Diffiety

In mathematics a diffiety, is a geometrical object introduced by Vinogradov (1984) playing the same role in the modern theory of partial differential equations as algebraic varieties play for algebraic equations.

In a bit more detail, a diffiety is the following: First, one defines Jet Bundles in which one can embed submanifolds that have attached to every point kth order tangent spaces (e.g. the first order is the usual tangent space at that point, the infinite order is the manifold itself at that point). These submanifolds of k-order Jets can then locally be parameterised by coordinates ${\displaystyle (x^{1},...,x^{n},u^{1}(x),...,u^{m}(x))}$ together with derivatives ${\displaystyle u_{j}^{i}:=\partial u^{i}(x)/\partial x^{j}}$ of the ${\displaystyle u^{i}}$ up to kth order. Thus these coordinates can naturally serve to parameterise solutions of a kth order PDE, say ${\displaystyle F(x^{i},u^{i},u_{j}^{i})=0}$.

The jet-submanifold that is locally described by these coordinates itself is by construction a geometric object and thus diffeomorphism invariant (which is not the case for one of its corresponding labels ${\displaystyle F(x^{i},u^{i},u_{j}^{i})=0}$ which might change its form upon a change of variables).
As differentiations of the PDE lead to higher order Jet spaces that represent the same PDE, one obtains a whole chain of Jets that represents one equation. This chain can then be studied with methods of cohomology. This generalises the concept of an algebraic variety as (the ideal of a) solution set of an algebraic equation to the (ideal of a) solution set of a PDE called diffiety. These diffieties are equipped with a contact structure needed for integration. The diffieties together with maps that preserve this contact structure are the objects and Morphisms of the Category of PDEs defined by Vinogradov. A thorough introduction to the topic is given in Vinogradov (2001).

Another way of generalizing ideas from algebraic geometry is differential algebraic geometry.