Symplectic Representation Theory

This proposal will apply powerful tools and techniques in geometry to solve certain problems in representation theory, a major branch of algebra interacting strongly with geometry and mathematical physics. Pure mathematics aims to abstract and distil the essence of familiar concepts: for instance, in the case of symmetries this leads to the definition of a group, the collection of symmetries of a given object. However, the mathematical definition is far more axiomatic, to the point that the underlying object that the group is describing all but disappears. In these cases it is important to try to recover this object, or more specifically to find all objects whose symmetries give rise to the group in question. This is the motivating idea behind representation theory. Despite this seemingly abstract problem, representation theory is crucially important in many areas of science such as physics (e.g. string theory / mirror symmetry), chemistry (study of molecular vibrations) and computer science, as well as being central for mathematics.

Algebra and geometry have been kindred spirits from the very conception of modern mathematics, with ideas and motivating problems passing to and fro all the time. For instance, continuous groups, the bedrock of Lie theory and modern representation theory, came to prominence thanks to Sophus Lie's program applying algebra to the study and classification of geometries. Since the conception of Lie theory, geometry has played a key role, time and again, in moving the subject forward. Conversely, commutative and homological algebra has been pivotal in the modern development of algebraic geometry, enabling the giants, such as Grothendieck, to rebuild the subject on firm mathematical foundations.

In this intradisciplinary proposal we aim once again to exploit powerful geometric results in the study of algebra, this time by developing the foundations of a theory of mixed Hodge structures on conic symplectic manifolds, thereby bringing the theory of mixed Hodge structures to bear on a host of (seemingly intractable) problems in representation theory. We also expect that the development of such a theory would also have myriad applications to the understanding of the geometry of conic symplectic manifolds.

The first key step to develop this theory of mixed Hodge structures on Deformatio-Quantization (DQ)-modules, is to generalise the construction of nearby and vanishing functors for D-modules to this setting. Secondly, we will use these functors to reconstruct the categories of interest as categories glued out of simpler subquotients. We also propose to develop a geometric analogue of Soergel's V-functor in this setting, allowing us to apply Rouquier's theory of quasi-hereditary covers to DQ-modules.

Through funding provided by the EPSRC first grant, my goal is to disseminate the underlying ideas, and mathematical philosophy, of my work to two key groups of people who lie outside the academic community immediately related to my work. The first group is that of Ph.D students, and early career postdocs, working either in representation theory or symplectic algebraic geometry. The second is that of secondary school pupils. Though the plan is to run workshops for both groups, these will be completely different in nature - the one for Ph.D students will involve lectures given by leading international academics in the subject, whilst the workshop for school pupils will be example based and student lead. In this way, I will be able to make a real, significant, impact on the people pipeline and skills base of the U.K. mathematics community.

Part I: Graduate Workshop

I will organize a workshop for later year graduate students and early career postdocs on "Representation theory of DQ-algebras", to be hosted at the University of Glasgow. There are two key goals for the workshop. The first goal being to teach students in representation theory the techniques and theory of DQ-algebras, so that they may apply this theory to solve problems in their area. Conversely, it will be the ideal opportunity to explore where their skills and knowledge can be used to tackle problems in the representation theory of DQ-algebras. The second goal is to educate early career researchers in algebraic geometry, in particular symplectic algebraic geometry, about the theory of DQ-algebras and how it can be used in geometry. As I have explained in the case for support, DQ-algebras are at the very forefront of current research in geometric representation theory, thus the workshop will take students to the cutting edge.
This will be a week long workshop, to be run in the summer of 2017. The three lecturers, all world leaders in the field and dynamic speakers, will each give a series of five lectures on their topics of expertise. In cooperation with the lecturers, I will run exercise classes in between the lectures, which will give the students an opportunity, via worked examples, to process and consolidate what they have heard in the lectures.

Part II: "The hidden beauty of symmetry" - an interactive journey

The aim of the workshop is to introduce school pupils to university level mathematics, and illustrate the applicability of "abstract mathematics" through a specific example - in this case through the abstract notion of a group. It is clear from talking to prospective, and also beginning, mathematics undergraduates that there is a great demand for workshops such as this, that help to illustrate and explain to pupils what proper mathematics is really all about, and how it can be used in the real world. For instance, it helps dispel the myth that mathematics is all about doing complicated, difficult, calculations and shows those of a more imaginative inclination that mathematics is something that they would find simulating too.

By the end of the day, students should have a clear intuitive grasp of the idea of an abstract group. This will be done by getting them to explore and learn about all the different places where groups naturally occur i.e. by taking them through many concrete examples. Emphasis will be given to real world examples, such as groups of transformations in 3D computer graphics, so that they appreciate the utility of the concept.