# Using the power of drawing to discern order in nature

When the 8^{th} European Congress of Mathematics (8ECM) gets underway in Portorož on the Adriatic coast of Slovenia this Sunday, it will be a big night for a certain ETH Zurich mathematics professor: Joaquim Serra will be presented with the European Mathematical Society’s EMS Prize. This honour is awarded only every four years, each time to ten researchers under the age of 36.

The award ceremony should have taken place in 2020, but like so many events was postponed due to the coronavirus pandemic. For 34-year-old Serra, who was born in Catalonia and raised in Barcelona, 2020 was a pivotal year. Despite the difficulties caused by the pandemic, he managed to climb the next rung on his scientific career ladder, bringing his years of studying and relocating as a postdoc to an end. After winning his EMS Prize in June, in September he was awarded an ERC Starting Grant, which provides promising, talented researchers with a springboard to research independence. And in December, the ETH Board appointed him Assistant Professor of Mathematics at ETH Zurich.

Serra was by no means destined to become a university mathematician. Few years ago, it looked as though he was headed for a career in the private sector. Serra grew up in a Catalan family and went to school and university in Barcelona. In middle school, he liked maths, but never dreamed that he would make a career of it: “Like most people, I couldn’t fathom what it would mean to be a mathematician,” he says.

It wasn’t until he took part in the International Mathematical Olympiad that he took a shine to mathematics. This competition for high-school students showed him that mathematics offered a world beyond mundane calculations, one that would allow him to creatively develop paths of reasoning. When the time came to go to university, he chose to pursue mathematics rather than physics or engineering sciences.

## Exposing the pure structures hidden in nature

Serra’s interest in scientific phenomena nevertheless continues to inform the way he approaches mathematics to this day. His area of research is partial differential equations. His current work concerns the mathematical equations of phase transitions arising in physics, chemistry, biology and economics. “As mathematicians, we try to look beyond artificial ornamentation and expose the pure structures in nature that we can’t see right away. We’re very glad whenever we discover order beneath the complexity of the phenomena we observe.”

Serra completed both his degree and his doctorate at the Universitat Politècnica de Catalunya (UPC) in Barcelona. It was there that he began collaborating with his doctoral thesis advisor Xavier Cabré, who will be among the main speakers in Portorož. Cabré’s group was researching the then emerging area of integro-differential equations. These equations govern such different phenomena as the prices of American options in finance or the macroscopic behaviour of gases. During this period, Serra learned how mathematicians go about developing new theories and proving theorems or mathematical statements.

After Serra completed his doctorate in 2014, he was at first unable to take that key career step for any researcher of going to work abroad. His son was just 18 months old at the time, and his wife was still studying to become a doctor of medicine. So Serra spent the next year and a half working as a consultant and business statistician at a company specialising in big data. Then he heard from a friend from his time as a doctoral candidate, Xavier Ros-Oton, who was then a postdoc of professor Alessio Figalli at Austin, that Figalli had been appointed to a professorship at ETH Zurich. Figalli is one of 11 winners of the EMS Prize who have gone on to win a Fields Medal, often referred to as the Nobel Prize of mathematics.

## When ice melts to form water

It soon became clear that Serra, Ros-Oton and Figalli shared an interest in researching those partial differential equations capable of describing the transformation and transition phenomena that are typical in both nature and economics. Such phenomena include ice melting to form water, liquid hardening to form crystal, biological cells or bacteria switching from active to inactive, and holders of financial contracts deciding whether or not to execute them. Then there is the emergence of energetically stable states such as that of soap bubbles. The mathematical challenge is that often the equations describing such phase transitions allow for a myriad of possible solutions, but only some of them, the stable states, are actually found in nature.

Once Serra’s wife had become a doctor, the couple decided the time had come for him to return to academic research. After a research visit to Berlin, Serra and his family moved to Switzerland in 2016. He took up a position at ETH, first as a postdoc in Figalli’s group and then, from 2018, working as an independent researcher funded by an “Ambizione” grant from the Swiss National Science Foundation.

Figalli and Serra share a great respect for the geometric approaches the Argentinian mathematician Luis Caffarelli uses to solve partial differential equations. In particular, Serra admires Caffarelli’s talent for using sketches to illuminate his proofs. “A mathematical proof is always a combination of logical statements. But sometimes, lurking behind the proof is a powerful geometric intuition. I find I can understand the proof much better if I translate the logical statements into drawings or sketches,” Serra says.

When ice melts to form water, for example, the individual atoms aren’t the only things that move. The boundary marking the transition from solid to liquid also undergoes dramatic movement and change. Solving the problem calls for a whole system of equations, for instance for an energy balance that defines the position of the “free” boundary that moves over time. This naturally makes describing free boundaries very difficult. A certain class of mathematical free boundary value problems have been known as the “Stefan problem” since the 19th century. But it wasn’t until the 1970s that Caffarelli made a breakthrough that continues to inspire and inform research to this day. Building on Caffarelli’s theory, the most recent work by Figalli, Ros-Oton and Serra provides answers for the first time to fundamental questions on Stefan’s problem that seemed completely inaccessible only some decades ago.

## The Sagrada Família of mathematics and the Sihlwald

The Stefan problem exemplifies that in mathematics, the search for a solution can take decades. In this regard, mathematical research is similar to the famous basilica in Serra’s hometown – the Sagrada Família, designed by Antoni Gaudí: although construction started in 1882 and has been going on for some 140 years, this long acclaimed masterpiece is not expected to be completed for quite some time.

The Serras are now a family of five and have settled into life in Gattikon near Zurich. Serra’s wife works at a hospital in Zurich and their older children, a boy and a girl, go to primary school. Serra himself must ensure he doesn’t fall behind: Apart from his baby son, he is the only member of his family who doesn’t yet speak fluent German. When not on the ETH campus, he enjoys the rural setting outside Zurich. Gattikon is just half an hour from the Sihlwald nature reserve. Given their surroundings, the family devotes a lot of time to hiking and bike rides, and in Switzerland they discovered the joys of skiing. “There aren’t many places in Europe that offer this combination of nature and a bustling, vibrant and culture-rich city. In Barcelona, you have to travel quite a distance before you’re really in nature,” Serra says.