Deformation theory

In mathematics, deformation theory is the study of infinitesimal conditions associated with varying a solution P of a problem to slightly different solutions P_ε, 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.

The most salient deformation theory in mathematics has been that of complex manifolds and algebraic varieties. This was put on a firm basis by foundational work of Kunihiko Kodaira and D. C. Spencer, after deformation techniques had received a great deal of more tentative application in the Italian school of algebraic geometry. One expects, intuitively, that deformation theory of the first order should equate the Zariski tangent space with a moduli space. The phenomena turn out to be rather subtle, though, in the general case.

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