In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a diffiety () is a
geometrical
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ca ...
object which plays the same role in the modern theory of
partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function.
The function is often thought of as an "unknown" to be sol ...
s that
algebraic varieties
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Mo ...
play for
algebraic equations
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mo ...
, that is, to encode the space of solutions in a more conceptual way. The term was coined in 1984 by
Alexandre Mikhailovich Vinogradov
Alexandre Mikhailovich Vinogradov (russian: Александр Михайлович Виноградов; 18 February 1938 – 20 September 2019) was a Russian and Italian mathematician. He made important contributions to the areas of differenti ...
as
portmanteau
A portmanteau word, or portmanteau (, ) is a blend of words[algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...](_blank)
the main objects of study (
varieties
Variety may refer to:
Arts and entertainment Entertainment formats
* Variety (radio)
* Variety show, in theater and television
Films
* ''Variety'' (1925 film), a German silent film directed by Ewald Andre Dupont
* ''Variety'' (1935 film), ...
) model the space of solutions of a system of algebraic equations (i.e. the zero locus of a set of
polynomials
In mathematics, a polynomial is an expression (mathematics), expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addition, subtrac ...
), 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
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considere ...
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 In the theory of differential forms, a differential ideal ''I'' is an ''algebraic ideal'' in the ring of smooth differential forms on a smooth manifold, in other words a graded ideal in the sense of ring theory, that is further closed under exter ...
.
An elementary diffiety will consist therefore of the ''infinite prolongation''
of a differential equation
, together with an extra structure provided by a special
distribution Distribution may refer to:
Mathematics
*Distribution (mathematics), generalized functions used to formulate solutions of partial differential equations
* Probability distribution, the probability of a particular value or value range of a vari ...
. 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
Variety may refer to:
Arts and entertainment Entertainment formats
* Variety (radio)
* Variety show, in theater and television
Films
* ''Variety'' (1925 film), a German silent film directed by Ewald Andre Dupont
* ''Variety'' (1935 film), ...
or
schemes are composed of irreducible
affine varieties
In algebraic geometry, an affine variety, or affine algebraic variety, over an algebraically closed field is the zero-locus in the affine space of some finite family of polynomials of variables with coefficients in that generate a prime ideal. ...
or
affine schemes
In commutative algebra, the prime spectrum (or simply the spectrum) of a ring ''R'' is the set of all prime ideals of ''R'', and is usually denoted by \operatorname; in algebraic geometry it is simultaneously a topological space equipped with th ...
, one defines a (non-elementary) diffiety as an object that ''locally looks like'' an elementary diffiety.
Formal definition
The formal definition of a diffiety, which relies on the geometric approach to differential equations and their solutions, requires the notions of jets of submanifolds, prolongations, and Cartan distribution, which are recalled below.
Jet spaces of submanifolds
For instance, for
one recovers just points in
and for
one recovers the
Grassmannian
In mathematics, the Grassmannian is a space that parameterizes all -Dimension, dimensional linear subspaces of the -dimensional vector space . For example, the Grassmannian is the space of lines through the origin in , so it is the same as the ...
of
-dimensional subspaces of
. More generally, all the projections
are
fibre bundles.
As a particular case, when
has a structure of
fibred manifold over an
-dimensional manifold
, one can consider submanifolds of
given by the
graphs
Graph may refer to:
Mathematics
*Graph (discrete mathematics), a structure made of vertices and edges
**Graph theory, the study of such graphs and their properties
*Graph (topology), a topological space resembling a graph in the sense of discre ...
of local
sections
Section, Sectioning or Sectioned may refer to:
Arts, entertainment and media
* Section (music), a complete, but not independent, musical idea
* Section (typography), a subdivision, especially of a chapter, in books and documents
** Section sig ...
of
. Then the notion of jet of submanifolds boils down to the standard notion of
jet of sections, and the
jet bundle
In differential topology, the jet bundle is a certain construction that makes a new smooth fiber bundle out of a given smooth fiber bundle. It makes it possible to write differential equations on sections of a fiber bundle in an invariant form. Je ...
turns out to be an
open
Open or OPEN may refer to:
Music
* Open (band), Australian pop/rock band
* The Open (band), English indie rock band
* ''Open'' (Blues Image album), 1969
* ''Open'' (Gotthard album), 1999
* ''Open'' (Cowboy Junkies album), 2001
* ''Open'' (YF ...
and
dense
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematically ...
subset of
.
