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?
Roughly, “integrating” a partial differential equation means to find a solution for it, usually understood with special properties, such as having prescribed boundary data (initial data), and/or having “closed form”, such as expression by quadratures (“integrable systems”) and/or having prescribed domain (e.g. over infinitesimal neighbourhoods in formal integrability). A PDE is “integrable” in a prescribed sense if it admits solutions in this prescribed sense.
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
Created on October 10, 2017 at 00:28:07. See the history of this page for a list of all contributions to it.