Why open notebook mathematics?
Collaboration is a wonderful thing.
- Collaboration means that someone else is excited about the same questions that excite you.
- Collaboration means that someone else values your input and ideas.
- Collaboration means that you can mix your ideas and specialist knowledge with someone else's, and overcome problems you might not solve alone.
- Collaboration means that, when something breaks or goes wrong, you have someone else to share in the misery and stress, and to help fix it.
How do collaborations get started?
In my experience, collaborations grow organically out of conversations around shared interests. You tell one another cool things you heard or read, you share/explain examples, you share open problems that have been driving you crazy. Eventually, you start making progress on one of these circles of ideas. Sometimes, you get a nice theorem, which you write up and publish. Sometimes, all you get is a better understanding of the ideas, or partial results. These you file away in the back of your mind, and revisit from time to time.
But how does the conversation start?
- Maybe you find yourself working in the same city as someone with the same interests.
- Maybe you meet at a conference.
- Maybe they're the external examiner for your PhD.
- Maybe they're your PhD supervisor.
- Maybe you serendipitously find out you're working on the same problem before scooping one another.
As an early career researcher, it can feel as if the world is full of very successful collaboration cliques, and it's difficult to break into them or start your own. Whether that's true or not, I think mathematics could be a less intimidating place if young researchers were able to see some of these conversations actually happening and were to feel as though they have both a right to contribute and a direct means of getting involved. If all conversations are happening by email or in offices or over coffee at conferences then there's a barrier.
An open research notebook lets me wax lyrical about open problems and circles of ideas which interest me. People who care can see some of these problems and know that I'm happy for them to contribute in any way (whether that's as small as asking questions for clarification and pointing out mistakes or as big as solving my questions with their own ideas).
There are other reasons I'm doing this, but that's the main one.
Here are some of the subsidiary reasons.
Grey literature, 1: Progress
In a collaboration-conversation, it's pretty rare that the first question you ask is the one you end up writing a paper about. The route from inspiration to discovery can be round-about. Along the way, you produce insights and results that don't quite go as far as you want them to, and never make it into the final paper. Maybe a whole project gets solved in principle, but no-one has the energy to write up the details.
I'm hoping that an open notebook will provide me with a way of keeping track of these idea fragments, so that other people who are interested in the same questions can see what I've tried. Perhaps this could save some wheel-reinvention and time.
Grey literature, 2: Problems
In some subjects, there are lists of open problems which help to direct the flow of mathematical discourse in interesting directions, and, occasionally, get cracked. The Kirby list and the Arnol'd list are two examples which come to mind.
In some subjects, there are monolithic conjectures which may not be solved for hundreds of years. The Riemann hypothesis, the Jacobian conjecture, the Hodge conjecture, the Langlands program, the Collatz conjecture, the Goldbach conjecture, and so on. Partial results, special cases or small improvements can be big news.
But the vast majority of open questions reside in our heads. Perhaps we tell our students about some exciting circle of questions, and they are passed down like folklore. Perhaps we exchange our favourite questions with our collaborators. But think about what that means for someone entering the field. They hear a couple of open problems from their PhD supervisor which they might work on. They hear about a bunch of huge famous conjectures which other people are making incremental progress on, around which a clique of specialists and their students may have accumulated. In conference/seminar talks they find out about problems that other people have already solved. Unless they are extremely lucky with their supervisor*, they might not really get the sense that their subject is thriving with a huge range of unanswered questions, any of which they could try to solve, or that it is connected with many other areas, any of which they could contribute to.
One of the purposes of this open notebook, then, is to shine a light on some of the questions that are lurking in my head so that other people can learn about them.
*I was. This was exactly the sense I got, and it was very exciting. My subject, symplectic geometry, grew out of a heady mix of dynamics, algebraic geometry, Lie theory, low-dimensional topology and gauge theory, but it's so technical these days that these historical connections are sometimes forgotten. For example, if you entered the subject today, you would be forgiven for thinking that Fukaya categories first appeared in the study of mirror symmetry; really, they grew out of Fukaya's early cogitations on the Atiyah-Floer conjecture about 3-manifold invariants. When you are aware of these connections, you can ask many more interesting questions.
Aren't you worried about someone stealing your ideas?
You can't really own an idea; they have lives of their own. Of course I would prefer people to join me in exploring the ideas I post about, but if someone figures out how to do something on their own, whether building on what I've posted or not, that's great. That's science.
Aren't you worried that you won't have any ideas to give to your PhD students if you post them all here?
What are "my PhD students"? They're people who happen to find themselves at my university and who chose me to help them on their journey through mathematics, specifically with the transition to research.
How can I help them? Certainly making them aware of open problems is part of what I do, but there are many other aspects, like helping them figure out how to express their own mathematical ideas, or giving them context for their subject, or introducing them to tools and results that might be useful. These other aspects take much more time than the simple act of telling them open problems. Anyway, they start generating their own questions quickly enough (and another aspect of my role is to help them figure out which of these questions is feasible, or can be modified into something feasible).
How can I be maximally useful as a mathematician? If I share my open problems with as many people as I can then perhaps I can help people who aren't "my students" too. And if enough other people try "open mathematics" then my students and I will learn about some interesting new problems in turn.
How do I get involved?
If you have something to say, raise an "issue" on the Gitlab repository (you'll need to create a Gitlab account to do this). Or just add a comment to one of the existing issues (if your contribution is relevant to that discussion).
If you want to add something to the project itself (e.g. edit one of the files), clone the git repository to your own open notebook, edit the files** and make a "merge request". That should spawn a page where we can discuss the changes if necessary (or I can just merge them if they're uncontroversial!).
And, of course, start your own repositories on your own open notebook, with your own circles of interesting problems.
**Note that when it comes to documents I am editing the "lzl" file and generating the tex automatically using a Python script of mine called lzl.py. You can obtain this Python script at https://github.com/jde27/lzl. So if you change the tex file directly I won't merge your changes (they'd just be overwritten by subsequent runs of lzl.py).