2025 Archives - Clay Mathematics Institute

Mehtaab Sawhney

Mehtaab Sawhney will receive his PhD from the Massachusetts Institute of Technology in 2024, under the supervision of Yufei Zhao.

While still a graduate student, Sawhney has achieved a stunning number of breakthroughs on fundamental problems across extremal combinatorics, probability theory, and theoretical computer science. He is a highly collaborative researcher whose partnership with Ashwin Sah has been particularly fruitful. His remarkable body of work has already transformed swathes of combinatorics. For example, working with Kwan, Sah and Simkin, he proved a 1973 conjecture of Erdős on the existence of high-girth Steiner triple systems; with Keevash and Sah he established the existence of subspace designs; with Jain and Sah he established sharp estimates for the singularity probability in a wide class of discrete random matrices; with Sah and Sahasrabudhe he showed the existence of the spectral distribution of sparse directed Erdős–Rényi graphs; and with Kwan, Sah and Sauermann, he developed highly novel tools in anti-concentration in order to prove the Erdős- McKay conjecture concerning edge statistics in Ramsey graphs.

Mehtaab was appointed as a Clay Research Fellow for a term of five years from 1 July 2024.

Photo: Mehtaab Sawhney

Ishan Levy

Ishan Levy will receive his PhD from the Massachusetts Institute of Technology in 2024, under the supervision of Michael Hopkins.

Levy is known for his deep and ingenious contributions to homotopy theory. His new techniques in algebraic K-theory have led to solutions of many old problems. In joint work with Burklund he established the rational convergence of the “Waldhausen tower” interpolating between the K-theory of the integers and one of the most important moduli spaces in the study of high dimensional manifolds, A(pt). He is most renowned for his work Ravenel’s “Telescope Conjecture.”

In the late 1970s Ravenel made a series of deep conjectures outlining a rich conceptual vision of stable homotopy. By the mid 1980s all but the Telescope Conjecture had been proved. For over 40 years this remained the most important problem in this part of homotopy theory. Levy’s methods in K-theory led him, Burklund, Hahn and Schlank, to the construction of counterexamples, and, in joint work with Burklund, Carmeli, Hahn, Schlank and Yanovski, to an estimate of the growth rate of the stable homotopy groups of spheres that was completely untouchable by previous methods.

Ishan was appointed as a Clay Research Fellow for a term of five years from 1 July 2024.

Photo: Archives of the Mathematisches Forschungsinstitut Oberwolfach

Huy Tuan Pham

Huy Tuan Pham will receive his PhD in 2023 from Stanford University, where he is advised by Jacob Fox.

Pham is a highly inventive and prolific researcher who has already made fundamental contributions to combinatorics, probability, number theory, and theoretical computer science. While still an undergraduate, he showed with Fox and Zhao that Green’s popular difference theorem, an extension of Roth’s theorem on arithmetic progressions in dense sets of integers, requires tower-type bounds – the first known application of Szemerédi’s regularity method that truly requires tower-type bounds. Subsequently, with Park, he proved the Kahn-Kalai conjecture on the location of phase transitions and Talagrand’s conjecture on selector processes; with Conlon and Fox, he solved various long-standing conjectures of Erdős in additive combinatorics concerning subset sums and Ramsey complete sequences; and with Cook and Dembo, he developed a quantitative nonlinear large deviations theory for random hypergraphs.

Huy Tuan Pham has been appointed as a Clay Research Fellow for five years beginning 1 July 2023.

Paul Minter

Paul Minter obtained his PhD in 2022 from the University of Cambridge, advised by Neshan Wickramasekera. Since then he has been a Veblen Research Instructor at Princeton University/IAS and a Junior Research Fellow at Homerton College, Cambridge.

Minter works in Geometric Measure Theory, tackling regularity and compactness questions for minimal hypersurfaces in Riemannian manifolds. He has advanced the subject by introducing powerful new techniques for analysing singularities of measure-theoretically defined minimal hypersurfaces with stable regular part, establishing, in particular, the uniqueness of classical tangent cones when they arise, and (with Wickramasekera) the uniqueness of tangent hyperplanes at branch points, in the absence of lower density classical singularities nearby. His remarkably general results – the first with no restrictions on multiplicity or the dimension of the hypersurface – provide a much sought-after extension to what is known about uniqueness of tangent cones and the asymptotic behaviour of minimal submanifolds near singularities. Applications include an understanding of the structure of area minimising hypersurfaces mod p, for any even integer p, near a singular point with a planar tangent cone.

Paul Minter has been appointed as a Clay Research Fellow for four years beginning 1 July 2023.

Photo: Dan Komoda, Institute for Advanced Study

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.

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.

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.

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