Stefan Kebekus - Project VD

Stream: Lean Together 2026

[Markus Himmel]

Discussion thread for the talk.

[Johan Commelin]

@Stefan Kebekus Would you mind sharing your slides? I'm interested in brainstorming further about differential operators.

[Sébastien Gouëzel]

I think we will want many flavors of differential operators. If you're doing analysis, you want them to act pointwise, on functions with a low degree of smoothness that might even not be fixed throughout the discussion, so the sheaf point of view is not viable there. But if you want to play with D-modules, you will need to switch to a more algebraic framework.

[Johan Commelin]

Hmmmzz, so even something like docs#TopCat.LocalPredicate is not really useful, because it's too sheafy?
(I'm also wondering whether it should take unbundled topological spaces as input.)

[Johan Commelin]

@Sébastien Gouëzel If the degree of smoothness is not fixed, does that immediately rule out sheaves?

[Sébastien Gouëzel]

A comment on the difficulty to define the field of meromorphic functions. This is very much related to the definition of the $L^p$ space in analysis, which is a space of equivalence classes of functions if you want to do functional analysis, but where you think of the elements as genuine functions when you write $f (x)$ (i.e., all the time). This has been a difficulty in the Carleson project, where you really want to use both points of view. In the end, what has worked best is to use the functions as true functions, and only project to equivalence classes when really necessary.

[Sébastien Gouëzel]

To be honest, I'm not really into sheaves, so don't listen to me too much here. For me, an issue with this point of view is that if you look at the Laplacian on C^2 functions, and the Laplacian on C^3 functions, they act on different sheaves so they're not directly related.

[Stefan Kebekus]

@Johan Commelin Please find my slides here:

• https://nextcloud.cplx.vm.uni-freiburg.de/index.php/s/Y2RsGdN7FYpFgSG

[Stefan Kebekus]

Sébastien Gouëzel schrieb:

A comment on the difficulty to define the field of meromorphic functions. This is very much related to the definition of the $L^p$ space in analysis, which is a space of equivalence classes of functions if you want to do functional analysis, but where you think of the elements as genuine functions when you write $f (x)$ (i.e., all the time). This has been a difficulty in the Carleson project, where you really want to use both points of view. In the end, what has worked best is to use the functions as true functions, and only project to equivalence classes when really necessary.

I agree. Eventually, we will probably have two objects, "meromorphic function, the function" and "meromorphic function, the field element". The task is then to devise an API that makes shifting hence and forth between the notions as painless as possible.

[Stefan Kebekus]

Sébastien Gouëzel schrieb:

To be honest, I'm not really into sheaves, so don't listen to me too much here. For me, an issue with this point of view is that if you look at the Laplacian on C^2 functions, and the Laplacian on C^3 functions, they act on different sheaves so they're not directly related.

I am a great fan of sheaves. C^3 functions are C^2, so that eventually we need a way to state compatibility. Also, all notions of differential operators I came across

• are local in nature
• are linear over a fixed ring (constants, functions coming from the base of a fibration, …)
• satisfy the iterated Leibniz rule whenever there is enough regularity to even state the rule.
  Even if we end up with different flavors of differential operators, I would expect them to have enough in common to warrant a common API that encapsulates these basic properties.

@Sébastien Gouëzel Please disagree. I will be more than happy to discuss.

[Sébastien Gouëzel]

My background in algebraic geometry is essentially inexistent, so please take what I say with a grain of salt. Still, I see at least two issues with the algebraic, sheafy, point of view.

First, in analysis one needs quite quickly to get out of the realm of differential operators, because they are too rigid (notably when you want to construct some kind of inverses). Instead one uses pseudo-differential operators, which are much more flexible while sharing a lot of the nice algebraic properties of differential operators -- except that they are not exactly local, only up to higher order terms.

Second, there is this weird fact. Take a nice space (say $\R$ or $\R^2$) and a finite differentiability (say $10$). Then, on the space of compactly supported $C^{10}$ functions on $\R$, there is a derivation at $0$ (i.e., a linear map $D : C^{10} \to \R$ satisfying $D(fg) = (Df) * g(0) + f(0) * Dg$) which is not coming from a tangent vector $v$ at $0$, i.e., which can not be written as $Df = f'(0) v$. This means that algebraic characterisations of differential operators only work well in $C^\infty$, but not in finite smoothness.

