Yilin Wang Receives the Salem Prize

Yilin Wang, Junior Professor of Mathematics at IHES, has been awarded the 2024 Salem Prize.

The Salem Prize, administered by the Institute for Advanced Study in Princeton, is awarded annually to a young mathematician who has made outstanding contributions to harmonic analysis and related fields. Named after the French mathematician Raphaël Salem, the prize honors his legacy as one of the great analysts of the 20th century.

Wang was awarded the Salem Prize for her “development of deep, novel connections between complex analysis, probability, and mathematical physics, particularly in relation to Teichmüller theory and Schramm-Loewner evolution.”

“Yilin Wang has uncovered many new features and approaches to study the Schramm-Loewner evolution that drives many important random structures in the complex plane. I personally look forward to seeing where her work takes the subject in the future,” says Terence Tao, Chair of the Scientific Committee of the 2024 Salem Prize.

In her doctoral thesis, Yilin Wang introduced a concept called the Loewner energy to quantify the roundness of simple planar curves. Intuitively, the Loewner energy measures the extent to which a curve deviates from being a perfect circle. Wang used this concept to study a unique class of random planar curves, known as Schramm-Loewner evolution (SLE), which model interfaces in 2D critical lattice models and conformal field theory (CFT).

The introduction of the Loewner energy enabled Wang to bridge SLE with Teichmüller theory and hyperbolic geometry. In particular, she discovered that the action of the SLE loop measure coincides with the Kähler potential of the universal Teichmüller space—an infinite-dimensional complex manifold that includes Teichmüller spaces of Riemann surfaces as complex submanifolds—where the study of the Kähler structure was first motivated by string theory. In collaboration with Fredrik Viklund, she worked on proving new results on the universal Teichmüller space inspired by results on SLEs and, more broadly, random conformal geometry.

“When Yilin’s first paper landed on my desk, I was immediately excited by it and quickly began exploring questions around the Loewner energy. Later, at a workshop, she explained her newly discovered links between the SLE world (which I knew well) and new areas for me, such as Teichmüller theory. These connections were strikingly beautiful, intriguing, and even mysterious. I knew this was a direction I wanted to pursue,” recalls Fredrik Viklund.

“We started an intense and productive collaboration. The first paper came together quickly, but the second required much harder work. Guided by her strong aesthetic sense, Yilin pushed us to achieve not only optimal results but also the most elegant proofs. In the end, our work paid off, resulting in what is perhaps my most satisfying paper,” Viklund adds.

In a recent preprint with Martin Bridgeman, Kenneth Bromberg, and Franco Vargas Pallete, who is currently a postdoctoral researcher at IHES, Wang and her co-authors were able to link the Loewner energy to the renormalized volume of hyperbolic 3-manifolds. This connection suggests a holographic principle for the Loewner energy, reminiscent of the conjectured $AdS_3/CFT_2$ correspondence in string theory, proposed by Juan Martín Maldacena.

Looking ahead, Yilin Wang aims to establish a broader holographic correspondence in the context of random conformal geometry using the probabilistic approach to conformal field theory.

“Congratulations to Yilin Wang for receiving the 2024 Salem Prize. Besides producing excellent research, Yilin Wang is a very active member of the Institute, and a driving force of the scientific activity at IHES. She perfectly embodies the cooperative and interdisciplinary spirit that we cultivate at IHES,” concludes Emmanuel Ullmo, Director of IHES.

Photo credit : Chris Peus / IHES