Geometric Representation Theory and W-algebras

In the early nineteenth century it was impossible to draw a distinction between mathematicians and physicists, since the greatest scientists worked on every important problem in these fields. One of the most influential polymaths of the era was Emmy Noether, who developed all of the foundational theories which inspired this research project. Her greatest contribution to physics was probably Noether's first theorem, which says that if you want to understand the conservation laws of the universe then it suffices to understand the symmetries of the universe. Conservation laws are the most fundamental laws of physics, giving us clues about the nature of matter, and the shape of space, and so Noether's theorem started a wave of discovery which has been growing and growing for over a hundred years, as mathematicians and physicist seek to understand the symmetries of the universe.

Today mathematicians and physicists are much easier to distinguish, however the subjects are still deeply intertwined. In modern day mathematical language, the study of symmetries is called representation theory and the goal of this project is to understand how Noether's algebraic structures can be expressed as symmetries. To rephrase this, my objective is to understand the representations of certain important families of algebras.

The ancient Greeks believed that all matter could be built up from indivisible pieces - the word "atom" literally means "indivisible" - and in the language of modern particle physics it is well-understood that all matter in the universe can be built up from the fundamental particles. In precisely the same way, the representations I seek to understand are also built from fundamental building blocks, known as irreducible representations. Can we describe these irreducible representations explicitly? Can we determine their structure and calculate their dimensions? In this research project we will answer these fundamental, elusive questions by relating each representation to an important geometric space, known as a symplectic leaf of a Poisson variety.

Some of the most important unanswered questions in this field pertain to algebras which we call "modular": this is because the underlying number system is not linear, like the real number line, but is circular like the numbers on the face of a clock. Questions in modular representation theory tend to be significantly harder due to the added complexity of the geometry and the arithmetic.

By working with tools on the interface between abstract algebra and geometry this project will make substantial exciting progress in some of the most challenging problems in modular representation theory, showing that Noether's wave of discovery is still growing on the ocean of mathematics.

Knowledge: This research will result in broad new bodies of knowledge by building bridges of understanding between the fields of algebra, geometry and conformal field theory, using the language of representation theory.

Impact Goal: To make substantial progress in the representation theory of Lie theoretic algebraic structures.

Impact actions: 1) I will solve interesting open problems in geometric representation theory using tools from algebra and Poisson geometry, combining every single thread of my previous research experience.
2) I will open up new fields of research by constructing and conducting in-depth studies of modular affine W-algebras.
3) By pursuing research objectives interacting with diverse fields of research, I will publish in high-profile general-interest journals.

People: The research communities in the fields of Lie theory, representation theory, Poisson geometry, and conformal field theory will be the primary point of impact from this project. My collaborators, project partners, visiting researchers and PDRA will be the ones most acutely affected, as well as the web of researchers with whom they are connected.

Impact Goal: To influence researchers in representation theory and surrounding fields with a synthesis of methodology, encouraging further interdisciplinary research.

Impact actions: 1) I will publicise my results to researchers in representation theory, Poisson geometry and beyond by giving talks at interdisciplinary conferences.
2) I will hold a workshop in 2022 inviting experts in geometric representation theory and vertex algebra theory to interact and discuss new bridges to Poisson geometry.
3) My collaborations and partnerships with strong researchers will allow me to disseminate my work widely by giving research seminars in a range of leading international institutions. The researchers who visit me will also give seminars at Kent leading to further transfer of knowledge.
4) I will write two survey articles towards the end of the proposed four-year fellowship, rendering my work more accessible to experts from surrounding fields.
5) I will embark on the supervision of a postdoctoral research associate at the University of Kent, collaborating on the third research program of my proposal. On top of this I will supervise Masters students in research-led projects in non-commutative geometry.

Society: Members of the general public who have an interest to understand the context and nature of contemporary mathematics will be given the opportunity to learn the background and objectives of my research.

Impact Goal: To educate final year school children and interested adults in the rudiments of Lie theory, leading to an understanding of some of my research objectives.

Impact actions: 1) I will attend the Royal Society's Public Engagement course and design a public engagement workshop based on my research objectives with the help of the dedicated outreach officer at the University of Kent. I will then conduct interactive workshops with final year school students.
2) I will give informal talks about my research at the Pint of Science Festival held annually in May.
3) I will collaborate with digital musician Daniel Ross to produce musical compositions inspired by mathematical concepts.

Timescales: The expected timescales for the the various research objectives of the project to be achieved are listed in my diagrammatic work plan. The immediate beneficiaries closely related to the body of research will experience impact soon after the results are released. The time takes for the research to influence their extended networks of researchers will be longer. My survey articles will be released in months 38 and 44 of the fellowship, rendering my results immediately more accessible. The general public will be effected by my research program at each of my outreach activities, which are planned in months 20, 26 and 32, as well as every May at the Pint of Science festival.