Yilin Wang: at the interface of probability and complex analysis
This portrait is part of a larger touring exhibition of Portraits of women mathematicians from Université Paris-Saclay, which is currently under production and will be launched in spring 2024.
Yilin Wang is a Junior Professor of Mathematics at the Institute of Advanced Scientific Studies (Institut des hautes études scientifiques, IHES). Positioning her research at the interface of complex analysis and probability theory, she works to highlight the connections between random conformal geometry, geometric function theory and Teichmüller theory. The first Junior Professor at IHES, she has recently been awarded an ERC Starting Grant from the European Research Council for her RaConTeich project.
Arriving in France from China for a 2-year scientific preparatory course for a competitive exam with the intention of becoming an engineer, Yilin Wang soon found herself drawn to the theoretical aspect of mathematics. "I was immediately fascinated by the perfection of the mathematical world in comparison to our imperfect, error-prone real world," she confirms. This intuition was confirmed when, during her competitive exams, she realised she found more pleasure in research-oriented exams than in those geared towards applications and engineering schools. She naturally joined the École Normale Supérieure (ENS) on Rue d'Ulm and took advantage of these years to explore different paths in mathematics. After obtaining a master's degree in probability and statistics, she went to the Swiss Federal Institute of Technology in Zurich (ETHZ) to do a thesis under the supervision of Wendelin Werner, winner of the Fields Medal in 2006. Armed with her PhD, in 2019, she was recruited to the Massachusetts Institute of Technology (MIT) as a C.L.E. Moore Instructor (a teaching position), then joined the Mathematical Sciences Research Institute in Berkeley (MSRI) in California to continue her postdoctoral research. In 2022, she joined IHES as a Junior Professor, becoming the first person to hold such a position at IHES.
Loewner energy as the key
A research project often begins with a new idea, a sort of cornerstone strong enough to form the foundation for subsequent work. Yilin Wang formulated this "new idea" during her thesis in Zürich. "At that time, I was interested in the asymptotic behaviour of simple random curves, such as SLE curves, derived from statistical mechanics." This research may seem fairly classic in the world of random conformal geometry, to which Yilin Wang belonged at the time... Until she introduced the notion of Loewner energy and showed that this quantity, useful in random conformal geometry, is also an important function for the geometry of spaces of curves, known as the universal Teichmüller space. "At the time, these two worlds - that of random curves and that of Teichmüller spaces - were far apart. But our research seemed to show that there was a link between these two fields," she explains. By establishing this first point of contact, Yilin Wang found the key to her future research.
Exploring the link between SLE and Teichmüller space
The researcher soon realised that the link between these two fields was probably stronger than the "simple" contact established during her thesis and worth exploring further. Subsequently, her research has consisted in exploring this contact and finding ways to connect these two worlds. "My ambition is to establish links between the fundamental concepts of random conformal geometry and Teichmüller theory by combining techniques from probability, complex analysis, representation theory and more." In 2023, Yilin Wang was awarded an ERC Starting Grant for this interfacing project, named RaConTeich and considered to be of major scientific interest. "Thanks to this funding, I will be setting up my own team, organising conferences and continuing my work, which I am convinced is likely to bring significant developments in both fields and provide new insights into several areas of mathematical physics."
Building bridges between communities
While Yilin Wang is well-versed in the world of random conformal geometry, her initial field of study, she needed to learn more about the concepts at work in Teichmüller theory. "When I was at MIT, I had the opportunity to attend a seminar at Harvard University on subjects related to Teichmüller theory and to meet specialists in this branch of mathematics, with whom I am still in contact today. It was my discussions with them that helped advance my thinking and structure my research." Given the impact of Yilin Wang's initial results, there are now researchers in the Teichmüller theories community interested in this link established between their field and probabilities. "An important and exciting part of my work today is therefore to build bridges between these two mathematical communities, who don't yet know each other very well, who don't necessarily speak the same language, but some of whose members feel that they have much to gain from this mutual contribution," emphasises Yilin Wang. The researcher is regularly invited to give thematic courses to PhD candidates working in both fields or to participate in conferences to present her work and discover that of her colleagues.
Little teaching but plenty of transmission
When asked about what she considers to be her "mission" as a mathematician, Yilin Wang's answer is clear: "Our job is not to prove that this or that theorem is true - theorems are always true - but to make the contribution of mathematics understandable to humans." By "humans", Yilin Wang is not primarily thinking of students - she admits that currently, research clearly takes precedence over teaching - but rather of herself and her peers. In addition to her efforts to bring together the various communities related to her subject, she also strives more broadly to contribute to the overall health of the mathematical community, by making an effort to popularise her research or by her involvement, for example, as an editor of the Journal of London Mathematical Society. "I sincerely believe that the quality of our interactions today will determine the quality of mathematics tomorrow."
Dialogue, a driving force
It is no surprise that Yilin Wang's first instinct, when reflecting on her young career, is to thank the many colleagues she has met along the way. "Whether it's the experienced researchers who, at the start of my research, showed an interest in my proposals and took the time to share their expertise with me when I was faced with difficult questions, or the postdoctoral fellows with whom I now find it so easy to work because they are so open, curious and fascinating.... They have all contributed to making me the researcher I am today and continue to be an invaluable driving force for me to move forward," concludes Yilin Wang.
