Please share a piece of advice for young researchers
and another one for more experienced ones.
As a PhD student, I don’t feel very qualified to give advice to experienced researchers, but my dubious advice to them would be to keep an open mind about their methods. Just like we are constantly trying to improve scientific knowledge, we should also be trying to increase the efficiency of the scientific process. The internet has changed the way we do research completely, and this process is still continuing. Unfortunately, change is slow because many people are set in their ways and stick to those tools they used as a young researcher.
My advice to other young researchers is to work on topics you yourself enjoy very much and not to worry too much about the rest. Thinking about it, this advice might be even more dubious than what I said for experienced researchers, but this at least is how I approach my research.
Which one is your favorite PhD comic?
It doesn’t have much to do with my research experience, but spending way too much time on mundane e-mails is one of my great weaknesses in general.
What sparked your interest in Mathematics?
Is there or was there an alternative to your career path?
It was clear from quite a young age that mathematics was something for me. And even before entering university I was particularly interested in dynamical systems. Only I did not realize at the time that this was a serious area of research. Or rather, I did not realize that the simple examples I read about in popular science books had important aspects in common with systems that are relevant to research. Already then I was fascinated about how simple rules can generate complicated dynamics, and complicated systems can be made sense of by understanding a few key properties. Of course, nowadays I have a very different opinion of what is simple and what is complicated, and the key properties I’m trying to understand are rooted much deeper in abstract mathematics, but the same philosophy still applies to much of the mathematics that I am interested in.
I’ve never really thought about an alternative career path. I’m not completely sure that I want to do research for the rest of my life, so it is reassuring to know that a degree in mathematics is highly valued in many corners of the job market, but at the moment I cannot think of any job I would enjoy more.
Which are your favorite sources for articles?
When I’m not looking for anything in particular, the arXiv is my favorite. I systematically check what is posted in a few categories. It is simply the best way to keep up to date with the current research. Or at least the best way to attempt to keep up to date.
When I am looking for something in particular, I go to whatever source Google Scholar points me to.
A researcher in the pre-internet days had to rely on journals to communicate the latest developments in the field, but for me the arXiv has completely taken over this job. Of course the journals still have an important role as a quality certificate for the content they publish, but unfortunately not all of them are doing a very good job at that. I’m eagerly waiting for the system of peer review to adapt to the internet age as well, although I don’t have a clear picture of what that would look like. Maybe collaborative reading can play a role in that.
Who are the contemporary researchers
that have influenced you the most?
Like for any young researcher, my PhD supervisor (Yuri Suris) is a major influence. The second name that comes to mind is Vladimir Arnold. He passed away in 2010 (I hope that still counts as contemporary), before I started my research career, so I never had any direct contact with him, but several of his books have had a major influence on me. The way he combines physical, geometric and abstract mathematical thinking in an intuitive and accessible way is a great example to me. I recommend his “mathematical methods of classical mechanics” to any student interested in the subject, and his “mathematical understanding of nature” to anyone who has the slightest interest in mathematics.
How would you explain the broader significance of your
work to non-researchers or academics from other disciplines?
The broader significance of my work is something the future will have to decide about. But I can say something about the broader significance of my field.
I work on integrable systems. Simply put they are systems that have too much structure. The systems can come for example from mechanics, geometry, or purely mathematical considerations. The structure they have can be as simple conserved quantities (like energy and momentum in mechanics) and symmetries (when some transformation turns every solution of the system into another solution), or it can be much more mathematically involved. The point is that they have so much structure that they have little freedom left to behave erratically. Once you understand the underlying structure, the behavior of an integrable system is suddenly easy to comprehend. Integrable systems are systems that at first sight look terribly complicated, but the mathematical background turns them into the simplest of things.
Integrable systems show up in many areas of physics, so understanding the mathematics of them is very helpful for modeling and analyzing physical systems. It might even help to understand the nature of our universe. My background in physics is too limited to say anything more specific about that, but I find it very comforting to know that what I am studying is related to the physical world.
How can you and other researchers in Mathematics
benefit from using PaperHive?
Integrable systems is a field where very different types of math tend to show up unexpectedly, so I am constantly trying to broaden my horizons. And I think the same holds for many of my colleagues. An active PaperHive discussion on review papers could make new fields more easily accessible. And perhaps even more important: it could provide the extra little bit of motivation that is necessary to spend enough time on a paper to really understand it.
Here are some topics that are currently on my list, and links to some relevant works on PaperHive:
- Hopf Algebras, partly because of their connection to Butcher Series in numerical analysis (https://paperhive.org/
documents/ETpcsPqSRvlA, https://paperhive.org/ documents/Hu8_rsylLYOf)
- Painleve equations, an important class of integrable systems that I know way too little about
- Dirac structures (https://paperhive.org/
documents/r7jr95w3ob-n, https://paperhive.org/ documents/hXElEcclt8b7), an often neglected common structure behind different instances of Lagrangian mechanics.
I have never been active on Stack Exchange or Mathoverflow and part of the reason for that is probably the issue from the PhD comic. I tend to over-think how to phrase my points and how to provide a good context. With in-document discussions the context is already there, which for me significantly reduces the barrier to engage in discussion.