Diffiety

In mathematics, a diffiety (/dəˈfəˌt/) is a geometrical object introduced by Alexandre Mikhailovich Vinogradov (see Vinogradov (1984a)) 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

Let be an -dimensional smooth manifold.

Two -dimensional submanifolds of are said to have the same -th order Jet at if they are tangent up to order .

(To be tangent up to order means that if one locally describes the submanifolds as images of sections, then the derivatives of those sections agree up to order .)

and have the same 1-jet while and have the same 3-jet.

One can show that being tangent up to order is a coordinate-invariant notion and an equivalence relation (see Saunders (1989) for instance). Therefore a jet is an equivalence class. We can use jets to define jet spaces.

The Jet Space is defined as the set of all jets of order of -dimensional submanifolds in at all points of , i.e.

One can show that Jet Spaces are naturally endowed with the structure of a smooth manifold (see Saunders (1989) again).

Differential equations

A differential equation is a submanifold of a Jet Space, .

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

The -jet prolongation of a submanifold , is the embedding given by
Furthermore, say that is a prolongation of the submanifold .

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 .

The -th prolongation of a differential equation is defined as

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 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.

An -plane at a point is defined to be a subspace of the tangent space of the form for any submanifold of (whose prolongation contains the point ).
The span of all -planes at a point is denoted by . The map
is called Cartan Distribution (on ).

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.

A generalized solution of a differential equation is defined to be an -dimensional submanifold that fulfills for all .

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.

If , then its Cartan Distribution is defined by

In this sense, the pair encodes the information about the (generalized) 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 (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 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.

If is a -th order differential equation, its elementary diffiety is the pair .

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 (non-elementary) diffiety as an object that locally looks like an elementary diffiety.

Suppose that is a generally infinite dimensional manifold equipped with a smooth function algebra and a finite dimensional distribution . A diffiety is a triple that is locally of the form where is a differential equation, denotes the class of infinitely differentiable functions on and locally means a suitable localization with respect to the Zariski topology corresponding to the algebra .

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 Vinogradov (2001).

Applications

Vinogradov sequence

The Vinogradov -spectral sequence (or, for short, Vinogradov sequence) is a spectral sequence related to the Cartan distribution which Vinogradov invented (see Vinogradov (1978)) 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. (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 (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 Krasilshchik (1999). Here will only be summarized which information is contained in the Vinogradov sequence.

  1. corresponds to action functionals constrained by the PDE and for , the corresponding Euler-Lagrange equation is .
  2. corresponds to conservation laws for solutions of .
  3. is interpreted as characteristic classes of bordisms of solutions of .
  4. There are still many terms awaiting an interpretation.

Secondary Calculus

Vinogradov developed a theory, which is known as secondary calculus (see Vinogradov (1984b), Vinogradov (1998), Vinogradov (2001)), 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 Vitagliano (2014).)

In Vitagliano (2009) 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.

References

  • Vinogradov, A. M. (1978), "A spectral sequence associated with a nonlinear differential equation and algebro-geometric foundations of Lagrangian field theory with constraints", Soviet Math. Dokl., 19: 144–148
  • Vinogradov, A. M. (1984a), "Local symmetries and conservation laws", Acta Applicandae Mathematicae, 2 (1): 21–78, doi:10.1007/BF01405491, MR 0736872
  • Vinogradov, A. M. (1984b), "The C-spectral sequence, Lagrangian formalism and conservation laws I, II", J. Math. Anal. Appl., 100: 1–129, doi:10.1016/0022-247X(84)90071-4
  • Saunders, D. J. (1989). The Geometry of Jet Bundles. London Mathematical Society Lecture Note Series. Cambridge University Press.
  • Vinogradov, A. M. (1998), "Introduction to secondary calculus", in M. Henneaux; I. S. Krasil’shchik; A. M. Vinogradov (eds.), Secondary Calculus and Cohomological Physics, Contemporary Mathematics, 219, American Mathematical Society, pp. 241–272, ISBN 978-0-8218-0828-3
  • Krasilshchik, I. S.; Vinogradov, A. M.; Bocharov, A. V.; Chetverikov, V. N.; Duzhin, S. V.; Khor’kova, N. G.; Samokhin, A. V.; Torkhov, Y. N.; Verbovetsky, A. M. (1999). Symmetries and Conservation Laws for Differential Equations of Mathematical Physics. Translations of Mathematical Monographs. American Mathematical Society.
  • Vinogradov, Aleksandr Mikhaĭlovich (2001), Cohomological analysis of partial differential equations and secondary calculus, AMS Bookstore, ISBN 978-0-8218-2922-6
  • Vitagliano, Luca (2009), "Secondary Calculus and the Covariant Phase Space", Journal of Geometry and Physics, 59 (4): 426–447, arXiv:0809.4164, Bibcode:2009JGP....59..426V, doi:10.1016/j.geomphys.2008.12.001
  • Vitagliano, Luca (2014), "On the Strong Homotopy Lie-Rinehart Algebra of a Foliation", Communications in Contemporary Mathematics, 16 (6): 1450007, arXiv:1204.2467, doi:10.1142/S0219199714500072
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