Warwick EPSRC Symposium on Derived Categories and Applications

The planned 2014-2015 Warwick EPSRC symposium is a year long concentrated activity on the theory and applications of derived categories. The subject of derived categories emerged in the second half of the 20th century as a distillation of the ideas of homological algebra, which calculates invariants of a topological space such as its "number of n-dimensional holes". While more high brow and abstract than the more primary methods of attaching invariants to a mathematical or physical object, the derived category has a number of important advantages, allowing us to see so-called "quantum symmetries" of manifolds that are inaccessible to more conventional theories. They are thus an essential ingredient of attempts to understand the mathematics of physically important theories such as string theory, mirror symmetry and supersymmetry.

Starting from the top, the more theoretical aspects covered during the year involve abstract notions such as higher category theory, DG enhancements and derived geometry. These are substantial generalisations of conventional geometry and category theory, and the theory is currently at the level of understanding and standardising the foundations of the new subject. This is an exciting stage in the development of a mathematical theory, but not one that can be convincingly explained in simple terms. Our symposium will run several schools and workshops at different levels expanding on these matters.

At the other extreme, derived categories feed back into explicit calculations that can be applied to give useful results describing the properties of usual objects of algebra, geometry and theoretical physics. For example, derived categories have provided by far the best treatment of the McKay correspondence, that relates the representation theory of a finite subgroup G in SL(2,CC) or SL(3,CC) with the topology of a resolution of the orbifold quotient CC^n/G. In a similar vein, our symposium will include workshops studying derived category approaches to the study of different moduli spaces and their invariants (such as the classical moduli spaces of vector bundles, or of algebraic curves).

In between these two extremes is a rich body of theories and problems in algebra, geometry and physics to which it is known or suspected that derived category methods can be applied. This includes issues arising from string theory, such as homological mirror symmetry, that works around the conjecture that the derived category mediates between the complex geometry of a Calabi-Yau 3-fold and the symplectic geometry of its mirror partner.

This proposal is for a symposium in pure mathematics, centred around fundamental problems of an abstract theoretical nature. Each component has huge potential impact in neighbouring branches of math.

Category theory is a core mathematical discipline with deep connections across math. Derived categories have important applications in geometry, representation theory and theoretical physics and other areas. The primary beneficiaries of this symposium will be researchers in algebra (especially noncommutative rings and representation theory), geometry (especially topology, algebraic geometry and symplectic geometry) and QFT (especially string theory, mirror symmetry and other quantum dualities).

A main overall thrust of the proposed symposium is to present new foundations and interpretations of homological algebra based on ideas around the Derived Category. The resulting clarification, simplifications and generalisations arising from these developments is likely to have very considerable impact on many areas of mathematics in the medium and longer term. In the more immediate future, the impact of our symposium is likely to be felt mostly in the research areas that already connect to derived categories, especially topology, symplectic and algebraic geometry, noncommutative algebra and theoretical physics (mirror symmetry, string theory, QFT and so on). Our workshops are designed to optimise delivery of this impact.

An idea that has been fertile over the last 40 years or so is that of a moduli space; it often happens that a problem in geometry, algebra, analysis or physics has solutions depending on parameters. In ideal cases, this setup may lead to a neat geometric space that parametrises our solutions in a neat way, together with a `tautological' universal object associating a point of the parameter space with the solution that it parametrises. This universal object then provides a rich palette of tools relating the geometry of the solutions with that of the moduli space and vice versa.

Our symposium contains several major new components that will have impact via moduli theory. For example, the systematic exploitation of the ideas of stability conditions in categories and the variation of stability has had notable recent successes, and has more to give in the near future.

Several of our workshops aim to treat these issues also in more general cases, where the moduli problem does not have such a neat solution for one or more technical reasons. Higher categories arise when the objects that one parametrises have additional symmetries -- one is led to parametrising categories and natural maps between them, rather than just objects in one category. This extra level of abstraction is needed to deal with many of the moduli problems of importance in applications to geometry, theoretical physics, noncommutative algebra and representation theory. Our workshops will develop these interesting technical topics, while laying down and disseminating methods that will be essential for future impact of national importance.

As with other recent Warwick EPSRC symposia, the 2014-2014 Symposium on Derived Categories and Applications will impact on the UK math community in the following ways:

1. Strengthening the UK knowledge base in category theory and links with related areas.

2. Bringing the top world specialists to the UK to promote collaboration with UK researchers, including postgraduate students and recent postdocs, and to enhance our research basis.

3. Supporting several leading UK groups in this active and vibrant area and enhancing their position at the cutting edge of research.

In particular, this project supports the drive on the part of UK universities to generate top-quality research. This is far from merely a matter of national prestige; no country can maintain economic success without a strong scientific base and outstanding universities and, in particular, without strong foundations in mathematical research.