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Leonard Gross
Leonard Gross (born February 24, 1931) is an American mathematician and Professor Emeritus of Mathematics at Cornell University. Gross has made fundamental contributions to mathematics and the mathematically rigorous study of quantum field theory. Education and career Leonard Gross graduated from James Madison High School in December 1948. He was awarded an Emil Schweinberg scholarship that enabled him to attend college. He studied at City College of New York for one term and then studied electrical engineering at Cooper Union for two years. He then transferred to the University of Chicago, where he obtained a master's degree in physics and mathematics (1954) and a Ph.D. in mathematics (1958). Gross taught at Yale University and was awarded a National Science Foundation Fellowship in 1959. He joined the faculty of the mathematics department of Cornell University in 1960. Gross was a member of the Institute for Advanced Study in 1959 and in 1983 and has held other visiting pos ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
New York City
New York, often called New York City or NYC, is the most populous city in the United States. With a 2020 population of 8,804,190 distributed over , New York City is also the most densely populated major city in the United States, and is more than twice as populous as second-place Los Angeles. New York City lies at the southern tip of New York State, and constitutes the geographical and demographic center of both the Northeast megalopolis and the New York metropolitan area, the largest metropolitan area in the world by urban landmass. With over 20.1 million people in its metropolitan statistical area and 23.5 million in its combined statistical area as of 2020, New York is one of the world's most populous megacities, and over 58 million people live within of the city. New York City is a global cultural, financial, entertainment, and media center with a significant influence on commerce, health care and life sciences, research, technology, education, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Loop Group
In mathematics, a loop group is a group of loops in a topological group ''G'' with multiplication defined pointwise. Definition In its most general form a loop group is a group of continuous mappings from a manifold to a topological group . More specifically, let , the circle in the complex plane, and let denote the space of continuous maps , i.e. :LG = \, equipped with the compact-open topology. An element of is called a ''loop'' in . Pointwise multiplication of such loops gives the structure of a topological group. Parametrize with , :\gamma:\theta \in S^1 \mapsto \gamma(\theta) \in G, and define multiplication in by :(\gamma_1 \gamma_2)(\theta) \equiv \gamma_1(\theta)\gamma_2(\theta). Associativity follows from associativity in . The inverse is given by :\gamma^:\gamma^(\theta) \equiv \gamma(\theta)^, and the identity by :e:\theta \mapsto e \in G. The space is called the free loop group on . A loop group is any subgroup of the free loop group . Examples An im ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs. The society is one of the four parts of the Joint Policy Board for Mathematics and a member of the Conference Board of the Mathematical Sciences. History The AMS was founded in 1888 as the New York Mathematical Society, the brainchild of Thomas Fiske, who was impressed by the London Mathematical Society on a visit to England. John Howard Van Amringe was the first president and Fiske became secretary. The society soon decided to publish a journal, but ran into some resistance, due to concerns about competing with the American Journal of Mathematics. The result was the '' Bulletin of the American Mathematical Society'', with Fiske as editor-in-chief. The de facto journal, as intended, was influential i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Academy Of Arts And Sciences
The American Academy of Arts and Sciences (abbreviation: AAA&S) is one of the oldest learned societies in the United States. It was founded in 1780 during the American Revolution by John Adams, John Hancock, James Bowdoin, Andrew Oliver, and other Founding Fathers of the United States. It is headquartered in Cambridge, Massachusetts. Membership in the academy is achieved through a thorough petition, review, and election process. The academy's quarterly journal, ''Dædalus'', is published by MIT Press on behalf of the academy. The academy also conducts multidisciplinary public policy research. History The Academy was established by the Massachusetts legislature on May 4, 1780, charted in order "to cultivate every art and science which may tend to advance the interest, honor, dignity, and happiness of a free, independent, and virtuous people." The sixty-two incorporating fellows represented varying interests and high standing in the political, professional, and commercial s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Guggenheim Fellow
Guggenheim Fellowships are grants that have been awarded annually since by the John Simon Guggenheim Memorial Foundation to those "who have demonstrated exceptional capacity for productive scholarship or exceptional creative ability in the arts." Each year, the foundation issues awards in each of two separate competitions: * One open to citizens and permanent residents of the United States and Canada. * The other to citizens and permanent residents of Latin America and the Caribbean. The Latin America and Caribbean competition is currently suspended "while we examine the workings and efficacy of the program. The U.S. and Canadian competition is unaffected by this suspension." The performing arts are excluded, although composers, film directors, and choreographers are eligible. The fellowships are not open to students, only to "advanced professionals in mid-career" such as published authors. The fellows may spend the money as they see fit, as the purpose is to give fellows " ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grigori Perelman
