RIGID STRUCTURE IN NONCOMMUTATIVE, GEOMETRIC & COMBINATORIAL PROBLEMS

The original research proposal had five objectives. Four have been to some extent completed by a number of mathematicians, including myself. Most of the outputs I list here are involved with these objectives, although there are a couple that I will mention that are involved in other topics.
Objective 1. "Establish a connection between Gaudin Model and Cherednik algebras of type A". This was carried out by Mukhin-Tarasov-Varchenko over a number of papers, and followed up by my PhD student (during part of the fellowship) Gwyn Bellamy. All predicted was essentially correct, and I have on-going work with one of the RAs from the fellowship which pushes this much further and is the subject of the next Objective.
Objective 2. "Confirm conceptually the generalised Calogero-Moser space conjectures of [22]". A book by Bonnafe-Rouquier is in press that sets out a generalisation of the conjectures of [22] to a geometric construction of left cells for complex reflection groups. Together with my RA, Adrien Brochier, we are writing up a paper which makes definitive progress on this problem, exactly through the methods described in the research proposal, plus several new ingredients: cactus group, asymptotic version of Drinfeld-Kohno theorem, theory of crystal bases.
Objective 3. "Develop tools of hamiltonian reduction, microlocal analysis and noncommutative geometry on the Hilbert scheme". This topic has moved rapidly in the last 6 years, with great progress in understanding my own localisation theorem, placing it in a more conceptual setting that applies to a wide class of symplectic singularities, particularly in work of McGerty-Nevins and Braden-Proudfoot-Webster. I proved what was referred in my proposal as a _key_ goal of the theory of Cherednik algebras by giving a new conceptual proof of the existence and properties of the Procesi bundle (this is publication "Macdonald positivity via the Harish-Chandra D-module"). My paper "Auslander-Gorenstein property for Z-algebras" establishes many basic homological properties for the noncommutative algebraic geometry of the Hilbert scheme, and the paper "Differential Operators and Cherednik Algebras" proves the fundamental theorem that allows comparison of the algebraic and geometric approaches to localisation for Hilbert schemes.
Objective 4. "Extend application to other important representation theoretic settings". Together with Losev, I wrote the most important paper on rational Cherednik algebras of type G(r,1,n), namely "On category O for cyclotomic rational Cherednik algebras". This paper confirmed 2 conjectures (of Etingof and of Rouquier) and unified several techniques from deformation quantization, algebraic combinatorics, Lie theory in order to prove the definitive results for this topic.
Objective 5. "Create a general definition of G-Hilb" There has been little progress on the topics mentioned within this objective.
Beyond these objectives I would also like to mention my paper, written jointly with Stephen Griffeth, on "Catalan numbers for complex reflection groups" which settled and proved a more general version of a beautiful conjecture of Bessis-Reiner on generalising of the classical Catalan numbers to complex reflection groups. This has been an impetus for the founding of a new topic "Rational Catalan combinatorics" which was recently the focus of a week-long conference at AIM in Palo Alto.
There were several developments arising from the Leadership Fellowship which are not in the form of outputs, but which I consider nonetheless to be very valuable.
Research Assistants: the grant supported a number of RAs, all of whom have gone to other academic jobs: Stephen Griffeth (permanent position in Talca, Chile); Will Donovan (postdoctoral fellowship at IPMU, Kavli Institute, Tokyo); Adrien Brochier (Whittaker Fellow at the University of Edinburgh), Francois Petit (Research associate, University of Luxembourg). The presence of these young researchers was a remarkably strong positive influence on the environment in the algebra and geometry group in Edinburgh. The group attracted a number of other young researchers, arriving on their own three-year fellowships (Maria Chlouveraki, Guillaume Pouchin), making the department one of the leading and most vibrant centres in Europe now for representation theory and algebraic geometry.
New hires and development of Hodge Institute: my Leadership Fellowship was a catalyst for the development of the algebra and geometry groups in Edinburgh, and there eventual unification as the Hodge Institute. This group now has twelve permanent members, and as a measure of its success four are currently enjoying five-year research fellowships sponsored by either EPSRC or ERC, and another has an ERC interview in about one month's time. The Hodge Institute is now an internationally leading presence in algebra and geometry, with a vital and sustainable postdoctoral and PhD programme.
PhD students: During the fellowship I have supervised five PhD students, three of whom have already graduated. Two have gone on to academic positions: Bellamy a permanent position in Glasgow, and Jenkins a research assistantship in Campinas, Brazil. The third, Spencer, is now a trainee actuary for Zurich, based in Swansea.
Personal profile: During the fellowship I gave an invited lecture at the ICM, served on the EPSRC Mathematics Programme SAT and on the REF Mathematics subpanel, and became one of the two principal editors for the Proceedings of the London Mathematical Society. I have also been the organiser for several important international conferences, including ones in Luminy, MSRI and Oberwolfach.
Public Engagement: I was awarded a sum of money for public engagement, and this was the basis for an exhibition for school-children at the Museum of Scotland at the Edinburgh International Science Festival on Moebius Geometry. The material produced for this is now used in the Science Faculty's Sci-Fun project, who travel to schools throughout the country to enthuse pupils with science.