A decomposition of a space E into a family of identical subspaces (fibers). The space of subspaces is called the base space. Fiber bundles arise naturally in many physical situations. The theory of fiber bundles has been applied to gauge theory in physics, and there is a lively interaction centered on ideas related to fiber bundles and gauge theories in which significant results have been contributed in both areas. The development of the mathematical theory of fiber bundles was begun in the 1930s and has numerous applications within mathematics.
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Graph theory
A branch of mathematics that belongs partly to combinatorial analysis and partly to topology. Its applications occur (sometimes under other names) in electrical network theory, operations research, organic chemistry, theoretical physics, and statistical mechanics, and in sociological and behavioral research. Both in pure mathematical inquiry and in applications, a graph is customarily depicted as a topological configuration of points and lines, but usually is studied with combinatorial methods. See also: Combinatorial theory; Topology
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Lie group
A topological group with only countably many connected components whose identity component is open and is an analytic group. An analytic group or connected Lie group is a topological group with the additional structure of a smooth manifold such that multiplication and inversion are smooth. Many groups that arise naturally as groups of symmetries of physical or mathematical systems are Lie groups. The study of Lie groups has applications to analytic function theory, differential equations, differential geometry, Fourier analysis, algebraic number theory, algebraic geometry, quantum mechanics, relativity, and elementary particle theory. See also: Group theory; Manifold (mathematics); Topology
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Topology
The branch of mathematics that studies the qualitative properties of spaces, as opposed to the more delicate and refined geometric or analytic properties. While there are earlier results that belong to the field, the beginning of the subject as a separate branch of mathematics dates to the work done 1895–1904 by Henri Poincaré. The ideas and results of topology have a central place in mathematics, with connections to almost all the other areas of the subject.