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)
homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
A differential equation (ordinary or partial) is called linear if the linear combination of any of its solutions is still a solution, hence if its space of solutions is a vector space.
Equivalently this means that the differential operator that corresponds to the differential equation is a linear operator.
Linear differential equations may be analyzed via harmonic analysis by applying Fourier transform to decompose solutions as superpositions of plane wave “harmonics” (e.g. Hörmander 90).
In physics a field theory whose equations of motion is a linear partial differential equation is called a free field theory.
Where a general (possibly non-linear) differential equation is equivalently an object in the slice category over the de Rham shape of the space of its free variables, a linear differential equation is more specifically a linear object in this slice. In the context of algebraic geometry these are the D-modules.
…
See also
Last revised on May 23, 2022 at 06:34:13. See the history of this page for a list of all contributions to it.