Pariah moonshine and math-writing

Getting mathematics into Nature (the journal) is next to impossible. Ask David Mumford and John Tate about it.

Last month, John Duncan, Michael Mertens and Ken Ono managed to do just that.

Inevitably, they had to suffer through a photoshoot and give their university’s PR-people some soundbites.

CAPTION

In the simplest terms, an elliptic curve is a doughnut shape with carefully placed points, explain Emory University mathematicians Ken Ono, left, and John Duncan, right. “The whole game in the math of elliptic curves is determining whether the doughnut has sprinkles and, if so, where exactly the sprinkles are placed,” Duncan says.

CAPTION

“Imagine you are holding a doughnut in the dark,” Emory University mathematician Ken Ono says. “You wouldn’t even be able to decide whether it has any sprinkles. But the information in our O’Nan moonshine allows us to ‘see’ our mathematical doughnuts clearly by giving us a wealth of information about the points on elliptic curves.”

(Photos by Stephen Nowland, Emory University. See here and here.)

Some may find this kind of sad, or a bad example of over-popularisation.

I think they do a pretty good job of getting the notion of rational points on elliptic curves across.

That’s what the arithmetic of elliptic curves is all about, finding structure in patterns of sprinkles on special doughnuts. And hey, you can get rich and famous if you’re good at it.

Their Nature-paper Pariah moonshine is a must-read for anyone aspiring to write a math-book aiming at a larger audience.

It is an introduction to and a summary of the results they arXived last February O’Nan moonshine and arithmetic.

Update (October 21st)

John Duncan send me this comment via email:

“Strictly speaking the article was published in Nature Communications (https://www.nature.com/ncomms/). We were also rejected by Nature. But Nature forwarded our submission to Nature Communications, and we had a great experience. Specifically, the review period was very fast (compared to most math journals), and the editors offered very good advice.

My understanding is that Nature Communications is interested in publishing more pure mathematics. If someone reading this has a great mathematical story to tell, I (humbly) recommend to them this option. Perhaps the work of Mumford–Tate would be more agreeably received here.

By the way, our Nature Communications article is open access, available at https://www.nature.com/articles/s41467-017-00660-y.”

Grothendieck seminar at the ENS

Next week, the brand new séminaire « Lectures grothendieckiennes » will kick off on Tuesday October 24th at 18hr (h/t Isar Stubbe).

There will be one talk a month, on a tuesday evening from 18hr-20hr. Among the lecturers are the ‘usual suspects’:

Pierre Cartier (October 24th) will discuss the state of functional analysis before Grothendieck entered the scene in 1948 and effectively ‘killed the subject’ (said Dieudonné).

Alain Connes (November 7th) will talk on the origins of Grothendieck’s introduction of toposes.

In fact, toposes will likely be a recurrent topic of the seminar.

Laurant Laforgue‘s title will be ‘La notion de vérité selon Grothendieck'(January 9th) and on March 6th there will be a lecture by Olivia Caramello.

Also, Colin McLarty will speak about them on May 3rd: “Nonetheless one should learn the language of topos: Grothendieck on building houses”.

The closing lecture will be delivered by Georges Maltsiniotis on June 5th 2018.

Further Grothendieck news, there’s the exhibition of a sculpture by Nina Douglas, the wife of Michael Douglas, at the Simons Center for Geometry and Physics (h/t Jason Starr).

It depicts Grothendieck as shepherd. The lambs in front of him have Riemann surfaces inserted into them and on the staff is Grothendieck’s ‘Hexenkuche’ (his proof of the Riemann-Roch theorem).

Grothendieck topologies as functors to Top

Either this is horribly wrong, or it must be well-known. So I guess I’m asking for either a rebuttal or a reference.

Take a ‘smallish’ category $\mathbf{C}$. By this I mean that for every object $C$ the collection of all maps ending in $C$ must be a set. On this set, let’s call it $y(C)$ for Yoneda’s sake, we can define a pre-order $f \leq g$ if there is a commuting diagram

$\xymatrix{D \ar[rr]^f \ar[rd]_h & & C \\ & E \ar[ru]_g &}$

A sieve $S$ on $C$ is the same thing as a downset in $y(C)$ with respect to this pre-order. Composition with $h : D \rightarrow C$ gives a map $h : y(D) \rightarrow y(D)$ such that $h^{-1}(S)$ is a downset (or, sieve) in $y(D)$ whenever $S$ is a downset in $y(C)$.

A Grothendieck topology on $\mathbf{C}$ is a function $J$ which assigns to every object $C$ a collection $J(C)$ of sieves on $C$ satisfying:

• $y(C) \in J(C)$,
• if $S \in J(C)$ then $h^{-1}(S) \in J(D)$ for every morphism $h : D \rightarrow C$,
• a sieve $R$ on $C$ is in $J(C)$ if there is a sieve $S \in J(C)$ such that $h^{-1}(R) \in J(D)$ for all morphisms $h : D \rightarrow C$ in $S$.

From this it follows for all downsets $S$ and $T$ in $y(C)$ that if $S \subset T$ and $S \in J(C)$ then $T \in J(C)$ and if both $S,T \in J(C)$ then also $S \cap T \in J(C)$.

In other words, the collection $\mathcal{J}_C = \{ \emptyset \} \cup J(C)$ defines an ordinary topology on $y(C)$, and the second condition implies that we have a covariant functor

$\mathbf{J} : \mathbf{C} \rightarrow \mathbf{Top}$ sending $C \mapsto (y(C),\mathcal{J}_C)$

That is, one can view a Grothendieck topology as a functor to ordinary topological spaces.

