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Atiyah–Singer Index Theorem
In differential geometry, the Atiyah–Singer index theorem, proved by Michael Atiyah and Isadore Singer (1963), states that for an elliptic differential operator on a compact manifold, the analytical index (related to the dimension of the space of solutions) is equal to the topological index (defined in terms of some topological data). It includes many other theorems, such as the Chern–Gauss–Bonnet theorem and Riemann–Roch theorem, as special cases, and has applications to theoretical physics. History The index problem for elliptic differential operators was posed by Israel Gel'fand. He noticed the homotopy invariance of the index, and asked for a formula for it by means of topological invariants. Some of the motivating examples included the Riemann–Roch theorem and its generalization the Hirzebruch–Riemann–Roch theorem, and the Hirzebruch signature theorem. Friedrich Hirzebruch and Armand Borel had proved the integrality of the  genus of a spin manifold, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Geometry
Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, linear algebra and multilinear algebra. The field has its origins in the study of spherical geometry as far back as classical antiquity, antiquity. It also relates to astronomy, the geodesy of the Earth, and later the study of hyperbolic geometry by Nikolai Lobachevsky, Lobachevsky. The simplest examples of smooth spaces are the Differential geometry of curves, plane and space curves and Differential geometry of surfaces, surfaces in the three-dimensional Euclidean space, and the study of these shapes formed the basis for development of modern differential geometry during the 18th and 19th centuries. Since the late 19th century, differential geometry has grown into a field concerned more generally with geometric structures on differentiable ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
 Genus
Â, â ( a- circumflex) is a letter of the Inari Sami, Skolt Sami, Romanian, Vietnamese and Mizo alphabets. This letter also appears in French, Friulian, Frisian, Portuguese, Turkish, Walloon, and Welsh languages as a variant of the letter " a". It is included in some romanization systems for Khmer, Persian, Balinese, Sasak, Russian, and Ukrainian. Berber languages "â" can be used in Berber Latin alphabet to represent . Emilian-Romagnol  is used to represent ːin Emilian dialects, as in Bolognese ''câna'' aːna"cane". Faroese , who translated the Gospel of Matthew into Faroese in 1823, used â to denote a non-syllabic a, as in the following example:  is not used in modern Faroese, however. French , in the French language, is used as the letter with a circumflex accent. It is a remnant of Old French, where the vowel was followed, with some exceptions, by the consonant . For example, the modern form ''bâton'' () comes from the Old French ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heat Equation
In mathematics and physics (more specifically thermodynamics), the heat equation is a parabolic partial differential equation. The theory of the heat equation was first developed by Joseph Fourier in 1822 for the purpose of modeling how a quantity such as heat diffuses through a given region. Since then, the heat equation and its variants have been found to be fundamental in many parts of both pure and applied mathematics. Definition Given an open subset of and a subinterval of , one says that a function is a solution of the heat equation if : \frac = \frac + \cdots + \frac, where denotes a general point of the domain. It is typical to refer to as time and as spatial variables, even in abstract contexts where these phrases fail to have their intuitive meaning. The collection of spatial variables is often referred to simply as . For any given value of , the right-hand side of the equation is the Laplace operator, Laplacian of the function . As such, the heat equation is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vijay Patodi
Vijay Kumar Patodi (12 March 1945 – 21 December 1976) was an Indian mathematician who made fundamental contributions to differential geometry and topology. He was the first mathematician to apply heat equation methods to the proof of the index theorem for elliptic operators. He was a professor at Tata Institute of Fundamental Research, Mumbai (Bombay). Education Patodi was a graduate of Government High School, Guna, Madhya Pradesh. He received his bachelor's degree from Vikram University, Ujjain, his master's degree from the Benaras Hindu University, and his Ph.D. from the University of Bombay under the guidance of M. S. Narasimhan and S. Ramanan at the Tata Institute of Fundamental Research. In the two papers based on his Ph.D. thesis, "Curvature and Eigenforms of the Laplace Operator" (''Journal of Differential Geometry''), and "An Analytical Proof of the Riemann-Roch-Hirzebruch Formula for Kaehler Manifolds" (also ''Journal of Differential Geometry''), Patodi made his ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Raoul Bott
Raoul Bott (September 24, 1923 – December 20, 2005) was a Hungarian-American mathematician known for numerous foundational contributions to geometry in its broad sense. He is best known for his Bott periodicity theorem, the Morse–Bott functions which he used in this context, and the Borel–Bott–Weil theorem. Early life Bott was born in Budapest, Kingdom of Hungary (1920–1946), Hungary, the son of Margit Kovács and Rudolph Bott. His father was of Austrian descent, and his mother was of Hungarian Jewish descent; Bott was raised a Catholic by his mother and stepfather in Bratislava, Czechoslovakia, now the capital of Slovakia. Bott grew up in Czechoslovakia and spent his working life in the United States. His family emigrated to Canada in 1938, and subsequently he served in the Canadian Forces, Canadian Army in Europe during World War II. Career Bott later went to college at McGill University in Montreal, where he studied electrical engineering. He then earned a PhD in math ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
René Thom
