Difference

In mathematics , a difference denotes a geometric object that was introduced by AM Vinogradov in 1984 and plays the same role in the modern theory of partial differential equations as an algebraic variety does in algebraic geometry .

A diffietity is constructed in somewhat more detail as follows: First, one defines jet spaces in which submanifolds can be embedded together with their tangent spaces up to the kth order . The resulting submanifolds in the jet spaces can then be parameterized locally by the coordinates and their derivatives of up to the k-th order. The coordinates defined in this way are therefore naturally used to parameterize a partial differential equation of the kth order, e.g. B. . ${\ displaystyle (x ^ {1}, \ dotsc, x ^ {n}, u ^ {1} (x), \ dotsc, u ^ {m} (x))}$ ${\ displaystyle u_ {j} ^ {i}: = \ partial u ^ {i} (x) / \ partial x ^ {j}}$ ${\ displaystyle u ^ {i}}$ ${\ displaystyle F (x ^ {i}, u ^ {i}, u_ {j} ^ {i}) = 0}$

The submanifold in jet space, which is described locally by these coordinates, is by construction a geometric object and therefore invariant under diffeomorphisms (which is not the case for the local "labels" , since these can change through coordinate transformations ). Since total derivatives of the differential equations lead to submanifolds in jet spaces of higher order, which however represent the same equations, one considers the Limes in jet space of infinite order. This generalizes the concept of an algebraic variety as the solution set of an ideal of an algebraic equation to the solution set of a “differential” ideal of a differential equation. The differential limit of the submanifolds together with a so-called Cartan distribution , which is required for integration, form a difference. Together with diffeomorphisms that maintain the distribution, differences form a category that is also called the category of differential equations and was also introduced by Vinogradov. An introduction to the topic is given, for example, in the book Cohomological analysis of partial differential equations and secondary calculus . ${\ displaystyle F (x ^ {i}, u ^ {i}, u_ {j} ^ {i}) = 0}$

Another way of generalizing ideas about algebraic geometry is through differential algebraic geometry .

Individual evidence

↑ AM Vinogradov: Local symmetries and conservation laws . Ed .: Acta Applicandae Mathematicae. 1984, p. 21-78 , doi : 10.1007 / BF01405491 .
↑ AM Vinogradov: Cohomological analysis of partial differential equations and secondary calculus . Ed .: AMS Bookstore. 2001, ISBN 978-0-8218-2922-6 ( google.com ).

Web links

The Diffiety Institute (no longer active in 2010, but contains useful information)
The Levi-Civita Institute (Successor to the above page with current information on the difference - summer schools.)
Geometry of Differential Equations

[1] AM Vinogradov: Local symmetries and conservation laws . Ed .: Acta Applicandae Mathematicae. 1984, p. 21-78 , doi : 10.1007 / BF01405491 .

[2] AM Vinogradov: Cohomological analysis of partial differential equations and secondary calculus . Ed .: AMS Bookstore. 2001, ISBN 978-0-8218-2922-6 ( google.com ).