Деформация теориясы: Математикалық өзгеристерді зерттеу

Branch of mathematics
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 a vector of small quantities. The infinitesimal conditions are the result of applying the approach of differential calculus to solving a problem with constraints. The name is an analogy to non rigid structures that deform slightly to accommodate external forces. 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 general form of a deformation is a flat map of complex analytic spaces, schemes, or germs of functions on a space. Grothendieck was the first to find this far reaching generalization for deformations and developed the theory in that context. The general idea is there should exist a universal family such that any deformation can be found as a unique pullback squareIn many cases, this universal family is either a Hilbert scheme or Quot scheme, or a quotient of one of them. For example, in the construction of the moduli of curves, it is constructed as a quotient of the smooth curves in the Hilbert scheme. If the pullback square is not unique, then the family is only versal.

One of the useful and readily computable areas of deformation theory comes from the deformation theory of germs of complex spaces, such as Stein manifolds, complex manifolds, or complex analytic varieties. For a Fano variety of positive dimension Mori showed that there is a rational curve passing through every point. The method of the proof later became known as Mori's bend and break. The rough idea is to start with some curve C through a chosen point and keep deforming it until it breaks into several components. Replacing C by one of the components has the effect of decreasing either the genus or the degree of C. So after several repetitions of the procedure, eventually we'll obtain a curve of genus 0, i. e. a rational curve. The existence and the properties of deformations of C require arguments from deformation theory and a reduction to positive characteristic.

The Serre–Tate theorem asserts, roughly speaking, that the deformations of abelian scheme A is controlled by deformations of the p divisible group consisting of its p power torsion points.

The so called Deligne conjecture arising in the context of algebras (and Hochschild cohomology) stimulated much interest in deformation theory in relation to string theory (roughly speaking, to formalise the idea that a string theory can be regarded as a deformation of a point particle theory). This is now accepted as proved, after some hitches with early announcements. Maxim Kontsevich is among those who have offered a generally accepted proof of this.