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Mohammed Abouzaid received his PhD from the University of Chicago in 2007 under the supervision of Paul Seidel. His thesis used techniques from tropical geometry to give a new approach to the homological mirror symmetry conjecture for toric varieties. He is interested in symplectic topology and its interactions with algebraic geometry and differential topology. Mohammed was apponted as a Clay Research Fellow for a term of five years beginning July 2007.
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Ivan Corwin received his PhD in 2011 from the Courant Institute at New York University under the supervision of Gerard Ben Arous. His interests include probability, mathematical physics and exactly solvable systems. One part of his research has been to compute exact formulas for the statistics of the solution to the Kardar-Parisi-Zhang non-linear stochastic PDE. This stochastic PDE and its statistics describe a large universality class of systems, including growth processes, interacting particle systems, directed polymers in random environments, and less directly random matrix theory. Ivan was appointed as a Clay Research Fellow for a term of four years beginning July 2012.
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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.
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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.
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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.
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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.
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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.
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Akhil Mathew received his PhD in 2017 from Harvard University under the supervision of Jacob Lurie. Previously he received an undergraduate degree from Harvard and studied at UC Berkeley for one year. His research focuses on homotopy theory, higher categories, and their applications, especially to derived algebraic geometry and algebraic K-theory. Some of his past work studied various generalizations of faithfully flat descent in stable homotopy theory and their role in describing certain invariants of structured ring spectra.
Akhil was appointed as a Clay Research Fellow for a term of five years beginning 1 July 2017.
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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.
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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.
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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.
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Georgios Moschidis obtained his PhD in 2018 from Princeton University, where he was advised by Mihalis Dafermos. Since then he has been a Miller Research Fellow at the University of California, Berkeley.
Moschidis’s work is focussed on mathematical problems arising in classical general relativity, the theory of gravity first formulated by Einstein in 1915. In this context, he is particularly concerned with instability phenomena. Moschidis gave the first rigorous account of the celebrated Friedman “ergo-sphere” instability, whose existence was conjectured in the early 1970’s. In a remarkable series of papers over the last few years, he has also addressed the problem of instability for anti-de Sitter (AdS) spacetime, the ground state solution to the Einstein equations with a negative cosmological constant. In contrast to the Friedman instability, the much sought-after instability in this context (which has been observed in numerical simulations) is non-linear in origin. Moschidis’s achievements include a rigorous proof of the instability of AdS in the case of the Einstein-Vlasov system with spherical symmetry. His groundbreaking proof contains ideas that transform the understanding of large-data phenomena in solutions of the Einstein equations.
Georgios Moschidis has been appointed as a Clay Research Fellow for a term of two years beginning 1 July 2021.