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)
The concept of diffiety (Vinogradov 81) reflects the concept of partial differential equation (generally non-linear) in analogy to how the concept of algebraic variety reflects that of polynomial equation:
A diffiety is the solution-locus $\mathcal{E}\hookrightarrow J^\infty_\Sigma(E)$ of a partial differential equation $D \Phi = 0$ regarded as an ordinary equation $\tilde D = 0$ on the jet bundle $J^\infty_\Sigma(E)$ of some bundle $E \overset{fb}{\to} \Sigma$.
Here $\Sigma$ is the space of free variables of the PDE, $E \overset{fb}{\to} \Sigma$ is the bundle of dependent variables, and a differential operator
on the space of smooth sections of $fb$ is represented by a bundle morphism
out of the jet bundle via jet prolongation $j^\infty_\Sigma \colon \Gamma_\Sigma(E) \to \Gamma_\Sigma(J^\infty_\Sigma(E))$ as $D \Phi = \tilde D (j^\infty_\Sigma(\Phi))$:
$\,$
For instance in Lagrangian field theory the bundle in question is a field bundle $E \overset{fb}{\to} \Sigma$, the partial differential equation is the Euler-Lagrange equation $\delta_{EL}\mathbf{L} = 0$, and its diffiety solution locus $\mathcal{E}$ inside the jet bundle $J^\infty_\Sigma(E)$ is called the shell of the field theory.
In Marvan 86 it was observed that Vinogradov’s formally integrable diffieties are equivalently the coalgebras over the jet comonad acting on locally pro-manifold-bundles (over a base space $\Sigma$ of free variables). This statement generalizes to the synthetic differential geometry of the Cahiers topos $\mathbf{H}$ (Khavkine-Schreiber 17), where the jet comonad is realized as the comonad corresponding to base change along the de Rham shape projection $\Sigma \overset{\eta_\Sigma}{\longrightarrow} \Im \Sigma$. By comonadic descent this implies that over formally smooth base spaces $\Sigma$ formally integrable diffieties are equivalently the bundles over the de Rham shape $\Im \Sigma$:
(Khavkine-Schreiber 17, thorem 3.52, theorem 3.60)
This makes manifest how diffieties are the analog in differential geometry of concepts in algebraic geometry: For $\Sigma$ a suitable scheme then a quasicoherent module over its de Rham shape $\Im \Sigma$ (“crystal”) is called a D-module and represents an algebraic linear partial differential equation, while a relative scheme over $\Im \Sigma$ is called a D-scheme and represents a general algebraic partial differential equation. See also at D-geometry for more on this.
Alexandre Vinogradov, Geometry of nonlinear differential equations, Journal of Soviet Mathematics 17 (1981) 1624–1649 (doi:10.1007/BF01084594)
Alexandre Vinogradov, Local symmetries and conservation laws, Acta Appl. Math., Vol. 2, 1984, p. 21 (MR85m:58192, doi)
Alexandre Vinogradov, Symmetries and conservation laws of partial differential equations: basic notions and results, Acta Appl. Math., Vol. 15, 1989, p. 3. MR91b:58282, doi
Alexandre 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
L. Vitagliano, Hamilton-Jacobi diffieties, arxiv/1104.0162
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
See also
Diffieties as coalgebras over the jet comonad are discussed in
Michal Marvan, A note on the category of partial differential equations, in Differential geometry and its applications, Proceedings of the Conference August 24-30, 1986, Brno (pdf)
Michal Marvan, thesis, 1989 (pdf, web)
Michal Marvan, On the horizontal cohomology with general coefficients, 1989 (web announcement, web archive)
Michal Marvan, section 1.1 of On Zero-Curvature Representations of Partial Differential Equations, (1993) (web)
Igor Khavkine, Urs Schreiber, Synthetic geometry of differential equations: I. Jets and comonad structure (arXiv:1701.06238)
XXI Summer Diffiety School School on Geometry of PDEs, July 19 - 31, 2018