Furher, the topos of sheaves on the site $(\mathbf{C},J)$ seems to fit in nicely. To a sheaf

$A : \mathbf{C}^{op} \rightarrow \mathbf{Sets}$

one associates a functor of flabby sheaves $\mathcal{A}(C)$ on $(y(C),\mathcal{J}_C)$ having as stalks

$\mathcal{A}(C)_h = Im(A(h))$ for all points $h : D \rightarrow C$ in $y(C)$

and as sections on the open set $S \subset y(C)$ all functions of the form

$s_a : S \rightarrow \bigsqcup_{h \in S} \mathcal{A}(C)_h$ where $s_a(h)=A(h)(a)$ for some $a \in A(C)$.

NaNoWriMo (1)

Some weeks ago I did register to be a participant of NaNoWriMo 2016. It’s a belated new-year’s resolution.

When PS (pseudonymous sister), always eager to fill a 10 second silence at family dinners, asked

(PS) And Lieven, what are your resolutions for 2016?

she didn’t really expect an answer (for decades my generic reply has been: “I’m not into that kinda nonsense”)

(Me) I want to write a bit …

stunned silence

(PS) … Oh … good … you mean for work, more papers perhaps?

(Me) Not really, I hope to write a book for a larger audience.

(PS) Really? … Ok … fine … (appropriate silence) … Now, POB (pseudonymous other brother), what are your plans for 2016?

If you don’t know what NaNoWriMo is all about: the idea is to write a “book” (more like a ‘shitty first draft’ of half a book) consisting of 50.000 words in November.

We’re 5 days into the challenge, and I haven’t written a single word…

Part of the problem is that I’m in the French mountains, and believe me, there always more urgent or fun things to do here than to find a place of my own and start writing.

A more fundamental problem is that I cannot choose between possible book-projects.

Here’s one I will definitely not pursue:

[section_title text=”The Grothendieck heist”]

“A group of hackers uses a weapon of Math destruction to convince Parisian police that a terrorist attack is imminent in the 6th arrondissement. By a cunning strategy they are then able to enter the police station and get to the white building behind it to obtain some of Grothendieck’s writings.

A few weeks later three lengthy articles hit the arXiv, claiming to contain a proof of the Riemann hypothesis, by partially dismantling topos theory.

Bi-annual conferences are organised around the globe aimed at understanding this weird new theory, etc. etc. (you get the general idea).

The papers are believed to have resulted from the Grothendieck heist. But then, similar raids are carried out in Princeton and in Cambridge UK and a sinister plan emerges… “

Funny as it may be to (ab)use a story to comment on the current state of affairs in mathematics, I’m not known to be the world’s most entertaining story teller, so I’d better leave the subject of math-thrillers to others.

Here’s another book-idea:

[section_title text=”The Bourbaki travel guide”]

The idea is to hunt down places in Paris and in the rest of France which were important to Bourbaki, from his birth in 1934 until his death in 1968.

This includes institutions (IHP, ENS, …), residences, cafes they frequented, venues of Bourbaki meetings, references in La Tribu notices, etc.

This should lead to some nice Parisian walks (in and around the fifth arrondissement) and a couple of car-journeys through la France profonde.

Of course, also some of the posts I wrote on possible solutions of the riddles contained in Bourbaki’s wedding announcement and the avis de deces will be included.

Here the advantage is that I have already a good part of the raw material. Of course it still has to be followed up by in-situ research, unless I want to turn it into a ‘virtual math traveler ’s guide’ so that anyone can check out the places on G-maps rather than having to go to France.

I’m still undecided about this project. Is there a potential readership for this? Is it worth the effort? Can’t it wait until I retire and will spend even more time in France?

Here’s yet another idea:

[section_title text=”Mr. Yoneda takes the Tokyo-subway”]

This is just a working title, others are “the shape of prime numbers”, or “schemes for hipsters”, or “toposes for fashionistas”, or …

This should be a work-out of the sga4hipsters meme. Is it really possible to explain schemes, stacks, Grothendieck topologies and toposes to a ‘general’ public?

At the moment I’m re-reading Eugenia Cheng’s “Cakes, custard and category theory”. As much as I admire her fluent writing style it is difficult for me to believe that someone who didn’t knew the basics of it before would get an adequate understanding of category theory after reading it.

It is often frustrating how few of mathematics there is in most popular maths books. Can’t one do better? Or is it just inherent in the format? Can one write a Cheng-style book replacing the recipes by more mathematics?

The main problem here is to find good ‘real-life’ analogies for standard mathematical concepts such as topological spaces, categories, sheaves etc.

The tentative working title is based on a trial text I wrote trying to explain Yoneda’s lemma by taking a subway-network as an example of a category. I’m thinking along similar lines to explain topological spaces via urban-wide wifi-networks, and so on.

But al this is just the beginning. I’ll consider this a success only if I can get as far as explaining the analogy between prime numbers and knots via etale fundamental groups…

If doable, I have no doubt it will be time well invested. My main problem here is finding an appropriate ‘voice’.

At first I wanted to go along with the hipster-gimmick and even learned some the appropriate lingo (you know, deck, fin, liquid etc.) but I don’t think it will work for me, and besides it would restrict the potential readers.

Then, I thought of writing it as a series of children’s stories. It might be fun to try to explain SGA4 to a (as yet virtual) grandchild. A bit like David Spivak’s short but funny text “Presheaf the cobbler”.

Once again, all suggestions or advice are welcome, either as a comment here or privately over email.

Perhaps, I’ll keep you informed while stumbling along NaNoWriMo.

At least I wrote 1000 words today…