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)
(properly supported peudo-differential operator)
A pseudo-differential operator $Q$ on a manifold $X$ is called properly supported if for each compact subset $K \subset X$ there exists a compact subset $K' \subset X$ such that for $u$ a distribution with support in $K$ it follows that the derivative of distributions $Q u$ has support in $K'$
and such that
(Hörmander 85 (18.1.21) recalled e.g. in Radzikowski 96. p. 8,9)
(differential operators are properly supported pseudo-differential operators)
Every ordinary differential operator $D$, regarded as a pseudo-differential operator, is properly supported (def. ), since differential operators do not increase the support of the functions they act on:
Johann Duistermaat, Lars Hörmander, Fourier integral operators II, Acta Mathematica 128, 183-269, 1972 (Euclid)
Lars Hörmander, The analysis of partial differential operators III, Springer 1985
Marek Radzikowski, Micro-local approach to the Hadamard condition in quantum field theory on curved space-time, Commun. Math. Phys. 179 (1996), 529–553 (Euclid)
Last revised on November 23, 2017 at 12:49:27. See the history of this page for a list of all contributions to it.