Diffiety

In mathematics, a diffiety () is a geometrical object introduced by Alexandre Mikhailovich Vinogradov (see ) playing the same role in the modern theory of partial differential equations as algebraic varieties play for algebraic equations.
Definition.
To define a diffiety, we need to take a geometric approach to the description of differential equations and their solutions. This requires the notions of jet spaces, prolongation, and Cartan distribution which will be introduced below. The reader familiar with those notions can directly proceed to the definition.
Jet Spaces.
One can show that "being tangent up to order formula_1" is a coordinate-invariant notion and an equivalence relation (see for instance).
Therefore a jet is an equivalence class. 
We can use jets to define jet spaces.
One can show that Jet Spaces are naturally endowed with the structure of a smooth manifold (see again).
Differential equations.
If one defines solutions as below, then this geometric definition of PDEs in local coordinates gives rise to expressions that are usually used to define PDEs and their solutions in mathematical analysis.
Prolongation.
Furthermore, one can define prolongations of equations, i.e. of submanifolds of Jet spaces.
To this end, consider a differential equation formula_2. One would like the formula_1-th prolongation of an equation of order formula_4 to be an equation of order formula_5, i.e. a submanifold of the jet space formula_6.
To achieve this, one first constructs the jet space formula_7 over formula_8-dimensional submanifolds of formula_9. As formula_9 is embedded in formula_11, one can always naturally embed formula_12 into formula_13. But since the latter is the space of repeated jets of submanifolds of formula_14, one can also always embed formula_6 into formula_16. As a result, when considering both formula_6 and formula_12 as subspaces of formula_16, their intersection is well-defined. This is used for the definition of the prolongation of formula_9.
Note however that such an intersection is not necessarily a manifold again (i.e. does not always exist in the category of smooth manifolds). One therefore usually requires formula_9 to be nice enough such that at least its first prolongation is indeed a submanifold of formula_22.
It can also be shown that this definition still makes sense, even when formula_1 goes to infinity.
Cartan distribution.
Note that below a distribution is not understood in the sense of generalized functions but is considered to be a subbundle of the tangent bundle as is usually done when considering distributions in differential geometry. 
The Cartan distribution is important in the algebro-geometric approach to differential equations because it allows to define generalized solutions of differential equations in purely geometric terms.
One can also look at the Cartan Distribution of a submanifold of formula_24 without the need to consider it inside formula_24. To do so, one defines the restriction of the Distribution to a submanifold of formula_24 as follows.
In this sense, the pair formula_27 encodes the information about the (generalized) solutions of the differential equation formula_9.
Definition of a diffiety.
In Algebraic Geometry the main objects of study are varieties which include all "algebraic consequences" of a system of algebraic equations. For example, if one considers the zero locus of a set of polynomials, then applying algebraic operations to this set (like adding those polynomials to each other or multiplying them with any other polynomial) will give rise to the same zero locus, i.e. one can actually consider the zero locus of the algebraic ideal of the initial set of polynomials.
Now in the case of differential equations, apart from applying algebraic operations, one additionally has the option to differentiate. Therefore, the differential analogue of a variety should be like a "differential ideal" and should include all "differential consequences". 
The natural object that does include the differential consequences of an equation formula_9 is its infinite prolongation formula_30. In general, it can be infinite dimensional. Additionally, one would like to pay attention to the geometric structure of the Cartan distribution defined above. Therefore, the pair formula_31 is defined as an elementary differential variety, or, for short, as an elementary diffiety. 
Note that when considering a differential equation formula_32, then one can show that the Cartan distribution formula_33 is exactly formula_8-dimensional unlike in the case of finitely many prolongations.
Elementary diffieties are geometric objects that play the same role in the theory
of partial differential equations as affine algebraic varieties do in the theory of algebraic equations. 
Just like varieties or schemes are composed of irreducible affine varieties or affine schemes, one can also define a (non-elementary) diffiety as an object that "locally looks like" an elementary diffiety. 
Maps that are said to "preserve the Cartan distribution" are smooth maps formula_35 which are such that the pushforward formula_36 at formula_37 acts as follows:
formula_38
Diffieties together with maps that preserve the Cartan distribution are the objects and Morphisms of the "Category of differential equations" defined by Vinogradov. 
A thorough introduction to the topic is given in .
Applications.
Vinogradov sequence.
The "Vinogradov formula_39-spectral sequence" (or, for short, "Vinogradov sequence") is a spectral sequence related to the Cartan distribution formula_39 which Vinogradov invented (see ) to calculate certain properties of the formal solution space of a differential equation. One may use diffieties to formulate it.
Assume that formula_41 is a diffiety.
Now define 
to be the algebra of differential forms over formula_43. Consider the corresponding de Rham complex:
Its cohomology groups formula_45 contain some structural information about the PDE. However, due to the Poincaré Lemma, they all vanish locally. In order to extract much more and even local information, one thus needs to take the Cartan distribution into account. This is what the Vinogradov sequence will facilitate. To this end, let 
be the submodule of differential forms formula_47 over formula_43 whose restriction to the distribution vanishes. This means
It is actually a so-called differential ideal since it is stable w.r.t. to the de Rham differential, i.e. formula_50.
Now let formula_51 be its formula_1-th power, i.e. the linear subspace of formula_53 generated by formula_54. Then one obtains a filtration
and since all ideals formula_56 are stable, this filtration completely determines a spectral sequence. (For more information on how spectral sequences work, see spectral sequence.) We denote this sequence by
The filtration above is finite in each degree, that means
If the filtration is finite in this sense, then the spectral sequence converges to the de Rham cohomology formula_59 (of the diffiety).
Therefore, one can now analyse the terms of the spectral sequence order by order. This is done for example in chapter 5 of . Here will only be summarized which information is contained in the Vinogradov sequence.
Secondary Calculus.
Vinogradov developed a theory, which is known as secondary calculus (see , , ), formalizing in cohomological terms the idea of a differential calculus on the space of solutions of a given system of PDEs, or, which is roughly the same, the space of integral manifolds of a given diffiety. In other words, secondary calculus provides substitutes for vector fields, differential forms, differential operators, etc., on a (generically) very singular space where these objects cannot be defined in the usual (smooth) way. (This summary was taken from the introduction of .)
In is analyzed the relationship between Secondary Calculus and the Covariant Phase Space (which is the solution space of the Euler-Lagrange equations associated to a Lagrangian field theory).
See also.
Another way of generalizing ideas from algebraic geometry is differential algebraic geometry.