Diffiety

In mathematics, a diffiety ( ) is a geometrical object which plays the same role in the modern theory of partial differential equations that algebraic varieties play for algebraic equations, that is, to encode the space of solutions in a more conceptual way. The term was coined in 1984 by Alexandre Mikhailovich Vinogradov as portmanteau from differential variety.

In mathematics, a diffiety ( ) is a geometrical object which plays the same role in the modern theory of partial differential equations that algebraic varieties play for algebraic equations, that is, to encode the space of solutions in a more conceptual way. The term was coined in 1984 by Alexandre Mikhailovich Vinogradov as portmanteau from differential variety. ==Intuitive definition== In algebraic geometry the main objects of study (varieties) model the space of solutions of a system of algebraic equations (i.e. the zero locus of a set of polynomials), together with all their "algebraic consequences". This means that, applying algebraic operations to this set (e.g. adding those polynomials to each other or multiplying them with any other polynomials) will give rise to the same zero locus. In other words, one can actually consider the zero locus of the algebraic ideal generated by the initial set of polynomials. When dealing with differential equations, apart from applying algebraic operations as above, one has also the option to differentiate the starting equations, obtaining new differential constraints. Therefore, the differential analogue of a variety should be the space of solutions of a system of differential equations, together with all their "differential consequences". Instead of considering the zero locus of an algebraic ideal, one needs therefore to work with a differential ideal. An elementary diffiety will consist therefore of the infinite prolongation \mathcal{E}^\infty of a differential equation \mathcal{E}\subset J^k(E,m) , together with an extra structure provided by a special distribution. Elementary diffieties play the same role in the theory of differential equations as affine algebraic varieties do in the theory of algebraic equations. Accordingly, just like varieties or schemes are composed of irreducible affine varieties or affine schemes, one defines a (non-elementary) diffiety as an object that locally looks like an elementary diffiety. ===Jet spaces of submanifolds=== Let E be an (m+e) -dimensional smooth manifold. Two m -dimensional submanifolds M , M' of E are tangent up to order k at the point p \in M \cap M' \subset E if one can locally describe both submanifolds as zeroes of functions defined in a neighbourhood of p , whose derivatives at p agree up to order k . One can show that being tangent up to order k is a coordinate-invariant notion and an equivalence relation. One says also that M and M' have same k -th order jet at p , and denotes their equivalence class by [M]^k_p or j^k_p M. The k -jet space of k -submanifolds of E , denoted by J^k(E,m) , is defined as the set of all k -jets of m -dimensional submanifolds of E at all points of E : J^k(E,m) := \{ [M]^k_p~|~p \in M,~ \text{dim}(M)=m, M \subset E \ \text{ submanifold} \} As any given jet [M]^k_p is locally determined by the derivatives up to order k of the functions describing M around p , one can use such functions to build local coordinates (x^i,u^j_\sigma) and provide J^k(E,m) with a natural structure of smooth manifold. For instance, for k=1 one recovers just points in E and for k=1 one recovers the Grassmannian of n -dimensional subspaces of TE . More generally, all the projections J^k(E) \to J^{k-1} E are fibre bundles. As a particular case, when E has a structure of fibred manifold over an n -dimensional manifold X , one can consider submanifolds of E given by the graphs of local sections of \pi: E \to X . Then the notion of jet of submanifolds boils down to the standard notion of jet of sections, and the jet bundle J^k(\pi) turns out to be an open and dense subset of J^k (E,m). ===Prolongations of submanifolds=== The k -jet prolongation of a submanifold M \subseteq E is j^k(M):M \rightarrow J^k(E,m), \quad p \mapsto [M]_p^k The map j^k(M) is a smooth embedding and its image M^{k}:= \text{im}(j^k(M)) , called the prolongation of the submanifold M , is a submanifold of J^k (E,m) diffeomorphic to M . ===Cartan distribution on jet spaces=== A space of the form T_\theta(M^{k}) , where M is any submanifold of E whose prolongation