René Frédéric Thom (; 2 September 1923 – 25 October 2002) was a French mathematician, who received the Fields Medal in 1958. He made his reputation as a topologist, moving on to aspects of what would be called singularity theory; he became world-famous among the wider academic community and the educated general public for one aspect of this latter interest, his work as the founder of catastrophe theory (later developed by Christopher Zeeman). Life and career René Thom grew up in a modest family in Montbéliard, Doubs and obtained a Baccalauréat in 1940. After the German invasion of France, his family took refuge in Switzerland and then in Lyon. In 1941 he moved to Paris to attend Lycée Saint-Louis and in 1943 he began studying mathematics at École Normale Supérieure, becoming agrégé in 1946. He received his PhD in 1951 from the University of Paris. His thesis, titled ''Espaces fibrés en sphères et carrés de Steenrod'' (''Sphere bundles and Steenrod squares' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Laurent C
Laurent may refer to: *Laurent (name), a French masculine given name and a surname **Saint Laurence (aka: Saint ''Laurent''), the martyr Laurent **Pierre Alphonse Laurent, mathematician **Joseph Jean Pierre Laurent, amateur astronomer, discoverer of minor planet (51) Nemausa * Laurent, South Dakota, a proposed town for the Deaf to be named for Laurent Clerc See also *Laurent series In mathematics, the Laurent series of a complex function f(z) is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansio ..., in mathematics, representation of a complex function ''f(z)'' as a power series which includes terms of negative degree, named for Pierre Alphonse Laurent * Saint-Laurent (other) * Laurence (name), feminine form of "Laurent" * Lawrence (other) {{Disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robion Kirby
Robion Cromwell Kirby (born February 25, 1938) is a Professor of Mathematics at the University of California, Berkeley who specializes in low-dimensional topology. Together with Laurent C. Siebenmann he developed the Kirby–Siebenmann invariant for classifying the piecewise linear structures on a topological manifold. He also proved the fundamental result on the Kirby calculus, a method for describing 3-manifolds and smooth 4-manifolds by surgery on framed links. Along with his significant mathematical contributions, he has over 50 doctoral students and is the editor of an influential problem list. Career He received his Ph.D. from the University of Chicago in 1965, with thesis "Smoothing Locally Flat Imbeddings" written under the direction of . He soon became an assistant professor at UCLA. While there he developed his " torus trick" which enabled him to solve, in dimensions greater than four (with additional joint work with Siebenmann), four of John Milnor's seven ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pontryagin Class
In mathematics, the Pontryagin classes, named after Lev Pontryagin, are certain characteristic classes of real vector bundles. The Pontryagin classes lie in cohomology groups with degrees a multiple of four. Definition Given a real vector bundle E over M, its k-th Pontryagin class p_k(E) is defined as :p_k(E) = p_k(E, \Z) = (-1)^k c_(E\otimes \Complex) \in H^(M, \Z), where: *c_(E\otimes \Complex) denotes the 2k-th Chern class of the complexification E\otimes \Complex = E\oplus iE of E, *H^(M, \Z) is the 4k-cohomology group of M with integer coefficients. The rational Pontryagin class p_k(E, \Q) is defined to be the image of p_k(E) in H^(M, \Q), the 4k-cohomology group of M with rational coefficients. Properties The total Pontryagin class :p(E)=1+p_1(E)+p_2(E)+\cdots\in H^*(M,\Z), is (modulo 2-torsion) multiplicative with respect to Whitney sum of vector bundles, i.e., :2p(E\oplus F)=2p(E)\smile p(F) for two vector bundles E and F over M. In terms of the individual Pon ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sergei Novikov (mathematician)
Sergei Petrovich Novikov ( Russian: Серге́й Петро́вич Но́виков ; 20 March 19386 June 2024) was a Soviet and Russian mathematician, noted for work in both algebraic topology and soliton theory. He became the first Soviet mathematician to receive the Fields Medal in 1970. Biography Novikov was born on 20 March 1938 in Gorky, Soviet Union (now Nizhny Novgorod, Russia). He grew up in a family of talented mathematicians. His father was Pyotr Sergeyevich Novikov, who gave a negative solution to the word problem for groups. His mother, Lyudmila Vsevolodovna Keldysh, and maternal uncle, Mstislav Vsevolodovich Keldysh, were also important mathematicians. Novikov entered Moscow State University in 1955 and graduated in 1960. In 1964, he received the Moscow Mathematical Society Award for young mathematicians and defended a dissertation for the ''Candidate of Science in Physics and Mathematics'' degree (equivalent to the PhD) under Mikhail Postnikov at Moscow ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
K-theory
In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is a cohomology theory known as topological K-theory. In algebra and algebraic geometry, it is referred to as algebraic K-theory. It is also a fundamental tool in the field of operator algebras. It can be seen as the study of certain kinds of invariants of large matrices. K-theory involves the construction of families of ''K''-functors that map from topological spaces or schemes, or to be even more general: any object of a homotopy category to associated rings; these rings reflect some aspects of the structure of the original spaces or schemes. As with functors to groups in algebraic topology, the reason for this functorial mapping is that it is easier to compute some topological properties from the mapped rings than from the original spaces or schemes. Examples of results gleaned from the K-theory approach include the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cobordism
In mathematics, cobordism is a fundamental equivalence relation on the class of compact space, compact manifolds of the same dimension, set up using the concept of the boundary (topology), boundary (French ''wikt:bord#French, bord'', giving ''cobordism'') of a manifold. Two manifolds of the same dimension are ''cobordant'' if their disjoint union is the ''boundary'' of a compact manifold one dimension higher. The boundary of an (n+1)-dimensional manifold W is an n-dimensional manifold \partial W that is closed, i.e., with empty boundary. In general, a closed manifold need not be a boundary: cobordism theory is the study of the difference between all closed manifolds and those that are boundaries. The theory was originally developed by René Thom for smooth manifolds (i.e., differentiable), but there are now also versions for Piecewise linear manifold, piecewise linear and topological manifolds. A ''cobordism'' between manifolds M and N is a compact manifold W whose boundary is th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