Grigori Yakovlevich Perelman ( rus, links=no, Григорий Яковлевич Перельман, p=ɡrʲɪˈɡorʲɪj ˈjakəvlʲɪvʲɪtɕ pʲɪrʲɪlʲˈman, a=Ru-Grigori Yakovlevich Perelman.oga; born 13 June 1966) is a Russian mathematician who is known for his contributions to the fields of geometric analysis, Riemannian geometry, and geometric topology. He is widely regarded as one of the greatest living mathematicians. In the 1990s, partly in collaboration with Yuri Burago, Mikhael Gromov, and Anton Petrunin, he made contributions to the study of Alexandrov spaces. In 1994, he proved the soul conjecture in Riemannian geometry, which had been an open problem for the previous 20 years. In 2002 and 2003, he developed new techniques in the analysis of Ricci flow, and proved the Poincaré conjecture and Thurston's geometrization conjecture, the former of which had been a famous open problem in mathematics for the past century. The full details of Perelman's work wer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poincaré Conjecture
In the mathematical field of geometric topology, the Poincaré conjecture (, , ) is a theorem about the characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dimensional space. Originally conjectured by Henri Poincaré in 1904, the Grigori Perelman's theorem concerns spaces that locally look like ordinary three-dimensional space but which are finite in extent. Poincaré hypothesized that if such a space has the additional property that each loop in the space can be continuously tightened to a point, then it is necessarily a three-dimensional sphere. Attempts to resolve the conjecture drove much progress in the field of geometric topology during the 20th century. The Perelman's proof built upon Richard S. Hamilton's ideas of using the Ricci flow to solve the problem. By developing a number of breakthrough new techniques and results in the theory of Ricci flow, Grigori Perelman was able to prove the Conjecture, and more than just t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fermion
In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks and leptons and all composite particles made of an odd number of these, such as all baryons and many atoms and nuclei. Fermions differ from bosons, which obey Bose–Einstein statistics. Some fermions are elementary particles (such as electrons), and some are composite particles (such as protons). For example, according to the spin-statistics theorem in relativistic quantum field theory, particles with integer spin are bosons. In contrast, particles with half-integer spin are fermions. In addition to the spin characteristic, fermions have another specific property: they possess conserved baryon or lepton quantum numbers. Therefore, what is usually referred to as the spin-statistics relation is, in fact, a spin statistics-quantum nu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boson
In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0,1,2 ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have odd half-integer spin (,, ...). Every observed subatomic particle is either a boson or a fermion. Bosons are named after physicist Satyendra Nath Bose. Some bosons are elementary particles and occupy a special role in particle physics unlike that of fermions, which are sometimes described as the constituents of "ordinary matter". Some elementary bosons (for example, gluons) act as force carriers, which give rise to forces between other particles, while one (the Higgs boson) gives rise to the phenomenon of mass. Other bosons, such as mesons, are composite particles made up of smaller constituents. Outside the realm of particle physics, superfluidity arises because composite bosons (bose particles), such as low temperature helium-4 atoms, follow Bose–Ein ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Constructive Quantum Field Theory
In mathematical physics, constructive quantum field theory is the field devoted to showing that quantum field theory can be defined in terms of precise mathematical structures. This demonstration requires new mathematics, in a sense analogous to classical real analysis, putting calculus on a mathematically rigorous foundation. Weak, strong, and electromagnetic forces of nature are believed to have their natural description in terms of quantum fields. Attempts to put quantum field theory on a basis of completely defined concepts have involved most branches of mathematics, including functional analysis, differential equations, probability theory, representation theory, geometry, and topology. It is known that a ''quantum field'' is inherently hard to handle using conventional mathematical techniques like explicit estimates. This is because a quantum field has the general nature of an '' operator-valued distribution'', a type of object from mathematical analysis. The existence th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logarithmic Sobolev Inequalities
In mathematics, logarithmic Sobolev inequalities are a class of inequalities involving the norm of a function ''f'', its logarithm, and its gradient \nabla f . These inequalities were discovered and named by Leonard Gross, who established them in dimension-independent form, in the context of constructive quantum field theory. Similar results were discovered by other mathematicians before and many variations on such inequalities are known. Gross proved the inequality: : \int_\big, f(x)\big, ^2 \log\big, f(x)\big, \,d\nu(x) \leq \int_\big, \nabla f(x)\big, ^2 \,d\nu(x) +\, f\, _2^2\log \, f\, _2, where \, f\, _2 is the L^2(\nu)-norm of f, with \nu being standard Gaussian measure In mathematics, Gaussian measure is a Borel measure on finite-dimensional Euclidean space R''n'', closely related to the normal distribution in statistics. There is also a generalization to infinite-dimensional spaces. Gaussian measures are nam ... on \mathbb^n. Unlike classical Sobolev in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abstract Wiener Space
The concept of an abstract Wiener space is a mathematical construction developed by Leonard Gross to understand the structure of Gaussian measures on infinite-dimensional spaces. The construction emphasizes the fundamental role played by the Cameron–Martin space. The classical Wiener space is the prototypical example. The structure theorem for Gaussian measures states that all Gaussian measures can be represented by the abstract Wiener space construction. Motivation Let H be a real Hilbert space, assumed to be infinite dimensional and separable. In the physics literature, one frequently encounters integrals of the form :\frac\int_H f(v) e^ Dv, where Z is supposed to be a normalization constant and where Dv is supposed to be the non-existent Lebesgue measure on H. Such integrals arise, notably, in the context of the Euclidean path-integral formulation of quantum field theory. At a mathematical level, such an integral cannot be interpreted as integration against a measure ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
