Viewing archives for Research Fellows

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
 

Xinyi Yuan

Xinyi Yuan received his PhD from Columbia University in 2008 under the supervision of Shou-Wu Zhang.  In his 2006 preprint, “Big Line Bundles over Arithmetic Varieties,” he proves an arithmetic analogue of a theorem of Siu and derives, among other consequences, a natural sufficient condition for when the orbit under the absolute Galois group is equidistributed.  Xinyi was apponted as a Clay Research Fellow for a term of three years beginning July 2008.

Tony Yue Yu

Tony Yue Yu received his PhD in 2016 from Université Paris Diderot under the supervision of Maxim Kontsevich and Antoine Chambert-Loir. He works on non-archimedean geometry, tropical geometry and mirror symmetry. He aims to build a theory of enumerative geometry in the setting of Berkovich spaces. Such a theory will give us a new understanding of the enumerative geometry of Calabi-Yau manifolds, as well as the structure of their mirrors. It is also intimately related to the theory of cluster algebras and wall-crossing structures.  Tony has been appointed as a Clay Research Fellow for a term of five years beginning 1 September 2016.

Photo by Jindie Mi.

Teruyoshi Yoshida

Teruyoshi Yoshida received his PhD in 2006 from Harvard University under the supervision of Richard Taylor. His mathematical interest is in the interface between automorphic forms and arithmetic algebraic geometry, with much of his work concerned with the geometric structure of Shimura varieties at places of bad reduction. Teruyoshi was appointed as a Clay Research Fellow for a term of three years beginning December 2007.

Alex Wright

Alex Wright received his PhD in 2014 from the University of Chicago under the supervision of Alex Eskin. His recent work concerns dynamics on moduli spaces and special families of algebraic curves that arise in this context. His interests include dynamics, geometry, and especially ergodic theory on homogenous spaces and Teichmüller theory. Alex received his BMath from the University of Waterloo in 2008. He was appointed as a Clay Research Fellow for a term of five years beginning 1 July 2014.

Miguel Walsh

Miguel Walsh was born in Buenos Aires, Argentina. He received his “Licenciatura” degree in 2010 from Universidad de Buenos Aires and his PhD from the same institution in 2012, under the supervision of Román Sasyk. During this period he held a CONICET doctoral fellowship. He is currently based at the University of Oxford. His research so far has focused on inverse problems in arithmetic combinatorics, the limiting behaviour of ergodic averages and the estimation of rational points on curves. Miguel has been appointed as a Clay Research Fellow for a term of four years beginning 1 July 2014.

Akshay Venkatesh

Akshay Venkatesh completed his undergraduate degree at the University of Western Australia, Perth, and received his PhD from Princeton University in 2002 under the direction of Peter Sarnak. His mathematical interests center around number theory and automorphic forms.  He is particularly interested inequidistribution questions on homogeneous spaces, and the interplay between ergodic and spectral techniques.  Akshay was appointed as a Clay Research Fellow for a term of two years beginning 2004.

András Vasy

András Vasy received his PhD from the Massachusetts Institute of Technology in 1997 under the supervision of Richard Melrose. The focus of his research program is partial differential equations in the presence of an additional geometric structure, of a higher rank type.  Examples include scattering theory both for N-body quantum Hamiltonians and on higher rank symmetric spaces, and wave propagation on manifolds with corners.  In 2002 he received the Alfred P. Sloan Research Fellowship.  András was a appointed as a Clay Research Fellow for a term of two years beginning 2004.

Jack Thorne

Jack Thorne was born in 1987 in Hereford, England. He received his BA at the University of Cambridge in England. He has since studied at Harvard University and Princeton University under the direction of Richard Taylor and Benedict Gross. He received his PhD in May 2012. His primary research interests are algebraic number theory and representation theory, and the diverse connections between these two subjects. Most recently he has been interested in using automorphy lifting techniques to establish new cases of the Fontaine-Mazur conjecture. Jack was appointed as a Clay Research Fellow for a term of five years beginning July 2012.