Enhancing Representation Theory, Noncommutative Algebra And Geometry Through Moduli, Stability And Deformations

The whole is far greater than the sum of its parts: a collection of objects exhibits many deeper structures than can be understood by simply investigating its constituent pieces. Our visual intuition endows us with a remarkably powerful tool to perceive the whole. This is geometry. Nonetheless, our deepest understanding couples this with order, with precision, and with calculation. This is algebra. The two viewpoints, when fused together, seek to explain both small-scale and large-scale behaviour, together, as one. It is often only by combining both perspectives that the most insightful understanding of either can be achieved.

Pioneered by the PI and coIs and others, the last two decades have seen a series of spectacular advances coming from our proposal's three main themes: stability (in representation theory and algebraic geometry), noncommutative deformations and enhancements (in noncommutative algebra and algebraic geometry), and moduli of complexes (Bridgeland stability, inspired by string theory). Each of these has individually resulted in some of the stand-out mathematical achievements of the last two decades. But all are reaching the limit of what they can achieve alone.

To take the next step, and to solve the pressing research questions, requires bringing together these approaches. This is what this Programme Grant will achieve. The PI and coIs, together with the mathematical expertise at our three institutions and the specialist collaboration of many mathematicians nationally and internationally whom we have enlisted, form an inspiring team with a unique expertise and breadth that straddles much of algebra and geometry. We are enthusiastic because we can now see the same structures arising independently and for separate reasons across different parts of mathematics, which suggests the existence of deep hidden connections. The history of science is filled with such examples, such as the discovery of the theory of symmetries (group theory) in mathematics and in quantum physics. Our own work brings several examples: wall-crossing arising independently in representation theory and in algebraic geometry; the use of noncommutative algebra found at the same time in geometric representation theory and the minimal model programme.

Everyone in our team has particular experience of applying their skills in creative and original ways to problems beyond our own specialism. We are therefore motivated not only by the progress that we expect to make on known unanswered questions, but also by the applications that we cannot yet predict. We believe that by pushing forward the mathematical state-of-the-art, and by reaching out to other disciplines, our proposal will maximise its potential, and through this it will shape and influence a broad range of future problems.

We have extensive plans to maximise the impact of our work in this proposal, within mathematics and more broadly in academia, as well as to the general public and policy makers. We are developing and synthesising exciting new areas in mathematics, many of which are only beginning to be uncovered now, and the potential benefits are substantial.

We envisage there being several different types of impact.

1. Within Pure Mathematics.

This grant will channel the individual expertise accumulated in our three institutions to extend and deepen the connections between algebra, geometry and topology, leading to new areas of research, new points of interaction, and substantial progress on important existing problems. We will host two international conferences, two workshops, and a research school in the UK; in conjunction with our Project Partners IPMU, Toronto, Tsinghua and UIC, we will apply for multiple sources of funding to organise a series of international conferences. This will be a way of reaching, and showcasing, our programme to a broader international audience than otherwise possible.

2. Knowledge Exchange.

We will work to make our research have the greatest possible impact beyond our own disciplines. We will build on our existing industrial relationships and contacts, including with the Innovation Centre for Data Science, DataLab, and the software company Wolfram, through Christian Korff in Glasgow. These relationships will be guided by the development of our research over the duration of the grant, and we will continually be on the lookout for new opportunities. To foster new links, we will host two sandpit events, which will allow us to bring together a broad range of potential users. We have already identified and contacted some potential partners, from the Centre for Signal and Image Processing at the Technology and Innovation Centre (TIC) in Glasgow city centre, to both TechCube and Codebase in Edinburgh. We will learn from previously successful sandpits hosted by the MIGSAA CDT in Edinburgh, and we will ensure that these conversations evolve into a genuine and sustainable two-way interaction.


3. Outreach.

We shall have a wide-ranging public outreach programme focussed around our research and its objectives, aimed at school children (through roadshows - we have experience of creating exhibits already) and the general public (through Science Festivals, social media and other traditional forms of media). We believe this will have a 3-fold effect: in the short term it will increase public awareness of the vitality and importance of mathematics as an end in itself and as the underpinning technology of much of the modern world; in the medium term it will encourage more young people to take up scientific or quantitative professions; and in the long term it will help to drive innovation in the UK economy. We also intend to harness this grant and its scope and vision to promote the case of mathematics to policy makers, at the local, devolved and national level, through our PI and Outreach Officer seeking opportunities to engage in the debate about the role core research plays to the long-term prosperity of our society.

4. The People Pipeline.

As an immediate impact on individuals, this grant will also create a new network of local, national and international collaborations, through our international Project Partners, which will persist well beyond the end of the grant, to the benefit of UK science. It will employ 6 PDRAs and at least 6 PhD students, all of whom will acquire broad mathematical expertise and be given unique training opportunities through our international partners. And through our sandpit events and internships, our PDRAs and PhD students will acquire the intradisciplinary expertise and the outward-facing skills necessary for both successful careers in academia, and the wider economy.