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D-geometry and Riemann-Hilbert problem. In order to make more visible one of the principal directions, where the series of entries which I am writing these days is heading to.
In connection to efforts related to D-geometry I rewrote great part of the entry analysis, to make it compatible both with that effort as with some of the main focuses of nlab. Urs, would you be so kind to read it through ?
Edit: I erased the query and wrote an extended new paragraph on "generalizations" at the end of distribution.
I rewrote great part of the entry analysis, to make it compatible both with that effort as with some of the main focuses of nlab. Urs, would you be so kind to read it through ?
I looked at it. This is an impressive collection of commented links.
I see that you are worried about certain aspects being underrepresented at the nLab. Certainly I know what you mean. Even in the subjects where we have a lot of activity, I feel we are only scatching the surface. Other topics are completely left in the dark, so far, that's true.
What can we do? We have only finite resources in time and energy. I necessarily type into the lab mostly only stuff that I need for myself, because there is not time to do more (I feel I already spend more time with typing stuff into the nLab apart from what I need for myself than is good for me). The idea of the Lab is that if everyone types in just the stuff he or she needs, then eventually with enough people participating this will still give a good comprehensive accound of many topics. Eventually.
So I suppose one thing we should do more is advertize more the activity on the Lab, to attract further people.
Given the past experience, I am thinking it would be good if we have a regular post series to the blog "THIS WEEK'S FINDS ON THE nLAB" or the like. Every week a brief summary of the latest activity. I suppose that might help incrementally get more and more people interested.
This is a good idea to have. Though a week might be a short time to be completely regular we could have it most of the weeks running...
However I am still less interested in random outside response as to the response of the people which are already close to us.
The idea of the Lab is that if everyone types in just the stuff he or she needs, then eventually with enough people participating this will still give a good comprehensive accound of many topics.
That is far too optimistic. And that would result only in learners writing entries on known stuff and not people who already know it. Of course for new research this is about what is expected, but we needs expositions of known stuff as well.
However I am still less interested in random outside response as to the response of the people which are already close to us.
But we also need more people. Currently there is not even half a dozen of regular contributors here
a week might be a short time to be completely regular
We could have "This Month's finds on the nLab".
But we also need more people. Currently there is not even half a dozen of regular contributors here
But there are some people who are close to us researchwise...or regularly talk at cafe and not at nlab. Many people say they will start using it at some point and they don't. I think the thematics is less the problem for these people than being convinced that it may be useful to them. The thematics is problem for other people, especially for physicists including mathematical physicists. They get scared with seeing infinity categories everywhere. That is why I started entries on classical physics (half a year ago), on various other parts of mathematics, redirect entries (e.g. geometry, analysis...) to make more connection to conventional textbooks and wider area to be familiar to more people.
But this last excursion into writing stubs on D-geometry is out of conviction that to do physics, one has to have contentful bridge to the toolbox of differential equations which are the main way how the basic equations of theoretical physics come about; and geometry has developed very fine tools in this area, with large progress in last 50 years (pseudodifferential operators, microlocal analysis, D-modules, perverse sheaves, works of Deligne (on the study of ODEs corresponding to regular connections), Hoermander and so on). Also more specifically the heat kernel kinds of expansion like in the work of Leinster and Willerton, point in this direction as well as the study of moduli spaces and Donaldson-Thomas invariants in works of Birdgeland, Kontsevich, Thomas, Soibelman and others with sophisticated lagrangian geometry. in geometry of quantum groups it is kind of mantra that lagrangian submanifolds are like "quantum points" and they have role in other approaches to the quantization. It seems to be most apparent in the D-module approach. There are other concrete motivations which I could list.
algebraic analysis. What do you think of it ?
algebraic analysis. What do you think of it ?
I think the POV that differential equations are to be thought of in terms of D-modules is most noteworthy. yes. Since we have a pretty good idea of what D-modules are in a very general oo-topos theoretic setup, this also provides a cool oo-categorification of the notion of differential equations, yes. This is worth a subsection at nPOV.
It seems you are not getting what the scale of the algebraic analysis program is, the introduction of D-modules is a SMALL part of the story, and they involve e.g. only the linear problems. Algebraic analysis includes the sheaf theoretic approach to distribution theory via hyperfunctions, the microlocalization concepts, the reduction of more complicated, incuding nonlinear, equations to the study of simpler on related spaces like X x X, resolutions and so on, the usage of canonical derived direct/proper direct/inverse image functors and so on...It takes much more to make the analysis canonical and Sato thinks that it is a historical accident that the foundaions of analysis are not made more canonical...Sato was very unhappy with the Schwarz approach to distributions via duals to Banach spaces and wanted a more canonical sheaf-theoretic approach, that is why hyperfunctions.
I gave D-geometry a brief Idea-sentence and two references.
(Thought we had that long ago, but just discovered that the entry was in a rather orphaned state.)
The idea section you put is nice and interesting and also a bit discuttable.
The term $\mathcal{D}$-geometry refers to a synthetic formulation of differential geometry with a focus on the geometry of de Rham spaces. A quasicoherent sheaf of modules over a de Rham space is called a D-module, short for module over the algebra of differential operators, which is where the term “D-geometry” derives from.
Let me see if I understand. You put “synthetic” because one uses systematically infinitesimal objects like the dual numbers, infinitesimal neighborhoods and alike. Is this correct ? Unimportant remark is that these were introduced by Grothendieck in algebraic geometry before synthetic differential geometry was put forward, but it is OK as this was the motivation of synthetic differential geometry; and most of the current advanced work in D-geometry are in the algebraic geometry language. De Rham space is of course just a restatement of this and it is not inherent in D-geometry to must work with de Rham space.
Now the discuttable part. All what is mentioned is OK for introducing D-modules and alike linear objects. These linear objects indeed do come from synthetic constructions with infinitesimal objects and neighborhoods. The restatement of D-modules is via crystals of quasicoherent sheaves. This all enriches usual geometry with linear differential operators and jet spaces. However, by D-geometry one means more (at least in the most advanced schools like Beilinson-Drinfeld): not only equipping the usual schemes and manifolds with their jet spaces and infinitesimal neighborhoods, but vastly generalizing the schemes and manifolds to D-schemes, i.e. crystals of affine schemes (rather than of quasicoherent sheaves) and similarly, diffieties. This bring nonlinear aspect which is not present or studied in synthetic differential geometry of Kock, Lawvere etc. It is rather a kind of noncommutative algebraic geometry arising from studying solutions of nonlinear differential equations. Thus it is not a reformulation of a geometry of manifolds and schemes, but a generalization to diffieties and D-schemes. This can not be obtained by passing to the de Rham space of some commutative variety.
What do you think about this ?
Gelfand introduced in early 1970s a toolbox of what he called formal geometry, where one studies infinitedimensional jet spaces, and related constructions like formal manifold in the sense of homological vector field, geometry related to cohomology of Lie algebras of formal vector fields and so on. Some ideas in noncommutative context are extended by Kontsevich in his work on deformation theory as well as earlier work (pdf) on formal noncommutative symplectic geometry. All this is closely related to the business of D-geometry, so I started a stub formal geometry, but recommend a discussion at MathOverflow (one of the article there puts the Lurie’s seminar on crystals as a main reference to emphasis the connection to D-geometry): MO: formal geometry.
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