nLab formally integrable 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

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

infinitesimal cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \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

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 08:47:12. See the history of this page for a list of all contributions to it.