Prolongations of submanifolds
The
-jet prolongation of a submanifold
is
The map
is a smooth
embedding
In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup.
When some object X is said to be embedded in another object Y, the embedding is gi ...
and its image
, called the prolongation of the submanifold
, is a submanifold of
diffeomorphic to
.
Cartan distribution on jet spaces
A space of the form
, where
is any submanifold of
whose prolongation contains the point
, is called an
-plane (or jet plane, or Cartan plane) at
. The Cartan distribution on the jet space
is the
distribution Distribution may refer to:
Mathematics
*Distribution (mathematics), generalized functions used to formulate solutions of partial differential equations
* Probability distribution, the probability of a particular value or value range of a vari ...
defined by
where
is the span of all
-planes at
.
Differential equations
A differential equation of order
on the manifold
is a submanifold
; a solution is defined to be an
-dimensional submanifold
such that
. When
is a fibred manifold over
, one recovers the notion of
partial differential equations on jet bundles and their solutions, which provide a coordinate-free way to describe the analogous notions of
mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
. While jet bundles are enough to deal with many equations arising in geometry, jet spaces of submanifolds provide a greater generality, used to tackle several PDEs imposed on submanifolds of a given manifold, such as
Lagrangian submanifolds and
minimal surfaces
In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below).
The term "minimal surface" is used because these surfaces originally arose as surfaces that ...
.
As in the jet bundle case, the Cartan distribution is important in the algebro-geometric approach to differential equations because it allows to encode solutions in purely geometric terms. Indeed, a submanifold
is a solution if and only if it is an
integral manifold In mathematics, certain systems of partial differential equations are usefully formulated, from the point of view of their underlying geometric and algebraic structure, in terms of a system of differential forms. The idea is to take advantage of ...
for
, i.e.
for all
.
One can also look at the Cartan distribution of a PDE
more intrinsically, defining
In this sense, the pair
encodes the information about the solutions of the differential equation
.
Prolongations of PDEs
Given a differential equation
of order
, its
-th prolongation is defined as
where both
and
are viewed as embedded submanifolds of
, so that their intersection is well-defined. However, such an intersection is not necessarily a manifold again, hence
may not be an equation of order
. One therefore usually requires
to be "nice enough" such that at least its first prolongation is indeed a submanifold of
.
Below we will assume that the PDE is formally integrable, i.e. all prolongations
are smooth manifolds and all projections
are smooth surjective submersions. Note that a suitable version of
Cartan–Kuranishi prolongation theorem Given an exterior differential system defined on a manifold ''M'', the Cartan–Kuranishi prolongation theorem says that after a finite number of ''prolongations'' the system is either ''in involution'' (admits at least one 'large' integral mani ...
guarantees that, under minor regularity assumptions, checking the smoothness of a finite number of prolongations is enough. Then the
inverse limit
In mathematics, the inverse limit (also called the projective limit) is a construction that allows one to "glue together" several related objects, the precise gluing process being specified by morphisms between the objects. Thus, inverse limits can ...
of the sequence
extends the definition of prolongation to the case when
goes to infinity, and the space
has the structure of a ''profinite-dimensional'' manifold.