contains the point \theta \in J^k (E,m) , is called an R -plane (or jet plane, or Cartan plane) at \theta . The Cartan distribution on the jet space J^k(E,m) is the distribution \mathcal{C} \subseteq T (J^k(E,m)) defined by \mathcal{C}:J^k(E,m)\rightarrow TJ^k(E,m),\qquad \theta\mapsto \mathcal{C}_\theta \subset T_\theta(J^k(E,m)) where \mathcal{C}_{\theta} is the span of all R -planes at \theta \in J^k(E,m) . One can also look at the Cartan distribution of a PDE \mathcal{E}\subset J^k(E,m) more intrinsically, defining \mathcal{C}(\mathcal{E}):=\{ \mathcal{C}_\theta\cap T_\theta(\mathcal{E})~|~\theta\in \mathcal{E}\} In this sense, the pair (\mathcal{E},\mathcal{C}(\mathcal{E})) encodes the information about the solutions of the differential equation \mathcal{E} . Given a differential equation \mathcal{E}\subset J^l(E,m) of order l , its k -th prolongation is defined as \mathcal{E}^k := J^k (\mathcal{E}, m)\cap J^{k+l}(E,m) \subseteq J^{k+l} (E,m) where both J^k (\mathcal{E}, m) and J^{k+l}(E,m) are viewed as embedded submanifolds of J^k (J^l (E, m), m) , so that their intersection is well-defined. However, such an intersection is not necessarily a manifold again, hence \mathcal{E}^k may not be an equation of order k+l . One therefore usually requires \mathcal{E} to be "nice enough" such that at least its first prolongation is indeed a submanifold of J^{k+1}(E, m) . Below we will assume that the PDE is formally integrable, i.e. all prolongations \mathcal{E}^k are smooth manifolds and all projections \mathcal{E}^k \to \mathcal{E}^{k-1} are smooth surjective submersions. Note that a suitable version of Cartan–Kuranishi prolongation theorem guarantees that, under minor regularity assumptions, checking the smoothness of a finite number of prolongations is enough. Then the inverse limit of the sequence \{ \mathcal{E}^k \}_{k \in \mathbb{N}} extends the definition of prolongation to the case when k goes to infinity, and the space \mathcal{E}^\infty has the structure of a profinite-dimensional manifold. ===Definition of a diffiety=== An elementary diffiety is a pair (\mathcal{E}^\infty,\mathcal{C}(\mathcal{E}^\infty)) where \mathcal{E}\subset J^k(E,m) is a k -th order differential equation on some manifold, \mathcal{E}^\infty its infinite prolongation and \mathcal{C}(\mathcal{E}^\infty) its Cartan distribution. Note that, unlike in the finite case, one can show that the Cartan distribution \mathcal{C}(\mathcal{E}^\infty) is m -dimensional and involutive. However, due to the infinite-dimensionality of the ambient manifold, the Frobenius theorem does not hold, therefore \mathcal{C}(\mathcal{E}^\infty) is not integrable A diffiety is a triple (\mathcal{O},\mathcal{F}(\mathcal{O}),\mathcal{C}(\mathcal{O})) , consisting of a (generally infinite-dimensional) manifold \mathcal{O} the algebra of its smooth functions \mathcal{F}(\mathcal{O}) a finite-dimensional distribution \mathcal{C}(\mathcal{O}) , such that (\mathcal{O},\mathcal{F}(\mathcal{O}),\mathcal{C}(\mathcal{O})) is locally of the form (\mathcal{E}^\infty,\mathcal{F}(\mathcal{E}^\infty),\mathcal{C}(\mathcal{E}^\infty)) , where (\mathcal{E}^\infty,\mathcal{C}(\mathcal{E}^\infty)) is an elementary diffiety and \mathcal{F}(\mathcal{E}^\infty) denotes the algebra of smooth functions on \mathcal{E}^\infty . Here locally means a suitable localisation with respect to the Zariski topology corresponding to the algebra \mathcal{F}(\mathcal{O}) . The dimension of \mathcal{C}(\mathcal{O}) is called dimension of the diffiety and its denoted by \mathrm{Dim}(\mathcal{O}) , with a capital D (to distinguish it from the dimension of \mathcal{O} as a manifold). ==Applications== ===Vinogradov sequence=== The Vinogradov \mathcal{C} -spectral sequence (or, for short, Vinogradov sequence) is a spectral sequence associated to a diffiety, which can be used to investigate certain properties of the formal solution space of a differential equation by exploiting its Cartan distribution \mathcal{C} . Given a diffiety (\mathcal{O},\mathcal{F}(\mathcal{O}),\mathcal{C}(\mathcal{O})) , consider the algebra of differential forms over \mathcal{O} \Omega(\mathcal{O}):=\sum_{i\ge 0} \Omega^i(\mathcal{O}) and the corresponding de Rham complex: C^\infty(\mathcal{O}) \longrightarrow \Omega^1(\mathcal{O}) \longrightarrow \Omega^2(\mathcal{O}) \longrightarrow \cdots Its cohomology groups H^i(\mathcal{O}):=\text{ker}(\text{d}_i)/\text{im}(\text{d}_{i-1}) 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 