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)
$(\esh \dashv \flat \dashv \sharp )$
dR-shape modality$\dashv$ dR-flat modality
$\esh_{dR} \dashv \flat_{dR}$
(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
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
Picard-Vessiot theory is the Galois theory of homogeneous linear ordinary differential equations. A version for homogeneous linear difference equations has also been developed. Alternatively, terms differential Galois theory and difference Galois theory have also been used.
Tannakian approach has been developed in
In the setup of differential abelian tensor categories over differential rings,
Galois theory for differential equations is one of the examples in
Difference equations are viewed from topos perspective (with sporadic recourse to Galois theory) in
…Hopf algebraic approach simplifies and generalizes the theory. Among other things the generalization gives a Picard-Vessiot theory for positive characteristic, for fields with not necessarily commuting derivations, and even for fields with (a set of) higher derivations.
Last revised on January 22, 2021 at 20:59:15. See the history of this page for a list of all contributions to it.