added some references on *constructive analysis* here.

at *normed division algebra* it used to say that “A normed field is either $\mathbb{R}$ or $\mathbb{C}$. ” I have changed that to “a normed field over $\mathbb{R}$ is…” and changed *normed field* from being a redirect to “normed division algebra” to instead being an entry on its own.

Thank you so much for responding or pointing me in the right direction if this forum isn't the place to find this help! ]]>

With Igor Khavkine we finally have a polished version of what is now “Part I” of a theory of variational calculus in a differentially cohesive $\infty$-topos. It’s now called:

*Synthetic geometry of differential equations*

*Part I. Jets and comonad structure*

We keep our latest version of the file **here**.

Comments are most welcome.

**Abstract**:

We give an abstract (synthetic) formulation of the formal theory of partial differential equations (PDEs) in synthetic differential geometry, one that would seamlessly generalize the traditional theory to a range of enhanced contexts, such as super-geometry, higher (stacky) differential geometry, or even a combination of both. A motivation for such a level of generality is the eventual goal of solving the open problem of covariant geometric pre-quantization of locally variational field theories, which may include fermions and (higher) gauge fields.

A remarkable observation of Marvan 86 is that the jet bundle construction in ordinary differential geometry has the structure of a comonad, whose (Eilenberg-Moore) category of coalgebras is equivalent to Vinogradov’s category of PDEs. We give a synthetic generalization of the jet bundle construction and exhibit it as the base change comonad along the unit of the “infinitesimal shape” functor, the differential geometric analog of Simpson’s “de Rham shape” operation in algebraic geometry. This comonad structure coincides with Marvan’s on ordinary manifolds. This suggests to consider PDE theory in the more general context of any topos equipped with an “infinitesimal shape” monad (a “differentially cohesive” topos).

We give a new natural definition of a category of formally integrable PDEs at this level of generality and prove that it is always equivalent to the Eilenberg-Moore category over the synthetic jet comonad. When restricted to ordinary manifolds, Marvan’s result shows that our definition of the category of PDEs coincides with Vinogradov’s, meaning that it is a sensible generalization in the synthetic context.

Finally we observe that whenever the unit of the “infinitesimal shape” ℑ\Im operation is epimorphic, which it is in examples of interest, the category of formally integrable PDEs with independent variables ranging in Σ is also equivalent simply to the slice category over ℑΣ. This yields in particular a convenient site presentation of the categories of PDEs in general contexts.

]]>trivia: is there established notation for the set of formal power series in $X$ that have vanishing coefficient of $X^0$ (vanishing constant term)?

]]>Epstein zeta function, just recording the definition and the two classical references by Kronecker and Siegel. Nominally, this is the page 13133 of the $n$Lab ;)

]]>New stub Weyl functional calculus redirecting also Weyl quantization. I would like to see ref.

- Lars Hörmander,
*The weyl calculus of pseudo-differential operators*, Comm. Pure Appl. Math.**32**, 3, 359–443, May 1979, doi,

but have no access to it (can anybody help?). I also added a sentence at Idea section of functional calculus reflecting that the previous definition there is not fitting functional calculi in the context of quantization, including Weyl’s case. One should do this generality discussion more carefully. the previous definition said that the functional calculus needs to be a homomorphism (from ordinary functions to operator functions). This is true for the functional calculus described in the entry, but not for the wider usage of the phrase like in Weyl functional calculus. Maybe we can resolve this in a better way.

]]>the entry *valuation* would deserve more clarification on that issue alluded to under “Sometimes one also…” and where the min-style definition appears the max-style definition should also appear.

The entry should say that at least with some qualification added, then a valued field is a normed field with multiplicative norm. – Or should it be semi-normed?

I could fiddle with it, but I feel I don’t quite get why the terminology here is so non-uniform that I am afraid I am missing something and maybe a more expert person should help.

In Scholze 11, remark 2.3 is a useful comment:

]]>The term valuation is somewhat unfortunate: If $\Gamma = \mathbb{R}_{\geq 0}$, then this would usually be called a seminorm, and the term valuation would be used for (a constant multiple of) the map $x \mapsto - log {\vert x \vert}$. On the other hand, the term higher-rank norm is much less commonly used than the term higher-rank valuation.

gave *rapidly decreasing function* its own entry, for completeness

Arnold Neumaier contacted me about the previous stubbiness of the entry *resurgence* and pointed me to his PO comment on the topic (here). I have put that into the Idea-section of the entry, equipped with some links.

The keyword *derivative* used to redirect to “differentiable map”. I found that less than useful for many purposes of linking to it, and so I have now split it off as a stand-alone entry. Presently this contains nothing than pointers to other entries. But it is already useful to see how many entries on variants of “derivative” we have, and to have a place to collect them all.

Created sampling theory, for the moment just recording some references of my interest.

We should also have Zak transform soon.

]]>at *distribution* there used to be a mentioning of “Colombeau algebras”. I have now removed that paragraph there, and have given it its stand-alone stub entry *Colombeau algebra*, expanding it slightly.

An expert might want to check. I haven’t actually looked into Colombeau algebras beyond a scanning of a review, and presently I don’t plan to delve into the topic. In fact their idea looks misguided to me.

All I mean to do here is to clean up the structure of the entry *distribution* (see also my comments in the thread on products of distibutions, here) while preserving what others had written before.

added example of uniform Cauchy sequences of (continuous) functions with values in a complete metric space: here

(possibly this is already, in more generality, in some other entry?)

]]>The term “bounded function” used to redirect to “bornological space”. I have given it its own entry.

(Noticed this when beginning to write out a proof at *Tietze extension theorem*.)

while writing out the proof of the fundamental product theorem in K-theory I had occasion to record that

]]>created *induced metric*, just for completeness

I noticed that presently *topological basis* redirects to *basis in functional analysis* instead of to the entry *topological base*. This seems dangerous. I’d like to change that redirect.

I noticed that the entry *analysis* is in a sad state. I now gave it an Idea-section (here), which certainly still leaves room for expansion; and I tried to clean up the very little that is listed at *References – General*

at *triangle inequality* the discussion of the interpretation in enriched category theory had been missing. I have added in a corrresponding section here and cross-linked with *Lawvere metric spaces*.

For completeness and for linking-purposes, I thought an entry *p-norm* was missing. So I created one.

I have added cross-links with the Idea-sections at *sequence space* and *Lebesgue space* and with the Examples-section at *normed vector space*. Created a stub for *Minkowski’s inequality*, so far containing essentially nothing but a pointer to Todd’s *p-norms (toddtrimble)*, which I vote for copying to there.

I was un-graying some links at *epsilontic analysis*. Among the titles of non-existing entries that are still being requested is

“classical analysis”

“topological property”.

Is it likely/desireable that we will have entries with these titles? Maybe we should change these links to point to “analysis” and to “topology” instead?

]]>added statement and proof that sequentially compact metric spaces are equivalently compact metric spaces

]]>added statement and proof that sequentially compact metric spaces are totally bounded

]]>added statement and proof to *Lebesgue number lemma*.

did the same for *sequentially compact metric spaces are equivalently compact metric spaces* but now the $n$Lab seems to have gone down before I could submit the edit.