|
Topological Modular Forms
In mathematics, topological modular forms (tmf) is the name of a spectrum that describes a generalized cohomology theory. In concrete terms, for any integer ''n'' there is a topological space \operatorname^, and these spaces are equipped with certain maps between them, so that for any topological space ''X'', one obtains an abelian group structure on the set \operatorname^(X) of homotopy classes of continuous maps from ''X'' to \operatorname^. One feature that distinguishes tmf is the fact that its coefficient ring, \operatorname^(point), is almost the same as the graded ring of holomorphic modular forms with integral cusp expansions. Indeed, these two rings become isomorphic after inverting the primes 2 and 3, but this inversion erases a lot of torsion information in the coefficient ring. The spectrum of topological modular forms is constructed as the global sections of a sheaf of E-infinity ring spectra on the moduli stack of (generalized) elliptic curves. This theory has rela ... [...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] |
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] |
Grothendieck Topology
In category theory, a branch of mathematics, a Grothendieck topology is a structure on a category ''C'' that makes the objects of ''C'' act like the open sets of a topological space. A category together with a choice of Grothendieck topology is called a site. Grothendieck topologies axiomatize the notion of an open cover. Using the notion of covering provided by a Grothendieck topology, it becomes possible to define sheaves on a category and their cohomology. This was first done in algebraic geometry and algebraic number theory by Alexander Grothendieck to define the étale cohomology of a scheme. It has been used to define other cohomology theories since then, such as ℓ-adic cohomology, flat cohomology, and crystalline cohomology. While Grothendieck topologies are most often used to define cohomology theories, they have found other applications as well, such as to John Tate's theory of rigid analytic geometry. There is a natural way to associate a site to an ordinary t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Étale Topology
In 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 ar ..., more specifically in algebra, the adjective étale refers to several closely related concepts: * Étale morphism ** Formally étale morphism * Étale cohomology * Étale topology * Étale fundamental group * Étale group scheme * Étale algebra Other * Étale (mountain) in Savoie and Haute-Savoie, France See also * Étalé space * Etail, or online commerce {{disambig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cohomology Theories
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed as a method of assigning richer algebraic invariants to a space than homology. Some versions of cohomology arise by dualizing the construction of homology. In other words, cochains are functions on the group of chains in homology theory. From its start in topology, this idea became a dominant method in the mathematics of the second half of the twentieth century. From the initial idea of homology as a method of constructing algebraic invariants of topological spaces, the range of applications of homology and cohomology theories has spread throughout geometry and algebra. The terminology tends to hide the fact that cohomology, a contravariant theory, is more natural than homology in many applications. At a basic level, this has to do with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological
Topology (from the Greek words , and ) is the branch of mathematics concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself. A topological space is a set endowed with a structure, called a ''topology'', which allows defining continuous deformation of subspaces, and, more generally, all kinds of continuity. Euclidean spaces, and, more generally, metric spaces are examples of topological spaces, as any distance or metric defines a topology. The deformations that are considered in topology are homeomorphisms and homotopies. A property that is invariant under such deformations is a topological property. The following are basic examples of topological properties: the dimension, which allows distinguishing between a line and a surface; compactness, which allows distinguishing between ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Presheaf
In mathematics, a sheaf (: sheaves) is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could be the ring of continuous functions defined on that open set. Such data are well-behaved in that they can be restricted to smaller open sets, and also the data assigned to an open set are equivalent to all collections of compatible data assigned to collections of smaller open sets covering the original open set (intuitively, every datum is the sum of its constituent data). The field of mathematics that studies sheaves is called sheaf theory. Sheaves are understood conceptually as general and abstract objects. Their precise definition is rather technical. They are specifically defined as sheaves of sets or as sheaves of rings, for example, depending on the type of data assigned to the open sets. There are also maps (or morph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Obstruction Theory
