“Knot theory is very intuitive and therefore pleasant to explain. You have a rope, you tie its ends together and you study the different knots it can be shaped into: some are trivial, some are equivalent to each other” — David Cimasoni explains, meaning that some knots can be untied or transformed into other knots. “In the end, once you formalize it, it comes down to a topological question that involves something called an invariant — in other words, a mathematical object assigned to each knot that doesn’t change when the knot is being deformed. This way, you can prove that two knots aren’t equivalent to each other if you show that their invariants aren’t equal. One fun fact is that defining these invariants can involve techniques from virtually any branch of mathematics."

Knot theory is very intuitive and therefore pleasant to explain. You have a rope, you tie its ends together and you study the different knots it can be shaped into.

As the name suggests, mathematical physics is the branch that studies mathematical models behind physical laws and phenomena. One model that David Cimasoni is currently working on is called the dimer model. According to the model, if you have a graph and a set of edges that doesn’t have common vertices yet covers all of them, it’s called a “perfect matching”. You need to find those perfect matchings and ways to count them: for example, if a graph can be embedded in a plane, there is an efficient way of counting them. For more general graphs, one can apply tools from knot theory, which particularly catches David Cimasoni’s interest.

About the author: Yuta Mikhalkin volunteers for Physics in the Science Olympiad media team after participating herself. She studies mathematics in the University of Geneva. 

David Cimasoni's area of research is not just of interest to mathematicians though; he mentions that knot theory, for instance, is of interest to molecular biologists for the insights it provides into how DNA molecules behave and interact with each other, and how enzymes act on entangled molecules. He has personally collaborated with a physicist who was studying light signals and how they could become knotted, providing insights into the mathematical aspect of the research. He comments that the intersection of mathematics and other topics is not just about the 'what', but the 'why': “A very good friend of mine works at Google now, and he's really trying to understand why the algorithms that drive AI work. Parameters can be tuned in order to improve the performance of machine learning models, but understanding the mathematics behind these decisions - visualizing what the geometric model is doing geometrically, as a sort of gradient descent on a manifold that finds good choices of local minima — is an important question too.”

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When people think about doing research, one common impression is that finding topics must be difficult. David Cimasoni explains that it’s actually not as difficult as it seems — most research ideas come from reading other people’s works, where open questions are almost always waiting to be explored. Although, occasionally, someone else might publish the same idea while you’re still working on it, which happened to David Cimasoni not long ago. Even though he still managed to publish his own paper on the topic, it made him realize how deeply we rely on external recognition for a sense of accomplishment.

Another aspect of research is that you’re working on a topic without really knowing in what direction to go or if there’s even an answer to your question. Or worse, the whole theory you spent so much time developing might just fall apart all of a sudden. No one’s really there to check that what you’re doing is right — you’re fully left on your own. “One year ago, a colleague and I published this paper, and about two months ago we noticed that there’s actually a mistake in it, and no one had seen it! So we had to write an email to the editor asking to block it and all. Fortunately, the mistake is now corrected and the main results of the article still hold true.” 

One year ago, a colleague and I published this paper, and about two months ago we noticed that there’s actually a mistake in it, and no one had seen it! So we had to write an email to the editor asking to block it and all. Fortunately, the mistake is now corrected and the main results of the article still hold true.

And what about teaching, the “burden” of a job in research? David Cimasoni primarily teaches undergraduate courses — often considered the least desirable — but he views this as a stimulating and meaningful part of his career. Whenever he hits a dead end in his research, which inevitably happens to everyone in the field, he finds reassurance in teaching, knowing it will always be valuable to someone out there: indeed, with an emphasis on clarity and structure, his lectures are particularly fascinating, and his well-written and precise lecture notes, even for courses he no longer teaches, are used and loved by many. And, contrary to what some might think, teaching is not nearly as boring as it seems. “It’s extremely easy to communicate art — you can just look at it or listen to it — but communicating math is not the same: it’s quite challenging and extremely interesting.”

When David Cimasoni was a student, and he once read, in a journal at EPFL, an interview with EPFL Professor Manuel Ojanguren. One thing he read in that interview struck him, and he still thinks about it today. “The question was: what is the main quality that one should have as a researcher? I thought he would obviously say you need to be smart. Instead, he said in French something like: Il faut avoir une très grande résistance à la frustration. You must be immensely resistant to frustration. And at the time, I just didn’t understand what he meant.” 

The question was: what is the main quality that one should have as a researcher? I thought he would obviously say you need to be smart. Instead, he said in French something like: Il faut avoir une très grande résistance à la frustration. You must be immensely resistant to frustration. And at the time, I just didn’t understand what he meant.

But today, the words hold much more meaning to him. In his words: “As a student, the exercises you are confronted with are often approachable in the sense that you are guaranteed to have solutions for them, and rarely are they open-ended even in the sense where you don’t know what your final answer is expected to be - and in any case, you know there will be an answer. During your master’s, questions you tackle become more open, but you’re still supervised by someone with expertise who has a good idea of how to solve it and who can ensure it gets done. In actual research, once you’re doing your PhD or after it, it’s much more difficult to know whether you’re going in the right direction!"

In actual research, once you’re doing your PhD or after it, it’s much more difficult to know whether you’re going in the right direction!  

So it’s not so much about being smart. It becomes a question of being tenacious, of not letting go, and of having the psychological ability to think to yourself "I can overcome this." Many times in his career, David Cimasoni saw people who were extremely smart, but unable to come to terms with the particularities of doing long-term problems. Conversely, he remarks that there are countless examples of people not considered prodigious by any means but who were able to reach the peaks of mathematics through perseverance and hard work, the most famous example of which is June Huh, the 2022 recipient of the Fields Medal (the most prestigious award in mathematics) who was famously rejected from almost every university he applied to for his PhD and did not obtain one until the age of 31, but proved to be a late bloomer and an outstanding mathematician.

In general, mathematics is currently at a crossroads: applied mathematics has become better and better funded, with recent advancements in artificial intelligence bringing in big external interest. Meanwhile, pure mathematics, which is often more abstract in nature and less readily connected to real-world applications, can find itself left behind at times. David Cimasoni points out that students who are concerned about studying pure mathematics should not worry that they are missing the pipeline towards research jobs at firms like Google and Amazon. Of course, a degree in a more applied topic provides a more direct route, but David Cimasoni course, a degree in a more applied topic provides a more direct route, but David Cimasoni remarks that he has a lot of colleagues that made the move from research into industry. "I have a friend, for example, who used to work in symplectic geometry and is now at Google. People hiring at these firms are smart enough to know that if someone has a Phd in pure mathematics, most probably they won't know everything about machine learning but they can pick it up very quickly." David Cimasoni’s concluding advice to any young budding mathematician is simple but meaningful: “Work hard, do what you love, and never stop trying!”

Work hard, do what you love, and never stop trying!