Algebraic and motivic vector bundles
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Speaker: Mike Hopkins (Harvard)
Speaker: Mike Hopkins (Harvard)
Research Fellow Peter Scholze has been awarded the 2015 Ostrowski Prize for his breakthrough work in arithmetic algebraic geometry. He developed the theory of perfectoid spaces and successfully applied the theory to address a number of difficult open questions. He proved Deligne’s weight monodromy conjecture for varieties that are nonsingular complete intersections in projective space; this represents the first major progress in the last 30 years toward’s Deligne’s conjecture. He also used perfectoid spaces to establish p-adic Hodge theory for rigid analtytic spaces. Further, with Weinstein, he showed that Rapoport-Zink spaces at infinite level are perfectoid spaces.
The Clay Mathematics Institute has launched PROMYS Europe in collaboration with Wadham College and the Mathematical Institute at the University of Oxford. PROMYS Europe is an extension of the very successful 26-year-old PROMYS program created by Glenn Stevens at Boston University. It is a challenging summer school designed to encourage very able and mathematically ambitious secondary school students to explore the creative world of mathematics. Carefully selected pre-university students from around Europe have gathered in Oxford for six weeks of rigorous mathematical activity.
Research Fellow James Maynard has been awarded a 2015 LMS Whitehead Prize for his spectacular results on gaps between prime numbers. He simplified and extended the work of Zhang on bounded gaps between primes, then made the most substantial advance for 75 years on how large the gap between consecutive primes can be, in particular answering a 10,000 dollar conjecture of Erdős.
The 2015 Clay Research award has been made jointly to Larry Guth and Nets Katz for their solution of the Erdős distance problem and for other joint and separate contributions to combinatorial incidence geometry. Their work is an important contribution to the understanding of the interplay between combinatorics and geometry.
Larry Guth is a professor of mathematics at MIT, having moved there from the Courant Institute in 2012. He received his PhD in 2005 from MIT under Tomasz Mrowka. Originally working in metric and systolic geometry, he has since made major contributions to harmonic analysis and combinatorics. In particular, he developed ideas introduced by Dvir to prove the multilinear Kakeya estimate (originally due to Bennett, Carberry and Tao) and then, with Bourgain, applied them to obtain new results on the ‘restriction conjecture’ in harmonic analysis.
Nets Katz has been a professor of mathematics at Caltech since 2013, having previously been at Indiana University. He received his PhD in 1993 from the University of Pennsylvania, under Dennis DeTurck. His earlier work was in harmonic analysis, but more recently he has diversified into combinatorics and PDEs. He has made important contributions in additive number theory, with Bateman on the ‘cap set problem’ and with Bourgain and Tao in their joint proof of the Bourgain–Katz–Tao sum product theorem.
The distance problem was posed by Paul Erdős in 1946. It concerns the distribution of distances determined by a set of n points in a metric space. At most how many times can the same distance (say the unit distance) occur? What is the minimum number of distinct distances that can occur?
In 2010, Guth and Katz published a spectacular breakthrough in which they gave a near-optimal answer to the second question for points in the plane, by proving an almost tight lower bound of the order of n / log n. Their paper built on a novel approach to the problem suggested by Elekes and Sharir, and also on the earlier work on the Kakeya problem
The first Clay Award for Dissemination of Mathematical Knowledge has been made to Étienne Ghys in recognition of his own important contributions to mathematical research and for his distinguished work in the promotion of mathematics.
Étienne Ghys is a CNRS Directeur de Recherche at ENS, Lyon. He has published outstanding work in his own fields of geometry and dynamics, both under his own name and under the collaborative pseudonym “Henri Paul de Saint Gervais”—contributions recognised by invitations to speak at the International Congress in 1990 and by his elevation to the French Académie des Sciences in 2004. He has also given invaluable service to the international mathematical community in many contexts, as a member of the program committee for the ICM in Hyderabad, as a member of the Fields Medal committee in 2014, and through service on many other bodies.
But it is through his work in the promotion of mathematics in France and elsewhere that he has become a legend. He has given numerous carefully crafted lectures to audiences ranging from school children to delegates at the International Congress in 2006, when he gave a beautiful and exceptionally clear plenary lecture on Knots and dynamics. He has enthusiastically embraced modern technology to aid the exposition of deep ideas, for example during his editorship of Images des mathématiques, which he transformed to an online publication in 2009, and which received more than five million visits over his five-year term of office. He himself has written more than 90 articles for Images, as well as a monthly column in Le Monde.
He created with others the Maison de mathématiques et informatique in Lyon and co-founded, with Dierk Schleicher, the International summer school of mathematics for young students. His series of films, produced with Aurélien Alvarez and Jos Leys and published as DVDs and online in many languages, has had a huge impact on high school students. The first, Dimensions, has been downloaded more than a million times.
The award will be presented after a public lecture by Professor Ghys in Oxford on October 1, 2015.
Image: F. Caterini
John V Pardon has been awarded a Clay Research Fellowship. He studied as an undergraduate at Princeton and is currently working with Yakov Eliashberg at Stanford University.
His most recent work concerns the construction of virtual fundamental cycles on moduli spaces of holomorphic curves in symplectic geometry. He is also interested in geometry and low-dimensional topology. John received his AB in Math from Princeton University in 2011. John has been appointed as a Clay Research Fellow for a term of five years beginning 1 July 2015.
James Maynard has been awarded a Clay Research Fellowship. He obtained his doctorate at Oxford in 2013 under the supervision of Roger Heath-Brown. He is currently a Fellow by Examination at Magdalen College, Oxford.
James is primarily interested in classical number theory, in particular the distribution of prime numbers. His research focuses on using tools from analytic number theory, particularly sieve methods, to study primes. James has been appointed as a Clay Research Fellow for a term of three years beginning 1 July 2015.