## Aravind Asok

Professor Aravind Asok has been appointed as a Clay Senior School to participate in the IAS/PCMI program Motivic Homotopy Theory.

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Professor Aravind Asok has been appointed as a Clay Senior School to participate in the IAS/PCMI program Motivic Homotopy Theory.

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Professor Fabien Morel has been appointed as a Clay Senior School to participate in the IAS/PCMI program Motivic Homotopy Theory.

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The Clay Mathematics Institute is pleased to announce that Ishan Levy and Mehtaab Sawhney have been awarded Clay Research Fellowships.

Ishan Levy** **will receive his PhD from the Massachusetts Institute of Technology in 2024, under the supervision of Michael Hopkins. Ishan has been appointed as a Clay Research Fellow for five years beginning 1 July 2024.

Mehtaab Sawhney** **will receive his PhD from the Massachusetts Institute of Technology in 2024, under the supervision of Yufei Zhao. Mehtaab has been appointed as a Clay Research Fellow for five years beginning 1 July 2024.

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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

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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

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Professor André Neves has been appointed as a Clay Senior Scholar to participate in New Frontiers in Curvature: Flows, General Relativity, Minimal Submanifolds, and Symmetry at the Simons Laufer Mathematical Research Institute.

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Professor Vincent Guedj has been appointed as a Clay Senior Scholar from August to December 2024 to participate in Special Geometric Structures and Analysis at the Simons Laufer Mathematical Research Institute.

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Professor Mikhail Kapranov has been appointed as a Clay Senior Scholar from January to May 2024 to participate in Noncommutative Algebraic Geometry at the Simons Laufer Mathematical Research Institute.

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Professor Bernd Ulrich has been appointed as a Clay Senior Scholar from January to May 2024 to participate in Commutative Algebra at the Simons Laufer Mathematical Research Institute.

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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.

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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

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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.