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Noncommutative geometry (NCG) is a branch of mathematics concerned with a geometric approach to noncommutative algebras, and with the construction of spaces that are locally presented by noncommutative algebras of functions, possibly in some generalized sense. A noncommutative algebra is an associative algebra in which the multiplication is not commutative, that is, for which does not always equal ; or more generally an algebraic structure in which one of the principal binary operations is not commutative; one also allows additional structures, e.g. topology or norm, to be possibly carried by the noncommutative algebra of functions.
In algebraic geometry, motives is a theory proposed by Alexander Grothendieck in the 1960s to unify the vast array of similarly behaved cohomology theories such as singular cohomology, de Rham cohomology, etale cohomology, and crystalline cohomology. Philosophically, a "motif" is the "cohomology essence" of a variety.
Daniel Gray Quillen was an American mathematician. He is known for being the "prime architect" of higher algebraic K-theory, for which he was awarded the Cole Prize in 1975 and the Fields Medal in 1978.
In mathematics, a Galois module is a G-module, with G being the Galois group of some extension of fields. The term Galois representation is frequently used when the G-module is a vector space over a field or a free module over a ring in representation theory, but can also be used as a synonym for G-module. The study of Galois modules for extensions of local or global fields and their group cohomology is an important tool in number theory.
In commutative algebra, a Gorenstein local ring is a commutative Noetherian local ring R with finite injective dimension as an R-module. There are many equivalent conditions, some of them listed below, often saying that a Gorenstein ring is self-dual in some sense.
Algebraic K-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Geometric, algebraic, and arithmetic objects are assigned objects called K-groups. These are groups in the sense of abstract algebra. They contain detailed information about the original object but are notoriously difficult to compute; for example, an important outstanding problem is to compute the K-groups of the integers.
In mathematics, monstrous moonshine, or moonshine theory, is the unexpected connection between the monster group M and modular functions, in particular, the j function. The initial numerical observation was made by John McKay in 1978, and the phrase was coined by John Conway and Simon P. Norton in 1979.
In mathematics, a comodule or corepresentation is a concept dual to a module. The definition of a comodule over a coalgebra is formed by dualizing the definition of a module over an associative algebra.
In algebraic geometry, local cohomology is an algebraic analogue of relative cohomology. Alexander Grothendieck introduced it in seminars in Harvard in 1961 written up by Hartshorne (1967), and in 1961-2 at IHES written up as SGA2 - Grothendieck (1968), republished as Grothendieck (2005). Given a function defined on an open subset of an algebraic variety, local cohomology measures the obstruction to extending that function to a larger domain. The rational function , for example, is defined only on the complement of on the affine line over a field , and cannot be extended to a function on the entire space. The local cohomology module detects this in the nonvanishing of a cohomology class . In a similar manner, is defined away from the and axes in the affine plane, but cannot be extended to either the complement of the -axis or the complement of the -axis alone ; this obstruction corresponds precisely to a nonzero class in the local cohomology module .
In mathematics, a Drinfeld module is roughly a special kind of module over a ring of functions on a curve over a finite field, generalizing the Carlitz module. Loosely speaking, they provide a function field analogue of complex multiplication theory. A shtuka is a sort of generalization of a Drinfeld module, consisting roughly of a vector bundle over a curve, together with some extra structure identifying a "Frobenius twist" of the bundle with a "modification" of it.
Floyd Leroy Williams is a North American mathematician well known for his work in Lie theory and, most recently, mathematical physics. In addition to Lie theory, his research interests are in homological algebra and the mathematics of quantum mechanics. He received his B.S.(1962) in Mathematics from Lincoln University of Missouri, and later his M.S.(1965) and Ph.D.(1972) from Washington University in St. Louis. Williams was appointed professor of mathematics at the University of Massachusetts Amherst in 1984, and has been professor emeritus since 2005. Williams' accomplishments earned him recognition by Mathematically Gifted & Black as a Black History Month 2019 Honoree.
In category theory, an abstract branch of mathematics, distributive laws between monads are a way to express abstractly that two algebraic structures distribute one over the other.
In mathematics, crystalline cohomology is a Weil cohomology theory for schemes X over a base field k. Its values Hn(X/W) are modules over the ring W of Witt vectors over k. It was introduced by Alexander Grothendieck (1966, 1968) and developed by Pierre Berthelot (1974).
In algebraic geometry, F-crystals are objects introduced by Mazur (1972) that capture some of the structure of crystalline cohomology groups. The letter F stands for Frobenius, indicating that F-crystals have an action of Frobenius on them. F-isocrystals are crystals "up to isogeny".
Hossein Zakeri is an Iranian mathematician. He, along with R. Y. Sharp, are the founders of generalized fractions, a branch in theory of commutative algebra which expands the concept of fractions in commutative rings by introducing the modules of generalized fractions. This topic later found applications in local cohomology, in the monomial conjecture, and other branches of commutative algebra.
Craig Lee Huneke is an American mathematician specializing in commutative algebra. He is a professor at the University of Virginia.
Barbara L. Osofsky is a retired professor of mathematics at Rutgers University. Her research concerns abstract algebra. Osofsky's contributions to mathematics include her characterization of semisimple rings in terms of properties of cyclic modules. Osofsky also established a logical equivalence between the continuum hypothesis and statements about the global dimension of associative rings.
David Kent Harrison was an American mathematician, specializing in algebra, particularly homological algebra and valuation theory.
Victoria Ann Powers is an American mathematician specializing in algebraic geometry and known for her work on positive polynomials and on the mathematics of electoral systems. She is a professor in the department of mathematics at Emory University.
Alexander A. Voronov is a Russian-American mathematician specializing in mathematical physics, algebraic topology, and algebraic geometry. He is currently a Professor of Mathematics at the University of Minnesota and a Visiting Senior Scientist at the Kavli Institute for the Physics and Mathematics of the Universe.
References
- 1 2 "Christel Rotthaus". Michigan State University. Retrieved Mar 11, 2015.
- ↑ Christel Rotthaus at the Mathematics Genealogy Project
- ↑ List of Fellows of the American Mathematical Society