Definition of a diffiety
An elementary diffiety is a pair
where
is a
-th order
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
on some manifold,
its infinite prolongation and
its Cartan distribution. Note that, unlike in the finite case, one can show that the Cartan distribution
is
-dimensional and
involutive. However, due to the infinite-dimensionality of the ambient manifold, the
Frobenius theorem does not hold, therefore
is not integrable
A diffiety is a triple
, consisting of
* a (generally infinite-dimensional) manifold
* the algebra of its smooth functions
* a finite-dimensional distribution
,
such that
is locally of the form
, where
is an elementary diffiety and
denotes the algebra of smooth functions on
. Here ''locally'' means a suitable localisation with respect to the
Zariski topology
In algebraic geometry and commutative algebra, the Zariski topology is a topology which is primarily defined by its closed sets. It is very different from topologies which are commonly used in the real or complex analysis; in particular, it is n ...
corresponding to the algebra
.
The dimension of
is called dimension of the diffiety and its denoted by
, with a capital D (to distinguish it from the dimension of
as a manifold).
Morphisms of diffieties
A morphism between two diffieties
and
consists of a smooth map
whose
pushforward
The notion of pushforward in mathematics is "dual" to the notion of pullback, and can mean a number of different but closely related things.
* Pushforward (differential), the differential of a smooth map between manifolds, and the "pushforward" op ...
preserves the Cartan distribution, i.e. such that, for every point
, one has
.
Diffieties together with their morphisms define the
category
Category, plural categories, may refer to:
Philosophy and general uses
* Categorization, categories in cognitive science, information science and generally
*Category of being
* ''Categories'' (Aristotle)
*Category (Kant)
*Categories (Peirce)
* ...
of differential equations.
Applications
Vinogradov sequence
The ''Vinogradov
-spectral sequence'' (or, for short, ''Vinogradov sequence'') is a
spectral sequence
In homological algebra and algebraic topology, a spectral sequence is a means of computing homology groups by taking successive approximations. Spectral sequences are a generalization of exact sequences, and since their introduction by , they hav ...
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
.
Given a diffiety
, consider the algebra of
differential forms
In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...
over
:
and the corresponding
de Rham complex
In mathematics, de Rham cohomology (named after Georges de Rham) is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapte ...
:
:
Its
cohomology groups
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...
contain some structural information about the PDE; however, due to the
Poincaré Lemma In mathematics, especially vector calculus and differential topology, a closed form is a differential form ''α'' whose exterior derivative is zero (), and an exact form is a differential form, ''α'', that is the exterior derivative of another diff ...
, 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
:
be the
submodule
In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a ring. The concept of ''module'' generalizes also the notion of abelian group, since the abelian groups are exactly the mod ...
of differential forms over
whose restriction to the distribution
vanishes, i.e.
:
Note that
is actually a
differential ideal In the theory of differential forms, a differential ideal ''I'' is an ''algebraic ideal'' in the ring of smooth differential forms on a smooth manifold, in other words a graded ideal in the sense of ring theory, that is further closed under exter ...
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
Filtration is a physical separation process that separates solid matter and fluid from a mixture using a ''filter medium'' that has a complex structure through which only the fluid can pass. Solid particles that cannot pass through the filter ...
:
and since all ideals
are stable, this filtration completely determines the following
spectral sequence
In homological algebra and algebraic topology, a spectral sequence is a means of computing homology groups by taking successive approximations. Spectral sequences are a generalization of exact sequences, and since their introduction by , they hav ...
:
:
The filtration above is finite in each degree, i.e. for every
:
so that the spectral sequence converges to the de Rham cohomology
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:
*
corresponds to
action functionals constrained by the PDE
. In particular, for
, the corresponding
Euler-Lagrange equation is
.
*
corresponds to
conservation laws
In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves over time. Exact conservation laws include conservation of energy, conservation of linear momentum, co ...
for solutions of
.
*
is interpreted as
characteristic classes
In mathematics, a characteristic class is a way of associating to each principal bundle of ''X'' a cohomology class of ''X''. The cohomology class measures the extent the bundle is "twisted" and whether it possesses sections. Characteristic classe ...
of
bordisms of solutions of
.
