Prix Schläfli 2018 in Mathematics: Livio Liechti

We could start with flamenco. Or with doughnuts. But neither of these would really help us to understand Livio Liechti's research. "On the spectra of mapping classes and the 4-genera of positive knots" is the title of the thesis which he submitted a year ago – and anybody who can visualise this has to be a member of a select circle of specialists. Whereby "visualise" is a fairly apt term. "I like the fact that the objects of my field of research are quite visual," says Liechti. He thinks of them three-dimensionally – and his mathematical thought processes also often work on this visual level, and not only in formulas, figures and logical sentences.

Livio Liechti
Immagine: Livio Liechti

It was exactly this picturesqueness that finally convinced Liechti to become a researcher. "When I started my studies, I was not at all sure that mathematics was the right field for me." The defining moment occurred during an Erasmus semester in Madrid, which centred around flamenco dancing on the one hand and a lecture on topological surfaces on the other. "Today I do mathematical research and don't dance the flamenco anymore," he says, summing up the result of his time spent in Madrid. He was too fascinated by the insight that "you can make something that is visual also completely precise" — that you can take something as ordinary as a doughnut, for example, and describe it in clear and precise statements that can fit into a rigorously formal straitjacket.

To this day, his research still spans the gap between the totally visual and pure mathematical abstraction. He has the distinct sense that the things he thinks about are real objects – "even though they exist in the universe of ideas rather than in the real world." And the knowledge gained through his work can definitely also be applied to specific problems, whether in number theory or in biology, for example polymers such as DNA that also tie into complicated knots. A part of his thesis describes how knots of a certain kind are very different from their mirror images — this could be important for bodily processes, for example, where what is termed chirality sometimes plays an important role. It is true that this is not a practical application, but there is still a link to things that exist in real life. He couldn’t say any more than this at present: "I don't know if there could be a practical benefit. Maybe not, or maybe only far in the future."

But his work is not motivated by practical applications, but rather by the transferability of a problem. He prefaces his doctoral thesis with a chapter in which the subsequent rigorous arguments are summarised freely and creatively, using an aesthetic example. In brief, the thesis on the one hand concerns the classification of surfaces and their mapping onto themselves, a kind of liquefaction of concrete structures that can happen in a specific way that can be described with mathematical logic. And on the other, it focuses on the question of how many ways there are to knot mathematical objects (simple answer: unlimited!) and about adding a "satisfactory order" to these. And here too, Liechti is driven by the same fascination for "how one can move from a visual picture to the very abstract". Which he then demonstrates very visually: everybody who ever touched a corded telephone or the headphones of a mobile phone knows that there is an unlimited number of twists.

Livio Liechti was awarded the Prix Schläfli 2018 in Mathematics by the Swiss Academy of Sciences for his doctoral thesis "On the spectra of mapping classes and the 4-genera of positive knots" at the University of Bern. He is now researching at the Institute of Mathematics of Jussieu in Paris.

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