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
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
A smooth function $q$ on a cotangent bundle (e.g. the symbol of a differential operator) is of order $m$ (and type $1,0$, denoted $q \in S^m = S^m_{1,0}$), for $m \in \mathbb{N}$, if on each coordinate chart $((x^i), (k_i))$ we have that for every compact subset $K$ of the base space and all multi-indices $\alpha$ and $\beta$, there is a real number $C_{\alpha, \beta,K } \in \mathbb{R}$ such that the absolute value of the partial derivatives of $q$ is bounded by
for all $x \in K$ and all cotangent vectors $k$ to $x$.
A Fourier integral operator $Q$ is of symbol class $L^m = L^m_{1,0}$ if
it is of the form
its principal symbol $q$ is of order $m$, in the above sense.
(Hörmander 71, def. 1.1.1 and first sentence of section 2.1)
The wave operator/Klein-Gordon operator on Minkowski spacetime is of class $L^2$, according to def. 1.
Last revised on November 23, 2017 at 10:53:48. See the history of this page for a list of all contributions to it.