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Ziquan Zhuang

Ziquan Zhuang obtained his PhD in 2019 from Princeton University, where he was advised by János Kollár. Since then he has been a Moore Instructor at Massachusetts Institute of Technology.

Zhuang is a remarkably prolific and inventive algebraic geometer who has already made a series of fundamental contributions to higher dimensional birational geometry. These include his landmark solution, with Liu and Xu, of the higher rank finite generation conjecture, which is the final step in the Yau-Tian-Donaldson Conjecture in the case of general Fano varieties. With Xu, Zhuang proved the positivity of the CM line bundle on the K-moduli space; with Ahmadinezhad, he invented a new framework to verify the K-stability of a large class of Fano varieties; and with Stibitz he proved striking results on birational superrigidity and K-stability of Fano varieties.

Ziquan was appointed as a Clay Research Fellow for a term of two years beginning 1 July 2022.

Hannah Larson

Hannah Larson will obtain her PhD in 2022 from Stanford University, where she has been advised by Ravi Vakil.

Displaying remarkable ingenuity, Larson has applied the modern techniques of degeneration and intersection theory to make significant advances in one of the classical areas of algebraic geometry – the geometry of complex curves and their moduli. Her papers bristle with surprising new ideas that attack classical problems. For example, searching for new perspectives on the space of vector bundles on the Riemann sphere, she proved striking results about the moduli space of curves and about stabilization forbranched covers of the sphere (with Canning), and extended Brill-Noether theory (which governs maps of general curves to projective space) to explain seemingly chaotic behaviour in the case of low-gonality curves (with E. Larson and Vogt).

Hannah was appointed as a Clay Research Fellow for a term of five years beginning 1 July 2022.

Alexander Petrov

Alexander Petrov will obtain his PhD in 2022 from Harvard University, where he has been advised by Mark Kisin.

Petrov has demonstrated exceptional creativity in proving surprising theorems concerning Galois representations and arithmetic local systems on algebraic varieties. Settling a conjecture of Litt, he proved that geometrically irreducible, arithmetic local systems on varieties over p-adic fields are essentially de Rham. He discovered a deep generalization of Belyi’s famous theorem, showing that any irreducible Galois representation which arises in the cohomology of an algebraic variety over a number field, appears in the space of algebraic functions on the fundamental group of the thrice punctured sphere. And he opened a new range of possibilities with counterexamples to a conjecture of Scholze on Hodge symmetry for rigid analytic varieties.

Alexander was appointed as a Clay Research Fellow for a term of five years beginning 1 July 2022.

Amol Aggarwal

Amol Aggarwal received his PhD in 2020 from Harvard University, where he was advised by Alexei Borodin. His research lies largely in probability theory and combinatorics, as well as their connections to mathematical physics, integrable systems, and dynamical systems. 

Aggarwal has already established himself as a powerful mathematician, resolving several longstanding conjectures of broad interest. His achievements to date include his proof of the local statistics conjecture for lozenge tilings, prescribing how local correlations for random tilings of large domains asymptotically depend on their boundary conditions. He also provided rigorous proofs for predicted phase transitions in the six-vertex model — a fundamental system from statistical mechanics — and for predicted asymptotic distributions in the one-dimensional asymmetric simple exclusion process, an important prototype for interacting particle systems. In a different direction, he proved the conjecture of Eskin and Zorich describing large genus asymptotics of the Masur-Veech volumes and the Siegel-Veech constants of moduli spaces of Abelian differentials. 

Amol was appointed as a Clay Research Fellow for a term of five years from 1 July 2020.

Antoine Song

Antoine Song received his PhD in 2019 from Princeton University, where he has been working under the guidance of Fernando Codá Marques. 

Song has already established himself as an expert in geometric analysis, solving longstanding problems of fundamental importance concerning the nature of minimal hypersurfaces in compact Riemannian manifolds. First he proved that in dimensions 3 to 7 the closed minimal hypersurface of least area in such a manifold is always embedded. Then, in joint work with Codá Marques and Neves, he showed that for generic metrics on closed manifolds in these dimensions, one can always find a sequence of minimal embedded hypersurfaces that become equidistributed in the sense that the average of the induced measures on the first n hypersurfaces in the sequence converges to the normalised volume measure on the ambient manifold as n tends to infinity. This was a dramatic improvement in the state of the art concerning a circle of problems inspired by Yau’s 1982 conjecture that every closed 3-dimensional Riemannian manifold contains infinitely many closed minimal surfaces. Building on work of Codá Marques and Neves, in 2018 Song proved Yau’s conjecture in complete generality.

Antoine was appointed as a Clay Research Fellow for a term of five years beginning 1 July 2019.

Maggie Miller

Maggie Miller obtained her PhD in 2020 from Princeton University, where she was advised by David Gabai. She is currently an NSF Postdoctoral Fellow at the Massachusetts Institute of Technology.

Miller has advanced the understanding of manifolds in dimensions 3 and 4 with her power and creativity, wielding a wide range of techniques — algebraic, combinatorial, geometric and topological. She has developed a theory of singular fibrations in 4-manifolds and used it to make significant progress on a 35 year old problem of Casson and Gordon: for a large class of fibered ribbon knots, she proves that the associated fibration of the 3-sphere extends over the closed complement of the ribbon disc in the 4-ball.

Her abundance of insight has made Miller a sought-after collaborator, working with a variety of co-authors to advance different aspects of low-dimensional topology: topological versus smooth isotopy for genus-1 surfaces in the 4-ball; taut foliations in 3-manifolds; concordance; trisections of 4-manifolds; diffeomorphisms of non-orientable 3-manifolds; and the use of knot Floer homology to give lower bounds on the bridge index of knots.

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

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.