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This paper gives an exposition of algebraic K-theory, which studies functors $K_{n} : Rings \to Abelian Groups$ , $n$ an integer. Classically $n = 0, 1$ introduced by Bass in the mid 60’s (based on ideas of Grothendieck and others) and $n = 2$ introduced by Milnor [Introduction to algebraic K-theory, Annals of Math. Studies, 72, Princeton University Press, 1971: Zbl 0237.18005]. These functors are defined and applications to topological K-theory (Swan), number theory, topology and geometry (the Wall finiteness obstruction to a CW-complex being finite,...

An introduction to Cartan Geometries

Sharpe, Richard (2002)

Proceedings of the 21st Winter School "Geometry and Physics"

A principal bundle with a Lie group $H$ consists of a manifold $P$ and a free proper smooth $H$ -action $P \times H \to P$ . There is a unique smooth manifold structure on the quotient space $M = P / H$ such that the canonical map $π : P \to M$ is smooth. $M$ is called a base manifold and $H \to P \to M$ stands for the bundle. The most fundamental examples of principal bundles are the homogeneous spaces $H \subset G \to G / H$ , where $H$ is a closed subgroup of $G$ . The pair $(𝔤, 𝔥)$ is a Klein pair. A model geometry consists of a Klein pair $(𝔤, 𝔥)$ and a Lie group $H$ with Lie algebra $𝔥$ . In this...

An introduction to gerbes on orbifolds

Ernesto Lupercio, Bernardo Uribe (2004)

Annales mathématiques Blaise Pascal

This paper is a gentle introduction to some recent results involving the theory of gerbes over orbifolds for topologists, geometers and physicists. We introduce gerbes on manifolds, orbifolds, the Dixmier-Douady class, Beilinson-Deligne orbifold cohomology, Cheeger-Simons orbifold cohomology and string connections.

An introduction to quantum sheaf cohomology

Eric Sharpe (2011)

Annales de l’institut Fourier

In this note we review “quantum sheaf cohomology,” a deformation of sheaf cohomology that arises in a fashion closely akin to (and sometimes generalizing) ordinary quantum cohomology. Quantum sheaf cohomology arises in the study of (0,2) mirror symmetry, which we review. We then review standard topological field theories and the A/2, B/2 models, in which quantum sheaf cohomology arises, and outline basic definitions and computations. We then discuss (2,2) and (0,2) supersymmetric Landau-Ginzburg...

An introduction to spectral and differential geometry in Carnot-Carathéodory spaces

Rumin, Michael (2005)

Proceedings of the 24th Winter School "Geometry and Physics"

An introduction to spherical orbit spaces.

McGowan, Jill, Searle, Catherine (2002)

International Journal of Mathematics and Mathematical Sciences

An introduction to symplectic topology

Claude Viterbo (1991)

Journées équations aux dérivées partielles

An introduction to the completeness of compact semi-riemannian manifolds

Miguel Sánchez (1994/1995)

Séminaire de théorie spectrale et géométrie

An introduction to the Einstein-Vlasov system

Alan Rendall (1997)

Banach Center Publications

An invariant of nonpositively curved contact manifolds

Thilo Kuessner (2011)

Open Mathematics

We define an invariant of contact structures and foliations (on Riemannian manifolds of nonpositive sectional curvature) which is upper semi-continuous with respect to deformations and thus gives an obstruction to the topology of foliations which can be approximated by isotopies of a given contact structure.

An isoembolic pinching theorem.

Christopher B. Croke (1988)

Inventiones mathematicae

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