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
Models
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 of a partial differential equation regarded as an ordinary equation on the jet bundle of some bundle .
Here is the space of free variables of the PDE, is the bundle of dependent variables, and a differential operator
on the space of smooth sections of is represented by a bundle morphism
out of the jet bundle via jet prolongation as :
For instance in Lagrangian field theory the bundle in question is a field bundle , the partial differential equation is the Euler-Lagrange equation , and its diffiety solution locus inside the jet bundle 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 of free variables). This statement generalizes to the synthetic differential geometry of the Cahiers topos (Khavkine-Schreiber 17), where the jet comonad is realized as the comonad corresponding to base change along the de Rham shape projection . By comonadic descent this implies that over formally smooth base spaces formally integrable diffieties are equivalently the bundles over the de Rham shape :
(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 a suitable scheme then a quasicoherent module over its de Rham shape (“crystal”) is called a D-module and represents an algebraic linear partial differential equation, while a relative scheme over 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
Last revised on December 11, 2017 at 11:26:30. See the history of this page for a list of all contributions to it.