2016 Abel Prize
Andrew Wiles has been awarded the 2016 Abel Prize “for his stunning proof of Fermat’s Last Theorem by way of the modularity conjecture for semistable elliptic curves, opening a new era in number theory.”
Andrew Wiles has been awarded the 2016 Abel Prize “for his stunning proof of Fermat’s Last Theorem by way of the modularity conjecture for semistable elliptic curves, opening a new era in number theory.”
Simion Filip has been awarded the Dynamical Systems Prize for Young Mathematicians by the Center for Dynamics and Geometry at Penn State University.
We start with a classical problem of Kummer and Darboux about describing surfaces that contain many circles and explain an answer that uses polynomial solutions of quadratic forms over arbitrary fields.
Speaker: János Kollár (Princeton)
Speaker: Bill Minicozzi (MIT)
Speaker: Manjul Bhargava (Princeton)
Abstract: Gauge theories are quantum field theories built directly out of local Lie group symmetry. Conversely, one can view many aspects of representation theory and harmonic analysis of Lie groups through the lens of gauge theory. We will explore the relation between centers and spectral decomposition in representation theory, on the one hand, and local operators and moduli spaces of vacua in gauge theory on the other (without assuming familiarity with either). Contemporary aspects of this relation featuring in the associated workshop include the interplay of geometric representation theory (in particular the geometric Langlands correspondence) with Seiberg-Witten geometry of supersymmetric gauge theories.
Speaker: David Ben-Zvi (UT, Austin)
The second PROMYS Europe summer school opened in Oxford on July 16. It runs until August 20.
For many years, CMI has supported PROMYS, the summer school founded by Glenn Stevens at Boston University. Each summer this brings together high school mathematicians from across America and beyond for six weeks of intense mathematical activity. Selection is highly competitive, the focus and achievements of the participants astonishing. The goal is to provide young mathematicians with the deep experience of mathematical discovery. For many participants, the experience is not a single event: there are opportunities to return for a second year or later on as counselors.
The success of the Boston Program inspired CMI to develop with Glenn Stevens a sister program in Oxford for European students, in partnership with Wadham College and the Mathematical Institute at the University of Oxford, and PROMYS Boston. After two trial years in which students were sent across the Atlantic to participate in the Boston summer school, and then return via a week of masterclasses in Oxford, PROMYS Europe ran for the first time at Wadham College in the summer of 2015.
Image: Julia Banfield
The Clay Mathematics Institute announces two 2016 Clay Research Awards.
The joint Award to Mark Gross and Bernd Siebert is made in recognition of their groundbreaking contributions to the understanding of mirror symmetry, in joint work generally known as the ‘Gross-Siebert Program’. It has its origins in surprising predictions of non-perturbative dualities in string theory: that the properties of certain interesting geometries, notably Calabi-Yau manifolds, are reflected in counter-intuitive ways in partner geometries (‘mirror manifolds’).
The Gross-Siebert program builds on an earlier, differential-geometric, proposal of Strominger, Yau, and Zaslow, in which the Calabi-Yau manifold is fibred by special Lagrangian tori, and the mirror by dual tori. The program’s central idea is to translate this into an algebro-geometric construction in an appropriate limit, involving combinatorial data associated with a degenerating family of Calabi-Yau manifolds. It draws on many areas of geometry, analysis and combinatorics and has made a deep impact on fields such as tropical and non-archimedean geometry, logarithmic geometry, the calculation of Gromov-Witten invariants, the theory of cluster algebras and combinatorial representation theory. Remarkable results independent of mirror symmetry are now emerging, notably in the geometric compactification of moduli spaces of K3 surfaces, in the construction of theta functions on Fano and Calabi-Yau varieties, and in proofs of Looijenga’s conjecture on the smoothability of certain surface cusps and of the positivity of Laurent coefficients conjecture.
The Award to Geordie Williamson is made in recognition of his groundbreaking work in representation theory and related fields.
In particular, the award recognises two major breakthroughs. First, his proof, with Ben Elias, of Soergel’s conjecture on bimodules associated to Coxeter groups. This established the combinatorial result that the coefficients of the Kazhdan-Lusztig polynomials are non-negative, as well as yielding a new proof of Kazhdan and Lusztig’s conjectured character formula for representations of complex semi-simple Lie algebras.
The second is the construction (building on earlier work with Ben Elias and Xuhua He) of counterexamples to the expected bounds in Lusztig’s conjectured character formula for rational representations of algebraic groups in positive characteristics that grow exponentially with the rank of the group.
The awards will be presented at the 2016 Clay Research Conference at Oxford on Wednesday, 28 September.
The Bodleian Library at Oxford holds on deposit a remarkable archive of papers of Lord Byron and his daughter, Ada Lovelace, who is celebrated for her pioneering work on programming Charles Babbage’s Analytical Engine. To mark her 200th birthday on 10 December 2015, the Clay Mathematics Institute announced a project to digitise her mathematical papers in the archive, which includes both sides of her extensive correspondence with Augustus De Morgan as well as other material that gives a fascinating insight into her journey into mathematics.
CMI has now completed the first stage of a project, by posting images and transcripts of the De Morgan correspondence and other papers.
Simion Filip and Tony Yue Yu (pictured) have been awarded Clay Research Fellowships. Simion and Tony were selected for their research achievements and their potential to become leaders in research mathematics. Each has been appointed for a term of five years.
Research Fellow Peter Scholze has won the Fermat Prize 2015 for his invention of perfectoid spaces and their application to fundamental problems in algebraic geometry and in the theory of automorphic forms.