Obstruction may refer to: Places * Obstruction Island, in Washington state * Obstruction Islands, east of New Guinea Medicine * Obstructive jaundice * Obstructive sleep apnea * Airway obstruction, a respiratory problem ** Recurrent airway obstruction * Bowel obstruction, a blockage of the intestines. * Gastric outlet obstruction * Distal intestinal obstruction syndrome * Congenital lacrimal duct obstruction * Bladder outlet obstruction Politics and law * Obstruction of justice, the crime of interfering with law enforcement * Obstructionism, the practice of deliberately delaying or preventing a process or change, especially in politics * Emergency Workers (Obstruction) Act 2006 Science and mathematics * Obstruction set in forbidden graph characterizations, in the study of graph minors in graph theory * Obstruction theory, in mathematics * Propagation path obstruction ** Single Vegetative Obstruction Model Sports * Obstruction (baseball), when a fielder illegally hinders a baser ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Charles Rezk
Charles Waldo Rezk (born 26 January 1969) is an American mathematician, specializing in algebraic topology and category theory. Education and career Rezk matriculated at the University of Pennsylvania in 1987 and graduated there in 1991 with B.A. and M.A. in mathematics. In 1996 he received his PhD from MIT with thesis ''Spaces of Algebra Structures and Cohomology of Operads'' and advisor Michael J. Hopkins. At Northwestern University Rezk was a faculty member from 1996 to 2001. At the University of Illinois he was an assistant professor from 2001 to 2006 and an associate professor from 2006 to 2014, and has been a full professor since 2014. He was at the Institute for Advanced Study in the fall of 1999, the spring of 2000, and the spring of 2001. He held visiting positions at MIT in 2006 and at Berkeley's MSRI in 2014. Since 2015 he has been a member of the editorial board of ''Compositio Mathematica''. Rezk was an invited speaker at the International Congress of Mathematicians ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mark Mahowald
Mark Edward Mahowald (December 1, 1931 – July 20, 2013) was an American mathematician known for work in algebraic topology. Life Mahowald was born in Albany, Minnesota in 1931. He received his Ph.D. from the University of Minnesota in 1955 under the direction of Bernard Russell Gelbaum with a thesis on ''Measure in Groups''. In the sixties, he became professor at Syracuse University and around 1963 he went to Northwestern University in Evanston, Illinois. Work Much of Mahowald's most important works concerns the homotopy groups of spheres, especially using the Adams spectral sequence at the prime 2. He is known for constructing one of the first known infinite families of elements in the stable homotopy groups of spheres by showing that the classes h_1h_j survive the Adams spectral sequence for j\geq 3. In addition, he made extensive computations of the structure of the Adams spectral sequence and the 2-primary stable homotopy groups of spheres up to dimension 64 together with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paul Goerss
Paul may refer to: People * Paul (given name), a given name, including a list of people * Paul (surname), a list of people * Paul the Apostle, an apostle who wrote many of the books of the New Testament * Ray Hildebrand, half of the singing duo Paul & Paula * Paul Stookey, one-third of the folk music trio Peter, Paul and Mary * Billy Paul, stage name of American soul singer Paul Williams (1934–2016) * Vinnie Paul, drummer for American Metal band Pantera * Paul Avril, pseudonym of Édouard-Henri Avril (1849–1928), French painter and commercial artist * Paul, pen name under which Walter Scott wrote ''Paul's letters to his Kinsfolk'' in 1816 * Jean Paul, pen name of Johann Paul Friedrich Richter (1763–1825), German Romantic writer Places *Paul, Cornwall, a village in the civil parish of Penzance, United Kingdom *Paul (civil parish), Cornwall, United Kingdom *Paul, Alabama, United States, an unincorporated community *Paul, Idaho, United States, a city *Paul, Nebraska, United Sta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Haynes Miller
Haynes Robert Miller (born January 29, 1948, in Princeton, New Jersey) is an American mathematician specializing in algebraic topology. Miller completed his undergraduate study at Harvard University and earned his PhD in 1974 under the supervision of John Coleman Moore at Princeton University with thesis ''Some Algebraic Aspects of the Adams–Novikov Spectral Sequence''. After his PhD, he became an assistant professor at Harvard and at Northwestern University, from 1977 at the University of Washington, and from 1984 a professor at the University of Notre Dame. Since 1986 he is a professor at the Massachusetts Institute of Technology (MIT). From 1992 to 1993 he was MIT's Chair of the Committee for Pure Mathematics, since 2004 Chair of the Undergraduate Mathematics Committee, and since 2005 MacVicar Faculty Fellow. His doctoral students at MIT include Brooke Shipley. In 1984 Miller proved the generalized Sullivan conjecture, independently of Jean Lannes and Gunnar Carlsson. In 19 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
