The 2017 Clay Research Award was made to Maryna Viazovska in recognition of her groundbreaking work on sphere-packing problems in eight and twenty-four dimensions. In particular, her innovative use of modular and quasimodular forms, which enabled her to prove that the E8 lattice is an optimal solution in eight dimensions.
The result had been suggested by earlier work of Henry Cohn and Noam Elkies, who had conjectured the existence of a certain special function that would force the optimality of the E8 lattice through an application of the Poisson summation formula. Viazovska’s construction of the function involved the introduction of unexpected new techniques and establishes important connections with number theory and analysis. Her elegant proof is conceptually simpler than that of the corresponding result in three dimensions.
She subsequently adapted her method in collaboration with Henry Cohn, Abhinav Kumar, Stephen Miller, and Danylo Radchenko to prove that the Leech lattice is similarly optimal in twenty-four dimensions.
The 2017 Clay Research Award was made to Aleksandr Logunov and Eugenia Malinnikova in recognition of their introduction of a novel geometric combinatorial method to study doubling properties of solutions to elliptic eigenvalue problems.
This has led to the solution of long-standing problems in spectral geometry, for instance the optimal lower bound on the measure of the nodal set of an eigenfunction of the Laplace-Beltrami operator in a compact smooth manifold (Yau and Nadirashvili’s conjectures).
The 2017 Clay Research Award was made to Eugenia Malinnikova and Aleksandr Logunov in recognition of their introduction of a novel geometric combinatorial method to study doubling properties of solutions to elliptic eigenvalue problems.
This has led to the solution of long-standing problems in spectral geometry, for instance the optimal lower bound on the measure of the nodal set of an eigenfunction of the Laplace-Beltrami operator in a compact smooth manifold (Yau and Nadirashvili’s conjectures).
The 2017 Clay Research Award was made to Jason Miller and Scott Sheffield in recognition of their groundbreaking and conceptually novel work on the geometry of Gaussian free field and its application to the solution of open problems in the theory of two-dimensional random structures.
The two-dimensional Gaussian free field (GFF) is a classical and fundamental object in probability theory and field theory. It is a random and Gaussian generalized function hdefined in a planar domain D. Despite its roughness and the fact that it is not a continuous function, it possesses a spatial Markov property that explains why it is the natural counterpart of Brownian motion when the time-line is replaced by the two-dimensional set D. Miller and Sheffield have studied what can be viewed as level-lines of h and more generally flow lines of the vector fields exp(iah), where a is any given constant. This framework, which they call imaginary geometry, allows them to embed many Schramm-Loewner Evolutions (SLE) within a given GFF. A detailed study of the way in which the flow lines interact and bounce off each other allowed Miller and Sheffield to shed light on a number of open questions in the area and to pave the way for further investigations involving new random growth processes and connections with quantum gravity.
The 2017 Clay Research Award was made to Scott Sheffield and Jason Miller in recognition of their groundbreaking and conceptually novel work on the geometry of Gaussian free field and its application to the solution of open problems in the theory of two-dimensional random structures.
The two-dimensional Gaussian free field (GFF) is a classical and fundamental object in probability theory and field theory. It is a random and Gaussian generalized function hdefined in a planar domain D. Despite its roughness and the fact that it is not a continuous function, it possesses a spatial Markov property that explains why it is the natural counterpart of Brownian motion when the time-line is replaced by the two-dimensional set D. Miller and Sheffield have studied what can be viewed as level-lines of h and more generally flow lines of the vector fields exp(iah), where a is any given constant. This framework, which they call imaginary geometry, allows them to embed many Schramm-Loewner Evolutions (SLE) within a given GFF. A detailed study of the way in which the flow lines interact and bounce off each other allowed Miller and Sheffield to shed light on a number of open questions in the area and to pave the way for further investigations involving new random growth processes and connections with quantum gravity.
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.
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 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.
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.
Aaron Pixton received his Ph.D. in 2013 from Princeton University under the supervision of Rahul Pandharipande. His research is in enumerative algebraic geometry. The topics he has worked on recently include the tautological ring of the moduli space of curves, moduli spaces of sheaves on 3-folds, and Gromov-Witten theory. Aaron has been appointed as a Clay Research Fellow for a term of five years beginning 1 September 2013.