# nLab linear differential equation

Contents

### Context

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

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

$(\esh \dashv \flat \dashv \sharp )$

• dR-shape modality$\dashv$ dR-flat modality

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

tangent cohesion

differential cohesion

singular cohesion

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

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

# Contents

## 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.

## References

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