higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
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 zero locus of a set of polynomials is an affine algebraic variety; mimicking this situation Alexandre Vinogradov has introduced a geometric object related to systems of nonlinear differential equations, the diffieties. Closely related notions in various flavours of geometry are the crystals of schemes (Grothendieck), D-varieties (Malgrange), D-schemes (Beilinson, Drinfeld) and differential algebras (Ritt, Kolchin).
A. M. Vinogradov, Local symmetries and conservation laws, Acta Appl. Math., Vol. 2, 1984, p. 21, MR85m:58192, doi
A. M. Vinogradov, Symmetries and conservation laws of partial differential equations: basic notions and results, Acta Appl. Math., Vol. 15, 1989, p. 3. MR91b:58282, doi
A. M. Vinogradov, Scalar differential invariants, diffieties and characteristic classes, in: Mechanics, Analysis and Geometry: 200 Years after, 379–414, MR92e:58244
G. Cicogna, G. Gaeta, Lie-point symmetries in bifurcation problems, Annales de l’institut Henri Poincaré (A) Physique théorique, 56 no. 4 (1992), p. 375-414 numdam
wikipedia: diffiety, secondary calculus and cohomological physics
Joseph Krasil'shchik’s webpage (with links to some papers) and wiki list of publications
Joseph Krasil'shchik, Alexander Verbovetsky, Homological methods in equations of mathematical physics, arxiv:math.DG/9808130, 150 p.
Joseph Krasil'shchik, Alexander Verbovetsky, Geometry of jet spaces and integrable systems, J. Geom. Phys. (2010), doi, arXiv:1002.0077
L. Vitagliano, Hamilton-Jacobi diffieties, arxiv/1104.0162