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)
For $(X, \pi)$ a Poisson manifold, a submanifold $S \hookrightarrow X$ is called coisotropic if the restriction of the contraction map with the Poisson tensor
to the conormal bundle $N^* S \hookrightarrow T^* S$ factors through the tangent bundle $T S$
Equivalently, $S\hookrightarrow X$ is coisotropic if the subalgebra of $C^\infty(X)$ of functions vanishing on $S$ is closed under the Poisson bracket.
A Poisson manifold induces a Poisson Lie algebroid, which is a symplectic Lie n-algebroid for $n = 1$. Its coisotropic submanifolds correspond to the Lagrangian dg-submanifolds (see there) of this Poisson Lie algebroid.
∞-Chern-Simons theory from binary and non-degenerate invariant polynomial
(adapted from Ševera 00)
Surveys include
The relation to the Poisson sigma-model is discussed in
Characterization in terms of leaves of Lagrangian foliation of the Poisson Lie algebroid is mentioned in
and discussed in more detail in section 7.2 of
Comments on higher algebra aspects are in the slides