Deformation theory | Wikipedia audio article
Channel: wikipedia tts
Date: 2018-12-28

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Deformation_theory


00:01:10 1 Deformations of complex manifolds
00:02:42 2 Functorial description
00:04:28 2.1 Technical remarks about infinitesimals
00:06:23 2.2 Motivation
00:11:01 2.3 Smooth pre-deformation functors
00:16:30 2.4 Tangent space
00:17:23 3 Applications of deformation theory
00:18:27 3.1 Bend-and-break
00:18:38 3.2 Arithmetic deformations
00:19:31 3.3 Deformations of abelian schemes
00:21:49 3.4 Galois deformations
00:22:18 4 Relationship to string theory
00:23:05 5 See also
00:23:40 6 References



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SUMMARY
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In mathematics, deformation theory is the study of infinitesimal conditions associated with varying a solution P of a problem to slightly different solutions Pε, where ε is a small number, or vector of small quantities. The infinitesimal conditions are therefore the result of applying the approach of differential calculus to solving a problem with constraints. One might  think, in analogy, of a structure that is not completely rigid, and that deforms slightly to accommodate forces applied from the outside; this explains the name.
Some characteristic phenomena are: the derivation of first-order equations by treating the ε quantities as having negligible squares; the possibility of isolated solutions, in that varying a solution may not be possible, or does not bring anything new; and the question of whether the infinitesimal constraints actually 'integrate', so that their solution does provide small variations. In some form these considerations have a history of centuries in mathematics, but also in physics and engineering. For example, in the geometry of numbers a class of results called isolation theorems was recognised, with the topological interpretation of an open orbit (of a group action) around a given solution. Perturbation theory also looks at deformations, in general of operators.