nLab linear differential equation

Contents

Context

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

differentiation, chain rule
differentiable function
infinitesimal space, infinitesimally thickened point, amazing right adjoint

V-manifolds

differentiable manifold, coordinate chart, atlas
smooth manifold, smooth structure, exotic smooth structure
analytic manifold, complex manifold
formal smooth manifold, derived smooth manifold

smooth space

diffeological space, Frölicher space
manifold structure of mapping spaces

Tangency

tangent bundle, frame bundle
- vector field, multivector field, tangent Lie algebroid;
- differential forms, de Rham complex, Dolbeault complex
local diffeomorphism, formally étale morphism
submersion, formally smooth morphism,
immersion, formally unramified morphism,
de Rham space, crystal
infinitesimal disk bundle

The magic algebraic facts

embedding of smooth manifolds into formal duals of R-algebras
smooth Serre-Swan theorem
derivations of smooth functions are vector fields

Theorems

Hadamard lemma
Borel's theorem
Boman's theorem
Whitney extension theorem
Steenrod-Wockel approximation theorem
Whitney embedding theorem
Poincare lemma
Stokes theorem
de Rham theorem
Hochschild-Kostant-Rosenberg theorem
differential cohomology hexagon

Axiomatics

Kock-Lawvere axiom

cohesion

(shape modality $\dashv$ flat modality $\dashv$ sharp modality)

$(\esh \dashv \flat \dashv \sharp )$
dR-shape modality $\dashv$ dR-flat modality

$\esh_{dR} \dashv \flat_{dR}$

infinitesimal cohesion

classical modality

tangent cohesion

differential cohomology diagram

differential cohesion

(reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)

$(\Re \dashv \Im \dashv \&)$

graded differential cohesion

fermionic modality $\dashv$ bosonic modality $\dashv$ rheonomy modality

$(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$

singular cohesion

\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{&#233;tale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }

Models

Models for Smooth Infinitesimal Analysis
- smooth algebra ( $C^\infty$ -ring)
- smooth locus
- Fermat theory
Cahiers topos
smooth ∞-groupoid
formal smooth ∞-groupoid
super formal smooth ∞-groupoid

Lie theory, ∞-Lie theory

Lie algebra, Lie n-algebra, L-∞ algebra
Lie group, Lie 2-group, smooth ∞-group

differential equations, variational calculus

D-geometry, D-module
jet bundle
variational bicomplex, Euler-Lagrange complex
Euler-Lagrange equation, de Donder-Weyl formalism,
phase space

Chern-Weil theory, ∞-Chern-Weil theory

connection on a bundle, connection on an ∞-bundle
differential cohomology
parallel transport, higher parallel transport, fiber integration in differential cohomology
holonomy, higher holonomy
gauge theory, higher gauge theory
Wilson line, Wilson surface

Cartan geometry (super, higher)

Klein geometry, (higher)
G-structure, torsion of a G-structure
Euclidean geometry, hyperbolic geometry, elliptic geometry
(pseudo-)Riemannian geometry
complex geometry
symplectic geometry
conformal geometry

Linear algebra

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…

models: topological, simplicial, localic, …

Idea

A differential equation (ordinary or partial) is called linear if the linear combination of any of its solutions is still a solution, hence if its space of solutions is a vector space.

Equivalently this means that the differential operator that corresponds to the differential equation is a linear operator.

Linear differential equations may be analyzed via harmonic analysis by applying Fourier transform to decompose solutions as superpositions of plane wave “harmonics” (e.g. Hörmander 90).

In physics a field theory whose equations of motion is a linear partial differential equation is called a free field theory.

Where a general (possibly non-linear) differential equation is equivalently an object in the slice category over the de Rham shape of the space of its free variables, a linear differential equation is more specifically a linear object in this slice. In the context of algebraic geometry these are the D-modules.

Examples

wave equation
Dirac equation
Klein-Gordon equation
Schrödinger equation
Tomonaga-Schwinger equation
…

Related concepts

superposition
D-module

References

Lars Hörmander, The analysis of linear partial differential operators, vol. I, Springer 1983, 1990