Many higher-order terms do not have an interpretation yet.
Variational bicomplex
As a particular case, starting with a fibred manifold
and its jet bundle
instead of the jet space
, instead of the ''
''-spectral sequence one obtains the slightly less general
variational bicomplex
In mathematics, the Lagrangian theory on fiber bundles is globally formulated in algebraic terms of the variational bicomplex, without appealing to the calculus of variations. For instance, this is the case of classical field theory on fiber b ...
.
More precisely, any
bicomplex determines two spectral sequences: one of the two spectral sequences determined by the variational bicomplex is exactly the Vinogradov ''
''-spectral sequence. However, the variational bicomplex was developed independently from the Vinogradov sequence.
Similarly to the terms of the spectral sequence, many terms of the variational bicomplex can be given a physical interpretation in
classical field theory
A classical field theory is a physical theory that predicts how one or more physical fields interact with matter through field equations, without considering effects of quantization; theories that incorporate quantum mechanics are called quantum ...
: for example, one obtains cohomology classes corresponding to action functionals, conserved currents, gauge charges, etc.
Secondary calculus
Vinogradov developed a theory, known as
secondary calculus, to formalise in cohomological terms the idea of a differential calculus on the space of solutions of a given system of PDEs (i.e. the space of integral manifolds of a given diffiety).
In other words, secondary calculus provides substitutes for functions, vector fields, differential forms, differential operators, etc., on a (generically) very singular space where these objects cannot be defined in the usual (smooth) way on the space of solution. Furthermore, the space of these new objects are naturally endowed with the same algebraic structures of the space of the original objects.
More precisely, consider the horizontal De Rham complex
of a diffiety, which can be seen as the leafwise
de Rham complex
In mathematics, de Rham cohomology (named after Georges de Rham) is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapte ...
of the involutive distribution
or, equivalently, the
Lie algebroid complex of the Lie algebroid
. Then the complex
becomes naturally a commutative
DG algebra
In mathematics, in particular abstract algebra and topology, a differential graded algebra is a graded associative algebra with an added chain complex structure that respects the algebra structure.
__TOC__
Definition
A differential graded a ...
together with a suitable differential
. Then, possibly tensoring with the normal bundle
, its cohomology is used to define the following "secondary objects":
* secondary functions are elements of the cohomology
, which is naturally a commutative DG algebra (it is actually the first page of the ''
''-spectral sequence discussed above);
* secondary vector fields are elements of the cohomology
, which is naturally a Lie algebra; moreover, it forms a graded Lie-Rinehart algebra together with
;
* secondary differential
-forms are elements of the cohomology
, which is naturally a commutative DG algebra.
Secondary calculus can also be related to the covariant
Phase Space
In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usually ...
, i.e. the solution space of the Euler-Lagrange equations associated to a
Lagrangian field theory
Lagrangian may refer to:
Mathematics
* Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier
** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
.
See also
*
Secondary calculus and cohomological physics In mathematics, secondary calculus is a proposed expansion of classical differential calculus on manifolds, to the "space" of solutions of a (nonlinear) partial differential equation. It is a sophisticated theory at the level of jet spaces and e ...
*
Partial differential equations on Jet bundles
*
Differential ideal In the theory of differential forms, a differential ideal ''I'' is an ''algebraic ideal'' in the ring of smooth differential forms on a smooth manifold, in other words a graded ideal in the sense of ring theory, that is further closed under exter ...
*
Differential calculus over commutative algebras In mathematics the differential calculus over commutative algebras is a part of commutative algebra based on the observation that most concepts known from classical differential calculus can be formulated in purely algebraic terms. Instances of this ...
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 EquationsDifferential Geometry and PDEs
{{Manifolds
Homological algebra
Partial differential equations
Differential geometry
Manifolds