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created differential equation just to link to Lawvere's two articles on that.
Peering through the Lawverian prose, say on page 10 of Toposes of laws of motion I seem to see the shadow of a theory of integration in smooth toposes.
The extension along the inclusion of an infinitesimal object into another object considered there is of the same general nature as the extensins along the inclusion of infnitesimal into all paths that I consider at integration of oo-Lie algebroid valued differential forms.
I mean, no big deal, just something I noticed.
added now some substance to differential equation:
the formulation of homogeneous first order ODEs in terms of just diagrams in a smooth topos.
One would think that this should be explicit in the articles by Lawvere cited, but apparently it is only implicit.
the puny section at synthetic differential geometry that used to be called "Related entries" I have expanded to a section constructions in synthetic differential geometry.
added to differential equation a brief stubby paragraph with a remark on differential equations as sub-tangent Lie algebroids and sub-Jet-D-Schemes.
Interesting. Are the differential equation which have solutions only for finite time included ? It seems that the solution is required for all $t$.
Well, it’s all solutions over the given parameter space.
So I think what you are looking for is the solution space itself regarded as sheaf on the category of parameter spaces:
$Sol(F) : \mathcal{A} \mapsto \{x \in A | F(t, \partial^i_t x) = 0\}$for any D-module $\mathcal{A}$.
People say the phrase that a D-module does not tell a differential equation but an invariant way of talking of its space of solutions. Is this fitting this ?
Yes, I think that’s what is meant.
The analogy to keep in mind is the relation of a (system of) ordinary equations and the corresponding variety. The former is a coordinate-description, the latter is the intrinsic object.
This is nice idea, much explored by Vinogradov, that it is a generalization of the classical algebraic geometry, but ODE and PDE experts would not quite agree. For the same differential equation one can consider various types of problems, initial value, boundary problems, look for weak solutions in various spaces etc. So in a sense this is like having an algebraic equation defining a variety over different rings, simultaneously. Probably somehow one can go in this direction here as well…
Today on the arXiv
We develop a new connection between Differential Algebra and Geometric Invariant Theory, based on an anti-equivalence of categories between solution algebras associated to a linear differential equation (i.e. differential algebras generated by finitely many polynomials in a fundamental set of solutions), and affine quasi-homogeneous varieties (over the constant field) for the differential Galois group of the equation. Solution algebras can be associated to any connection over a smooth affine variety. The spectrum of a solution algebra is an algebraic fiber space over the base variety, with quasi-homogeneous fiber. We also discuss the relevance of this result in Transcendental Number Theory.
So in a sense this is like having an algebraic equation defining a variety over different rings, simultaneously.
Of course, algebraic geometers (since at least Weil) do such things, but with various formal devices about the field of definition.
I have created a stub in my lab partial differential equation (zoranskoda) which just gives hint on how the subject is wide (even in my, nonexpert, and narrow vision). On the other hand, in the main $n$Lab, partial differential equation redirects to differential equation. I spent quite much time thinking about this in last few weeks, and somehow think that this is not the optimal choice yet in fact. I think that the creator of the entry (I suppose Urs) had a quite reasonable idea that PDE is general differential equation, while ODE is a rather special kind, so a differential equation should be about the general case. There are several objections however to this point of view (I am writing this to provoke further thinking and diversification of grand scheme of differential equation related entries in $n$Lab):
As a rule PDE people do not include ODE into their subject; this is, of course, not the logical division, but methodological, so $n$Lab may to some extent (as focusing on logical and architectural structure of mathematics) ignore it, sacrificing some useful methodological divisions.
I also think that the term differential equation includes also many generalizations which are not considered PDE-s. For example, fractional differential equations in the sense of utilitizing fractional derivatives, stochastic differential equations (Ito and Stratonovich; though stochastic PDE is also a used term, I do not know how general), formal differential equations, including over other fields, noncommutative formal differential equations, differential equations with operator coefficients etc. also think that the partial derivatives are mainly about the finite dimensional situation, and various infinite dimensional concepts like variational derivatives are not really included. For $n$Lab it is important to have in mind second quantized differential equations from QFT, where in addition to infinite-dimensionality one has identical particles, i.e. the extension of operators to multiparticle Hilbert spaces; problems with singularity of fields etc.
The beautiful POV on differential equations in the entry currently s interesting and should be generalized. I mean, the definition is about the solutions in smooth terms, quite general from a point of view of a space considered, but rather meager from the point of view of (generalized) function space involved in a solution. Thus, maybe it should be a small paragraph with more details referred to a future separate entry on differential equation in cohesive topos or alike. I know that topos people like Marta Bunge considered certain spaces of distributions which have something in common with distributions in the sense of Sobolev and Schwartz; maybe one can consider something like those to define weak solutions in cohesive topos ? This would be very interesting. I also remind of Sullivan’s ideas that derived geometry should be useful in fluid mechanics quoted in homological algebra in the finite element method: Dennis Sullivan, Algebra, topology and algebraic topology of 3D ideal fluids, arxiv/1010.2721 (has some ideas on need of a version of higher geometry, hence of homological methods in particular, at the very end of the article)
Difference equations do not properly belong to the differential equations, but I suspect that they do, when the subject is considered in appropriate generalized, noncommutative, sense. For example, doing D-modules on quantum groups leads to certain difference equations.
For a related mind teaser let me mention something related. I have asked in MathOverflow a question on new frameworks in PDE-s and for about a couple of weeks, no answer, nor even a comment (though quite many upvotes for the question)
http://mathoverflow.net/questions/129631/recent-fundamental-new-directions-in-pdes
Isn’t “an algebraic equation defining a variety over different rings simultaneously” what people nowadays call a ’scheme’?
Yes, if understood as a functor. I do not know of an analogous site to $Aff$ such that $M$-points for a $D$-scheme would give a functor and such that different cases of $M$ correspond to something what are in special standard cases spaces of weak solutions as distributions, as elements of Sobolev spaces, hyperfunctions and alike. I am not saying that it is impossible to find a fairly good site for this, containing the information enabling a dozen of most important examples of functional spaces of solutions to be defined this (or similar) way, but so far it has not been done.
added pointer to:
Also divided the References into “General” and “Via jet bundles and D-modules” and took the liberty of adding to the latter. pointer to Khavkine-Schreiber and to Wellen (Cherubini).
added these pointers:
Cliff P. Burgess, Primer on Partial Differential Equations for Physicists (1990) [pdf]
Bijan Kumar Bagchi, Partial Differential Equations for Mathematical Physicists, Routledge (2020) [ISBN:9780367227029]
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