nLab formally integrable 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)

Variational calculus

variational calculus

Contents

Idea

A partial differential equation is said to be formally integrable (e.g. Goldschmidt 67, def. 7.2) if it is integrable at least over infinitesimal neighbourhoods (aka “formal neighbourhoods”, whence the name).

References

• Hubert Goldschmidt, Integrability criteria for systems of nonlinear partial differential equations, Journal of Differential Geometry 1 (1967) 269–307 (Euclid)

• Maciej Zworski, Numerical linear algebra and solvability of partial differential equations, Communications in Mathematical Physics 229 (2002) 293–307

• Batu Güneysu, Markus Pflaum, The profinite dimensional manifold structure of formal solution spaces of formally integrable PDE’s (arXiv:1308.1005)

A synthetic discussion in terms of differential cohesion is given in

Last revised on February 8, 2019 at 03:47:12. See the history of this page for a list of all contributions to it.