The Homological Minimal Model Programme (Extension)

A surprisingly large proportion of the natural world, with its incredibly rich structure, systems and complex lifeforms, can be understood using scientific principles that are either underpinned, controlled, or can be approximated by, mathematical objects called polynomials. These fundamental objects are built on solid, long-standing mathematical foundations, and have the advantage of being able to describe relationships with both precision and with grace. Their deceptively simple form, however, often masks a deep and complicated underlying geometry, which in turn often exhibits very counter-intuitive behaviour.

This proposal lies within the framework of polynomials, and their attached geometric varieties. It seeks to answer a series of open questions in birational surgeries, their classification, and enumerative questions by using newly constructed noncommutative invariants, and using the additional structure that these encode to distinguish geometric objects to a much finer degree.

The first part of this proposal seeks classification of geometric structures via noncommutative techniques, and in process proposes an ADE classification of certain Jacobi algebras. This, and previous work, strongly suggests results in other areas, and the second part of the project involves these, from generating sets of the pure braid groups, to new combinatorial tilings of the plane which conjecturally control many structures in both algebra and geometry, with strong links to Coxeter groups. The third part unifies and generalises into a wider framework, which involves understanding enumerative and structural questions for both singular flops, and for flips.

The primary foreseeable impact of this project will be measured through its scientific outputs and influence, within mathematics, through the work of many researchers worldwide. It will produce the highest quality, internationally leading research. Whilst in the long-term, fundamental research generates huge impacts outside academia, at this early stage this potential impact is difficult and impossible to gauge, and so the short-term focus is to maximise the scientific impact and dissemination, through the twelve specific strategies outlined in the Pathways to Impact.

The proposal has additional impact, and benefit, through:

(1) Establishing and maintaining a unique world leading activity. The scientific case, built on my pioneering work with noncommutative techniques in geometry, has put the UK on the world map in this area.

(2) It uniquely links several areas where the UK is world-leading. I now have a substantial track record in solving problems in the wide range of subjects that the proposal interacts with, including the world-leading activities of (amongst others) Bridgeland, Joyce, Thomas, Corti, Smith and Smoktunowicz.

(3) It will help me to build a new world-leading group in this area in Glasgow.

Impact will be further achieved as the proposal comprehensively meets all three of the key strategies laid out in the 2015 Strategic Review. It will Build Leadership through supporting me in this crucial juncture in my career, and also through attracting the best international researchers as RAs, and through training them in its unifying perspective will equip them with broad expertise; it supports Balancing Capability principally by strengthening under-represented algebraic directions such as commutative algebra, but also maintaining and strengthening capability throughout a wide range of its portfolio; it will help Accelerating Impact of the highest quality research by a fast dissemination of results within the academic community, through the twelve specific strategies laid out in the Pathways to Impact.

As my track record outlined in the Case for Support demonstrates, there are significant and naturally-occurring interdisciplinary interactions in my work, and my unifying principle that noncommutative algebra controls many geometric processes has seen spectacular progress over the last four years. This fellowship extension would allow me to really press home these research advances when there are obvious timelines advantages, and to truly maximise its impact.