Diffiety
In mathematics, a diffiety is a geometrical object introduced by Alexandre Mikhailovich Vinogradov 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 is a coordinate-invariant notion and an equivalence relation.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.
Instead of considering jets of submanifolds as above, it often suffices to define jets of sections of a fibered manifold.
In this case one can describe those submanifolds that are horizontal to the projection as images of sections of the fibered manifold. A jet of sections is then an equivalence class of sections that are tangent up to order at some point. This gives rise to the definition of a Jet Bundle which is a slightly less general construction than a Jet Space. For more information, have a look at the Wikipedia page on Jet bundles.
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. One would like the -th prolongation of an equation of order to be an equation of order, i.e. a submanifold of the jet space.
To achieve this, one first constructs the jet space over -dimensional submanifolds of. As is embedded in, one can always naturally embed into. But since the latter is the space of repeated jets of submanifolds of, one can also always embed into. As a result, when considering both and as subspaces of, their intersection is well-defined. This is used for the definition of the prolongation of.
Note however that such an intersection is not necessarily a manifold again. One therefore usually requires to be nice enough such that at least its first prolongation is indeed a submanifold of.
It can also be shown that this definition still makes sense, even when 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 without the need to consider it inside. To do so, one defines the restriction of the Distribution to a submanifold of as follows.
In this sense, the pair encodes the information about the solutions of the differential equation.
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 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 is its infinite prolongation. 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 is defined as an elementary differential variety, or, for short, as an elementary diffiety.
Note that when considering a differential equation, then one can show that the Cartan distribution is exactly -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 diffiety as an object that locally looks like an elementary diffiety.
Maps that are said to preserve the Cartan distribution are smooth maps which are such that the pushforward at acts as follows:
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 -spectral sequence is a spectral sequence related to the Cartan distribution which Vinogradov invented to calculate certain properties of the formal solution space of a differential equation. One may use diffieties to formulate it.Assume that is a diffiety.
Now define
to be the algebra of differential forms over.
Consider the corresponding de Rham complex:
Its cohomology groups
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 over
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..
Now let be its -th power, i.e. the linear subspace of generated by.
Then one obtains a filtration
and since all ideals are stable, this filtration completely determines a 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 .
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.
- corresponds to action functionals constrained by the PDE and for, the corresponding Euler-Lagrange equation is.
- corresponds to conservation laws for solutions of.
- is interpreted as characteristic classes of bordisms of solutions of.
- There are still many terms awaiting an interpretation.
Similarly to the terms of the spectral sequence, many of its terms can be given a physical interpretation if one considers the diffiety of a classical field theory. For example, one obtains cohomology classes corresponding to action functionals, conserved currents, gauge charges and other important notions in one unifiedly organized scheme.
Since the Wikipedia article on the variational bicomplex is currently quite short, the reader is instead referred to the for more information.
Secondary Calculus
Vinogradov developed a theory, which is known as secondary calculus, 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 very singular
space where these objects cannot be defined in the usual way.
In is analyzed the relationship between Secondary Calculus and the Covariant Phase Space.