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In mathematics, the Cartan model is a differential graded algebra that computes the equivariant cohomology of a space.
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In mathematics, the Cartan model is a differential graded algebra that computes the equivariant cohomology of a space.
In mathematics, equivariance is a form of symmetry for functions from one space with symmetry to another. A function is said to be an equivariant map when its domain and codomain are acted on by the same symmetry group, and when the function commutes with the action of the group. That is, applying a symmetry transformation and then computing the function produces the same result as computing the function and then applying the transformation.
Daniel Gray "Dan" Quillen was an American mathematician.
In mathematics, the plus construction is a method for simplifying the fundamental group of a space without changing its homology and cohomology groups. It was introduced by Michel Kervaire (1969), and was used by Daniel Quillen to define algebraic K-theory. Given a perfect normal subgroup of the fundamental group of a connected CW complex , attach two-cells along loops in whose images in the fundamental group generate the subgroup. This operation generally changes the homology of the space, but these changes can be reversed by the addition of three-cells.
In mathematics, a bundle gerbe is a geometrical model of certain 1-gerbes with connection, or equivalently of a 2-class in Deligne cohomology.
Mathai Varghese is a mathematician at the University of Adelaide. His most influential contribution to date is the Mathai–Quillen formalism, which he formulated together with Daniel Quillen, and which has since found applications in index theory and topological quantum field theory. He was appointed a full professor in 2006. He was appointed Director of the Institute for Geometry and its Applications in 2009. In 2011, he was elected a Fellow of the Australian Academy of Science. In 2013, he was appointed the Elder Professor of Mathematics at the University of Adelaide, and was elected a Fellow of the Royal Society of South Australia. In 2017, he was awarded an ARC Australian Laureate Fellowship. In 2021, he was awarded the prestigious Hannan Medal and Lecture from the Australian Academy of Science, recognizing an outstanding career in Mathematics.
In mathematics, equivariant cohomology is a cohomology theory from algebraic topology which applies to topological spaces with a group action. It can be viewed as a common generalization of group cohomology and an ordinary cohomology theory. Specifically, the equivariant cohomology ring of a space with action of a topological group is defined as the ordinary cohomology ring with coefficient ring of the homotopy quotient :
In mathematics, twisted K-theory is a variation on K-theory, a mathematical theory from the 1950s that spans algebraic topology, abstract algebra and operator theory.
Graeme Bryce Segal FRS is an Australian mathematician, and professor at the University of Oxford.
In mathematics, elliptic cohomology is a cohomology theory in the sense of algebraic topology. It is related to elliptic curves and modular forms.
The Atiyah–Segal completion theorem is a theorem in mathematics about equivariant K-theory in homotopy theory. Let G be a compact Lie group and let X be a G-CW-complex. The theorem then states that the projection map
In mathematics, the Mathai–Quillen formalism is an approach to topological quantum field theory introduced by Atiyah and Jeffrey (1990), based on the Mathai–Quillen form constructed in Mathai and Quillen (1986).
Michèle Vergne (born August 29, 1943 in L’Isle-Adam, Val d´Oise) is a French mathematician, specializing in analysis and representation theory.
In differential geometry, an equivariant differential form on a manifold M acted upon by a Lie group G is a polynomial map
In mathematics, given an action of a group scheme G on a scheme X over a base scheme S, an equivariant sheafF on X is a sheaf of -modules together with the isomorphism of -modules
In differential geometry, the Kirwan map, introduced by British mathematician Frances Kirwan, is the homomorphism
The Bredon cohomology, introduced by Glen E. Bredon, is a type of equivariant cohomology that is a contravariant functor from the category of G-complex with equivariant homotopy maps to the category of abelian groups together with the connecting homomorphism satisfying some conditions.
In differential geometry, the localization formula states: for an equivariantly closed equivariant differential form on an orbifold M with a torus action and for a sufficient small in the Lie algebra of the torus T,
In algebraic geometry, a GKM variety is a complex algebraic variety equipped with a torus action that meets certain conditions. The concept was introduced by Mark Goresky, Robert Kottwitz, and Robert MacPherson in 1998. The torus action of a GKM variety must be skeletal: both the set of fixed points of the action, and the number of one-dimensional orbits of the action, must be finite. In addition, the action must be equivariantly formal, a condition that can be phrased in terms of the torus' rational cohomology.
In algebraic geometry, the cohomology of a stack is a generalization of étale cohomology. In a sense, it is a theory that is coarser than the Chow group of a stack.
In mathematics, homotopy theory is a systematic study of situations in which maps come with homotopies between them. It originated as a topic in algebraic topology but nowadays is studied as an independent discipline. Besides algebraic topology, the theory has also been used in other areas of mathematics such as algebraic geometry and category theory.
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