## Proofs and papers in the 21st century

It’s strange how mathematics has evolved over the last few centuries, but the way research mathematics is presented hasn’t changed in a meaningful way. Sure, since the advent of internet, it is easier and faster to propagate your work to the mathematical community, but the form in which this happens seems quite archaic, considering the tools available nowadays. What I mean is the following: since the 15th-16th century, people were able to spread their ideas to an audience broader than ever before, due to the invention of printing. This practice of putting your mathematical research into print has carried all the way to today, with the only difference that printed journals are nowadays also available online. Although the opportunities offered by the online medium are enormous, it seems to me that no meaningful changes have been made since the transition from paper to screen. One of the few features is that you are now able to quickly click-through to references and citations using hyperlinks (which is a feature that actually could use improvement in my opinion, see later). Other than that, the linear structure of text on paper (read from top to bottom) has been preserved, as is the lack of interactivity.

Although I realize that many mathematicians like to complain that there is no need to change what works (I also prefer blackboards to smartboards for teaching purposes, I admit), I would like to propose a thought experiment on how research mathematics is presented and how it could be improved. One of the first links you will stumble upon in a quick Google search is Leslie Lamport‘s “How to write a 21st century proof” (pdf). In this note, Leslie proposes a new style of proofs, which admittedly, seems a bit extreme the first time you come across it. However, extreme does not necessarily imply bad or useless. Let me give you some motivation why change is not always bad. As Leslie puts it himself in “How to write a proof”, an earlier version of the aforementioned essay:

Mathematical notation has improved over the past few centuries. In the
seventeenth century, a mathematician might have written

There do not exist four positive integers, the last being greater than two, such that the sum of the first two, each raised to the power of the fourth, equals the third raised to that same power.

How much easier it is to read the modern version

There do not exist positive integers $x, y, z$, and $n$, with $n > 2$, such that $x^n+y^n=z^n$.

Surely, he has a point. The added value of using notation and symbols to shorten the language in which you want to convey your mathematical idea cannot be understated. It is this same evolution of natural language to a more structured language that Leslie (and I) would like to see in proofs. I will refer to his Section 2 for an explicit example of how he proves things nowadays.

Personally, my frustration with the current limitations of static pdfs arose while reading a paper, and having to scroll all the way down for the nth time to check a reference, only to scroll all the way up. I’m sure this has to be a familiar feeling for other people. It made me reflect on the shortcomings of the current state, and try to think of some improvements, so here’s two improvements which should be feasible in the near future.

#### References on hover

The first suggestion is related to my own frustration. It is already possible to click through on references in recent articles, however, wouldn’t it be nice to have a little pop up box appear when hovering over an article in text. In the last few years, YouTube introduced a similar concept for their videos, allowing you to hover the mouse over the timeline at the bottom of a video and see a preview of what will be shown at that specific time. It should not be that hard to replicate a similar ‘textbox-on-hover’ when hovering over a reference, no?

#### Hypertext

I started thinking more about the way a paper is presented in a pdf and came across the idea of hypertext, or what is apparently called accordions. A Google search revealed that Leslie Lamport had had this idea when I was still in diapers. I will show a rough sketch of a similar example to demonstrate the merit of the idea. It gives the opportunity for quick first readings, skip things you already know or understand but also delve into proof steps that might be unclear. You can write for several audiences (e.g. students and more advanced researchers) at the same time while still catering to their respective needs (e.g. students might need more explanations or motivations on why and how certain things are done). However, there is also a certain drawback to this method. It forces authors to more concise and precise, which is not necessarily a bad thing, but surely there’s a lot more work involved. Secondly, opinions on the depth of detail that should be given are inevitably going to differ so clear guidelines on just how much should be explained might not be feasible. It might improve a lot of texts, but new technology will not suddenly solve the problem of badly-written papers as this is a classical PEBCAK problem.

Without further ado, let me demonstrate the simple example which I recently needed in my own research. The implementation is very rough and also doesn’t allow for LaTeX typesetting, so I’ll have to ask to use your imagination a little. Ideally, there would also be some form of hierarchy as described in Leslie’s paper, but again my limitations with HTML prevent me to show this. In short, we want to prove that a certain group acts transitively on a set of points in the affine plane $\mathrm{AG}(2,q)$, $q$ an odd prime power, which could be an exercise in an undergraduate course.

## G2R2: the aftermovie

Wow. As I was interviewed before camera, I figured that there would be some footage afterwards, but the quality of this clip is much more than I expected. All credit goes to the Novosibirsk State University and their wonderful team. It deserves to be shared, so here it is, starring yours truly!

## G2R2: week 2

Onto the second week.

