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Yang Li

Yang Li received his PhD in 2019 from Imperial College London, under the guidance of Simon Donaldson and Mark Haskins. He has already made significant contributions to the understanding of Calabi-Yau  metrics in complex differential geometry and Riemannian manifolds with exceptional holonomy.  In a series of three papers, he studied the behaviour of Calabi-Yau metrics on 3-folds with holomorphic fibrations, when the fibres have small volume.  He discovered a new complete Calabi-Yau metric on ℂ3 with singular tangent cone at infinity and showed that this gives a model for the behaviour around the critical points of the fibration, resolving an important question in the field.  

In more recent work, he considers  special Lagrangian fibrations, of the kind appearing in the Strominger-Yau-Zaslow  picture of Mirror Symmetry, and obtained new  models for the metric around singular fibres using a powerful combination of techniques from geometry and analysis.  Yang’s wider body of work includes many results on Yang-Mills connections, and the solution of the Plateau problem for maximal submanifolds.

Yang was appointed as a Clay Research Fellow for a term of four years from 1 August 2020. 

Alexander Smith

Alexander Smith obtained his PhD in 2020 from Harvard University, where he was advised by Noam Elkies and Mark Kisin. He currently holds a National Science Foundation Postdoctoral Fellowship at the Massachusetts Institute of Technology.

Alexander Smith has already solved major problems in number theory and forged a novel research programme that combines ideas from combinatorics and probability with sophisticated aspects of number theory. Conditional on the Birch and Swinnterton-Dyer conjecture, he has largely settled an influential conjecture of Goldfeld from 1979 describing how ranks should be distributed in quadratic twist families of elliptic curves. This significantly advances our understanding of the ancient problem of deciding which numbers are congruent in the sense of being the area of a right-angle triangle with rational side lengths. Smith has also determined the exact distribution of the Sylow 2-subgroups of the class groups of imaginary quadratic fields, thereby proving the Cohen-Lenstra-Gerth conjectures from the 1980s. In recognition of these achievements, he was awarded the inaugural David Goss Prize in Number Theory.

Alexander Smith was appointed as a Clay Research Fellow for a term of four years beginning 1 July 2021.

Lisa Picirillo

Lisa Piccirillo obtained her PhD in 2018 from the University of Texas at Austin, where she was advised by John Luecke. She held an NSF Postdoctoral Fellowship at Brandeis University before moving to the Massachusetts Institute of Technology, as a Moore Instructor then Assistant Professor.

Piccirillo is renowned for her work in low-dimensional topology. Most famously, she proved that the Conway knot is not slice, i.e., there is no 2-disc smoothly embedded in the 4-ball that intersects the bounding 3-sphere in the Conway knot. The sliceness of the Conway knot had been an open problem for over 50 years, gaining iconic status as a test problem. Many sophisticated tools were developed in the area to tackle problems of this type, drawing on diverse parts of mathematics, including gauge theory, representation theory and symplectic topology. Piccirillo’s proof involves an ingenious fusion of these tools with inspired, direct topological constructions. In recognition of this achievement, she was awarded the inaugural Maryam Mirzakhani New Frontiers Prize in 2020. She continues to demonstrate the power of her techniques by solving longstanding problems in low dimensional topology.

Lisa was appointed as a Clay Research Fellow for a term of three years beginning 1 July 2021.