and introduce a more sophisticated sequence. To this end, let \mathcal{C}\Omega(\mathcal{O})=\sum_{i\ge 0}\mathcal{C}\Omega^i(\mathcal{O}) \subseteq \Omega(\mathcal{O}) be the submodule of differential forms over \mathcal{O} whose restriction to the distribution \mathcal{C} vanishes, i.e. \mathcal{C}\Omega^p(\mathcal{O}) := \{ w \in \Omega^p (\mathcal{O}) \mid w(X_1,\cdots,X_p)=0 \quad \forall~X_1,\ldots,X_p\in \mathcal{C}(\mathcal{O}) \}. Note that \mathcal{C}\Omega^i(\mathcal{O}) \subseteq \Omega^i(\mathcal{O}) is actually a differential ideal since it is stable w.r.t. the de Rham differential, i.e. \text{d}(\mathcal{C}\Omega^i(\mathcal{O}))\subset \mathcal{C}\Omega^{i+1}(\mathcal{O}) . Now let \mathcal{C}^k\Omega(\mathcal{O}) be its k -th power, i.e. the linear subspace of \mathcal{C}\Omega generated by w_1 \wedge \cdots \wedge w_k,~w_i\in \mathcal{C}\Omega . Then one obtains a filtration \Omega(\mathcal{O}) \supset \mathcal{C}\Omega(\mathcal{O}) \supset \mathcal{C}^2\Omega(\mathcal{O}) \supset \cdots and since all ideals \mathcal{C}^k\Omega are stable, this filtration completely determines the following spectral sequence: \mathcal{C}E(\mathcal{O})=\{E^{p,q}_r,\text{d}_r^{p,q}\}\qquad\text{where}\qquad E^{p,q}_0 := \frac{\mathcal{C}^p\Omega^{p+q}(\mathcal{O})}{\mathcal{C}^{p+1}\Omega^{p+q}(\mathcal{O})},\qquad\text{and}\qquad E_{r+1}^{p,q} := H(E_{r}^{p,q},d_r^{p,q}). The filtration above is finite in each degree, i.e. for every k \geq 0 \Omega^k(\mathcal{O}) \supset \mathcal{C}^1\Omega^k(\mathcal{O}) \supset \cdots \supset \mathcal{C}^{k+1}\Omega^k(\mathcal{O})= 0, so that the spectral sequence converges to the de Rham cohomology H(\mathcal{O}) of the diffiety. One can therefore analyse the terms of the spectral sequence order by order to recover information on the original PDE. For instance: E_1^{0,n} corresponds to action functionals constrained by the PDE \mathcal{E} . In particular, for \mathcal{L}\in E_1^{0,n} , the corresponding Euler-Lagrange equation is \text{d}_1^{0,n} \mathcal{L}=0 . E_1^{0,n-1} corresponds to conservation laws for solutions of \mathcal{E} . E_2 is interpreted as characteristic classes of bordisms of solutions of \mathcal{E} . Many higher-order terms do not have an interpretation yet. More precisely, consider the horizontal De Rham complex \overline{\Omega}^\bullet(\mathcal{O}) := \Gamma (\wedge^\bullet \mathcal{C(O)}^*) of a diffiety, which can be seen as the leafwise de Rham complex of the involutive distribution \mathcal{C(O)} or, equivalently, the Lie algebroid complex of the Lie algebroid \mathcal{C(O)} . Then the complex \overline{\Omega}^\bullet(\mathcal{O}) becomes naturally a commutative DG algebra together with a suitable differential \overline{d} . Then, possibly tensoring with the normal bundle \mathcal{V} := T\mathcal{O}/\mathcal{C(O)} \to \mathcal{O} , its cohomology is used to define the following "secondary objects": secondary functions are elements of the cohomology \overline{H}^\bullet(\mathcal{O}) = H^\bullet (\overline{\Omega}^\bullet(\mathcal{O}),\overline{d}) , which is naturally a commutative DG algebra (it is actually the first page of the \mathcal{C} -spectral sequence discussed above); secondary vector fields are elements of the cohomology \overline{H}^\bullet(\mathcal{O}, \mathcal{V}) = H^\bullet (\overline{\Omega}^\bullet(\mathcal{O} \otimes \mathcal{V}),\overline{d}) , which is naturally a Lie algebra; moreover, it forms a graded Lie-Rinehart algebra together with \overline{H}^\bullet(\mathcal{O}) ; secondary differential p -forms are elements of the cohomology \overline{H}^\bullet(\mathcal{O}, \wedge^p \mathcal{V}^*) = H^\bullet (\overline{\Omega}^\bullet(\mathcal{O} \otimes \wedge^p \mathcal{V}^*),\overline{d}) , which is naturally a commutative DG algebra. Secondary calculus can also be related to the covariant Phase Space, i.e. the solution space of the Euler-Lagrange equations associated to a Lagrangian field theory. ==See also== Secondary calculus and cohomological physics Partial differential equations on Jet bundles Differential ideal Differential calculus over commutative algebras Another way of generalizing ideas from algebraic geometry is differential algebraic geometry. ==References== ==External links== The Diffiety Institute (frozen since 2010) The Levi-Civita Institute (successor of above site) Geometry of Differential Equations Differential Geometry and PDEs

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