The first week consisted of six days of classes and conferences talks, for which we were rewarded with one free day, including a visit to the local zoo. Here, you have the opportunity to see a liger (lion-tiger hybrid) and supposedly, a liliger has also been born here! Although it sounds intriguing, the actual animal resembled Garfield more than anything, being just a big, fat cat. Maybe they overfeed, maybe genetics are at play, who knows.

The liger at Novosibirsk Zoo

In any case, our brains got some well-deserved rest, which was needed for the upcoming classes.

## G2R2: week 1

Well, I severely overestimated the available time I would have the last few weeks to write a recap. I’ll do it anyway, but not so fresh from memory. Since both weeks contained a lot of material, I will split it in two posts.

The first week had lectures from Roman Nedela and Sasha Mednykh, who replaced a sick Gareth Jones. Personally, these lectures were an added bonus on top of the lectures of the second week, which were my real goal, as they lie much closer to my field of research. But, one can always pick up interesting ideas from another domain, so of we went.

## G2R2: first week

So classes have started, leaving very little time for other things. Especially since the lectures are interwoven with conference talks, keeping us busy from 10am to 10pm. Anyway, a quick write-up from the cultural program of the first week.

## Summer conferences in 2018

This year, I was fortunate enough to attend several conferences again. I haven’t written about them, secretly hoping someone else would, but this hope seems to be in vain.

Combinatorics 2018 was great fun, in lovely former casino in Arco, near the Garda lake in Italy. I gave a presentation about some work in progress (still need to work out more details before I post anything about it here), and got some great response. I haven’t had much time implement the suggestions, but it’s definitely high on the to do list. Then I went to Budapest, for Building Bridges II, a conference in honor of László Lovász’ 70th birthday. So far, I have only seen Gil Kalai’s blog post describing some of the presentations. There were some great talks to be found here, among those of Noga Alon, Lex Schrijver, Laci Babai and others. Good news for the people who couldn’t make it: almost all lectures were recorded and can be viewed on the website! It seems however that Babai’s talk is not among those, a shame, as I found it to be one of the highlights of the first day.

One regret is that this last conference took place in the same week as the Symmetry vs Regularity conference in Pilsen. Unlike Laci Babai, who could attend two different conferences in a single week, I had to make a choice. The Dutch saying “choosing is losing” definitely applied here. Probably, the topics in this conference lie closer to my research interests, but I stuck with the other conference for other reasons.

The next few weeks I will however focus again on one of my research interests (that of algebraic combinatorics), in the G2R2 summer school in Novosibirsk, Russia. This three-week event combines a summer school with the G2R2 conference. The G2 conferences are a yearly summer school + conference combination, and this year the topics are Groups, Graphs, Representations and Relations (hence the name). I intend to write a small recap after every week. The first week is dedicated to a cultural program with visits to the Budker Institute of Nuclear Physics and several musea. Other participants went on a hiking tour towards the Altai mountains, but I was a tad too late in registering for this activity (procrastination is not a good habit). The second week, we will have lectures by Roman Nedela and Gareth Jones, although the latter might be replaced by Alexander Mednykh due to personal reasons. Personally, I’m mostly looking forward to the third week, when Akihiro Munemasa and Mikhail Muzychuk will talk about some topics in algebra. You can find a more detailed overview of the topics on the website of the summer school.

Come back in a few days for a recap of the cultural activities!

## Triangle-free induced subgraphs of the unitary polarity graph

That is the long title of an article which is joint work with Francesco Pavese. Last week, I received the news that it has been accepted for publication, my very first publication! In the article, we continue some work that was started in an earlier preprint with Leo Storme, which ironically, is still under review.

I have posted about this topic in an earlier blog post, but I’ll give a quick recap here.

Roughly speaking the question is the following: consider a finite projective plane of order $q$ and a unitary polarity $\perp$ of this plane (see [1] for all about this). A polar triangle with respect to $\perp$ is a triple of points $x_1,x_2,x_3$ such that $x_i^\perp = x_jx_k$ for any triple $i,j,k$ of distinct indices. We are interested in large sets of points such that no triple from this set forms a polar triangle. This is a particular instance of the classical forbidden configuration problem, which has appeared in many forms in the realm of extremal combinatorics.

Using a combination of algebraic graph theory techniques and geometrical constructions, we manage to obtain upper bounds for general projective planes, and lower bounds for the Desarguesian and Figueroa plane, which asymptotically match! It was the first time I got to use eigenvalue interlacing [2] in a new environment, so I’m quite content with the result.

It has been accepted to the European Journal of Combinatorics, but until its actual publication, you can check it out on the arXiv.

References

[1] D. Hughes, F. Piper, Projective planes, Graduate Texts in Mathematics, Springer-Verlag New York-Berlin, 1973.

[2] W.H. Haemers, Interlacing Eigenvalues and Graphs, Linear Algebra Appl.,
226/228:593–616, 1995.

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