets. For example, a projection of a real algebraic set along a coordinate axis need not be a real algebraic set, but it is always a semialgebraic set: this is the Tarski–Seidenberg theorem. Related fields are o-minimal theory ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Branko Grünbaum
Branko Grünbaum (; 2 October 1929 – 14 September 2018) was a Croatian-born mathematician of Jewish descentBranko Grünbaum Hrvatska enciklopedija LZMK. and a professor at the in . He received his Ph.D. in 1957 from Hebrew University of Jerusalem. Life< ...
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Convex Polytopes
''Convex Polytopes'' is a graduate-level mathematics textbook about convex polytopes, higher-dimensional generalizations of three-dimensional polyhedron, convex polyhedra. It was written by Branko Grünbaum, with contributions from Victor Klee, Micha Perles, and Geoffrey Colin Shephard, G. C. Shephard, and published in 1967 by John Wiley & Sons. It went out of print in 1970. A second edition, prepared with the assistance of Volker Kaibel, Victor Klee, and Günter M. Ziegler, was published by Springer-Verlag in 2003, as volume 221 of their book series Graduate Texts in Mathematics. ''Convex Polytopes'' was the winner of the 2005 Leroy P. Steele Prize for mathematical exposition, given by the American Mathematical Society. The Basic Library List Committee of the Mathematical Association of America has recommended its inclusion in undergraduate mathematics libraries. Topics The book has 19 chapters. After two chapters introducing background material in linear algebra, topology, and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jürgen Richter-Gebert
Jürgen or Jurgen is a popular masculine given name in Germany, Estonia, Belgium and the Netherlands. Notable people named Jürgen include: A * Jürgen Ahrend (1930–2024), German organ builder *Jürgen Alzen (born 1962), German race car driver * Jürgen Arndt, East German rower * Jürgen Aschoff (1913–1998), German physician and biologist B * Jürgen Barth (born 1947), German engineer and racecar driver * Jürgen Bartsch (1946–1976), German serial killer * Jurgen Van den Broeck (born 1983), Belgian cyclist *Jürgen von Beckerath (1920–2016), German Egyptologist * Jürgen Berghahn (born 1960), German politician * Jürgen Bertow (born 1950), East German rower * Jürgen Blin (1943–2022), West German boxer * Jürgen Bogs (born 1947), German football manager * Jürgen Brähmer (born 1978), German boxer * Jürgen Bräuninger, South African composer and professor * Jürgen Budday (born 1948), German conductor C * Jürgen Cain Külbel (born 1956), German journalist and inves ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Polytopes
''Convex Polytopes'' is a graduate-level mathematics textbook about convex polytopes, higher-dimensional generalizations of three-dimensional polyhedron, convex polyhedra. It was written by Branko Grünbaum, with contributions from Victor Klee, Micha Perles, and Geoffrey Colin Shephard, G. C. Shephard, and published in 1967 by John Wiley & Sons. It went out of print in 1970. A second edition, prepared with the assistance of Volker Kaibel, Victor Klee, and Günter M. Ziegler, was published by Springer-Verlag in 2003, as volume 221 of their book series Graduate Texts in Mathematics. ''Convex Polytopes'' was the winner of the 2005 Leroy P. Steele Prize for mathematical exposition, given by the American Mathematical Society. The Basic Library List Committee of the Mathematical Association of America has recommended its inclusion in undergraduate mathematics libraries. Topics The book has 19 chapters. After two chapters introducing background material in linear algebra, topology, and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Michael Kapovich
Michael Kapovich (also ''Misha Kapovich'', Михаил Эрикович Капович, transcription Mikhail Erikovich Kapovich, born 1963) is a Russian-American mathematician. Kapovich was awarded a doctorate in 1988 at the Sobolev Institute of Mathematics in Novosibirsk with thesis advisor Samuel Leibovich Krushkal and thesis "Плоские конформные структуры на 3-многообразиях" (Flat conformal structures on 3-manifolds, Russian lang. thesis). Kapovich is now a professor at University of California, Davis, where he has been since 2003. His research deals with low-dimensional geometry and topology, Kleinian groups, hyperbolic geometry, geometric group theory, geometric representation theory in Lie groups, , and configuration spaces of arrangements and mechanical linkages. in 2006 in Madrid he was an Invited Speaker at the International Congress of Mathematicians with talk ''Generalized triangle inequalities and their applications''. He i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ravi Vakil
Ravi D. Vakil (born February 22, 1970) is a Canadian-American mathematician working in algebraic geometry. He is the current president of the American Mathematical Society. Education and career Vakil attended high school at Martingrove Collegiate Institute in Etobicoke, Ontario, where he won several mathematical contests and olympiads. After earning a BSc and MSc from the University of Toronto in 1992, he completed a PhD in mathematics at Harvard University in 1997 under Joseph Daniel Harris, Joe Harris. He has since been an instructor at both Princeton University and Massachusetts Institute of Technology, MIT. Since the fall of 2001, he has taught at Stanford University, becoming a full professor in 2007. ''The Rising Sea: Foundations of Algebraic Geometry'', a mathematical textbook about algebraic geometry by Ravi Vakil, will be published in 2025, although drafts have already been available online ever since he began to write in 2010. Contributions Vakil is an algebraic geome ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and as an end to obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics and from evolutionary biology to computer science. Combinatorics is well known for the breadth of the problems it tackles. Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and geometry, as well as in its many application areas. Many combinatorial questions have historically been considered in isolation, giving an ''ad hoc'' solution to a problem arising in some mathematical context. In the later twentieth century, however, powerful and general theoretical methods were developed, making combinatorics into an independent branch of mathematics in its own right. One of the oldest and most accessible parts of combinatorics ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Laurent Lafforgue
Laurent Lafforgue (; born 6 November 1966) is a French mathematician. He has made outstanding contributions to Langlands' program in the fields of number theory and Mathematical analysis, analysis, and in particular proved the Langlands conjectures for the Automorphism#Automorphism group, automorphism group of a function field. The crucial contribution by Lafforgue to solve this question is the construction of compactifications of certain moduli stacks of shtukas. The mathematical proof, proof was the result of more than six years of concentrated efforts. In 2002 at the 24th International Congress of Mathematicians in Beijing, China, he received the Fields Medal together with Vladimir Voevodsky. Biography Laurent Lafforgue has two brothers, Thomas and Vincent Lafforgue, Vincent, both mathematicians. Thomas is now a teacher in a ''classe préparatoire aux grandes écoles'' at Lycée Louis le Grand in Paris and Vincent Lafforgue, Vincent a CNRS directeur de recherches at the Insti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
