synthetic differential geometry
Introductions
from point-set topology to differentiable manifolds
geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry
Differentials
Tangency
The magic algebraic facts
Theorems
Axiomatics
(shape modality $\dashv$ flat modality $\dashv$ sharp modality)
$(ʃ \dashv \flat \dashv \sharp )$
dR-shape modality$\dashv$ dR-flat modality
$ʃ_{dR} \dashv \flat_{dR}$
(reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)
$(\Re \dashv \Im \dashv \&)$
fermionic modality$\dashv$ bosonic modality $\dashv$ rheonomy modality
$(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$
Models
Models for Smooth Infinitesimal Analysis
smooth algebra ($C^\infty$-ring)
differential equations, variational calculus
Euler-Lagrange equation, de Donder-Weyl formalism?,
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
principle of extremal action, Euler-Lagrange equations, de Donder-Weyl formalism?
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).
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.