Simion Filip will receive his PhD in June 2016 from the University of Chicago under the supervision of Alex Eskin. He is interested in the connections between dynamical systems and algebraic geometry, in particular between Teichmüller dynamics and Hodge theory. His recent interests also involve K3 surfaces and their special geometric properties. Simion has been appointed as a Clay Research Fellow for a term of five years beginning 1 July 2016.
Photo by Eduard Duryev.
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 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.
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.
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 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 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.
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.
Ewain Gwynne obtained his PhD in 2018 from the Massachusetts Institute of Technology, under the supervision of Scott Sheffield. Since then he has held the Herschel Smith Fellowship at the University of Cambridge, and is a Junior Research Fellow at Trinity College Cambridge.
Gwynne is a remarkably productive and ingenious researcher with broad interests across probability, especially conformal probability, Schramm-Loewner evolution, Liouville quantum gravity, and random geometry in dimension 2. He has already made landmark contributions to these areas, often developing collaborations with other leading talents in the field. For example, he made breakthroughs with Ding on the fractal dimension of Liouville quantum gravity, with Miller on self-avoiding walks and percolation interfaces on random planar maps, and with Sun on the Fortuin-Kasteleyn model on random planar maps.
Ewain was appointed as a Clay Research Fellow for a term of four years beginning 1 July 2019.
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.
The winner of the 2021 Clay Research Award is Bhargav Bhatt from the University of Michigan. The award recognises his groundbreaking achievements in commutative algebra, arithmetic algebraic geometry, and topology in the p-adic setting.
His profound contributions include the development, in joint work with M. Morrow and P. Scholze, of a unified p-adic cohomology theory (prismatic cohomology) and, in joint work with J. Lurie, a p-adic Riemann-Hilbert functor. Striking applications of this work include Bhatt’s resolution of longstanding problems in commutative algebra, in particular concerning the Cohen-Macaulay property and Kodaira vanishing up to finite covers. These results have in turn fuelled startling progress on the minimal model program in mixed characteristic.
Photo: Katie Grillaert
Speaker: Camillo De Lellis (IAS)