[Kevin Buzzard]

Doesn't the fact that d on C^2 functions is not d on C^3 functions simply translate into the statement that there's d on C^2 and d on C^3 and inclusions of sheaves from C^3 to C^2 and then there's some commutative square of morphisms of sheaves?

It's the non-locality comment which is scaring me a lot more though. The sheaf axiom is precisely "the property which defines me can be checked locally" and a morphism of sheaves involves a lot of commutative diagrams which again say that the information in the morphism is of a local nature. What is an example of a pseudodifferential operator being "not exactly local"?

[Sébastien Gouëzel]

If $f$ and $g$ coincide around $0$, then you may have $L f (0)\ne L g (0)$ when $L$ is a pseudo-differential operator.

[Moritz Doll]

Kevin Buzzard said:

What is an example of a pseudodifferential operator being "not exactly local"?

Basically every pseudodifferential operator that is not a differential operator is non-local, so you could take $\mathcal{F}^{-1} (1 + |\xi|^2)^{s/2} \mathcal{F}$ from my talk for $s \not \in 2\mathbb{N}$. This is related to the fact that a distribution supported at a single point has to be a sum of derivatives of the Dirac delta.
The feature of a pseudodifferential operator is that it preserves local regularity, i.e., if a distribution is smooth at a point, then A f (for A psido) is smooth at that point, or even better microlocal regularity (where you measure regularity in directions (using the Fourier transform)).

[Kevin Buzzard]

It sounds like sheaf theory is not set up to handle pseudodifferential operators!

[Junyan Xu]

I also just realized, the classic Sheaves on Manifolds (Kashiwara--Schapira) in algebraic analysis doesn't actually cover pseudo-differential operators.
image.png

[Stefan Kebekus]

@Sébastien Gouëzel Thanks for pointing this out. I learned something! With that in mind, do you think it might make sense to concentrate on potential implementations of "differential operators" for now, and leave "pseudodifferential operators" for later? It might already be an achievement if we can come up with a sane API that covers the classical cases (including Lie derivatives of functions and differential forms, the Laplacian on a Riemann manifold, del and delBar on complex manifolds, ...)

[Sébastien Gouëzel]

It would definitely make sense to build first (or separately) a theory for differential operators. For me, the main question is: does it act on unbundled or bundled objects? I.e., would a differential operators be defined on all functions (or on all sections of a given vector bundle) or only on the ones that are global and with some degree of smoothness? I have a preference for the unbundled design, because it is more flexible and easier to localize (i.e., if a function is $C^2$ around a point but not globally, then its laplacian is well behaved around this point -- which is not something one can get if the laplacian is only defined on the space of globally $C^2$ functions). So I'd advocate that the main API should be unbundled, but of course one should also deduce bundled versions from them.

[Patrick Massot]

Note that in our differential geometry branch, Michael and I defined covariant derivatives on vector bundles and we did you the unbundled approach, for the reasons indicated by Sébastien.

[Antoine Chambert-Loir]

About differential operators and sheaves, remember the theorem of Peetre, https://en.wikipedia.org/wiki/Peetre_theorem

[Stefan Kebekus]

@Antoine Chambert-Loir Unable to remember because I was not aware of the theorem :frown:

Thanks for pointing this out! I will definitely carry that with me into the upcoming discussions.

[Anatole Dedecker]

Antoine Chambert-Loir said:

About differential operators and sheaves, remember the theorem of Peetre, https://en.wikipedia.org/wiki/Peetre_theorem

Doesn't this suffer from the issue mentionned by Sébastien that it doesn't work for finite regularities?

[Sébastien Gouëzel]

I think it works in finite regularity, see Theorems A and B in https://www.degruyterbrill.com/document/doi/10.1515/forum-2013-0159/html (reference provided by chatGPT, I haven't checked the proofs)

[Moritz Doll]

It seems like you can track the regularity needed in the Schwartz kernel theorem (Hörmander does that in the proof, but states the result only for smooth), and then it you end up with a proof for finite regularity (the kernel theorem reduces the result to a fact about distributions with support at a point).

[Anatole Dedecker]

Okay nice, I had only ever heard about it in the smooth case, nice to know.

[Bas Spitters]

@Stefan Kebekus The link to the slides does not work for me. It's behind a login.

[Stefan Kebekus]

@Bas Spitters Apologies, I accidentally moved the file while cleaning up after the talk. Here's a new link

• https://nextcloud.cplx.vm.uni-freiburg.de/index.php/s/m8qay7PdZT6QXc5