Bridgeland stability and the moveable cone

There are many instances in life when two numbers might be the same and yet there is no relation between the objects that are being counted. For example, the number of coins 1p, 2p, 5p,..., £2 in British currency, and the number of legs on a healthy spider. However, such coincidences in mathematics are sometimes merely the shadow of a much more interesting relation that exists on a deeper level between different structures. When this can be explained, we not only understand the original coincidence but, better yet, we can translate our understanding from one structure to the other. Put simply, we uncover the dictionary between two languages that we might only partially understand.

One such example is provided by a positive number that appears in two apparently different mathematical contexts: one is equal to the number of certain types of symmetry that are defined in a rather abstract, algebraic way; and the other is obtained by counting certain geometrically-defined objects. In fact, this coincidence can be explained quite beautifully by a relation known as the "McKay correspondence". Roughly speaking, this correspondence describes in a very concrete way in which two rather abstract objects (called triangulated categories), one defined in terms of algebra and the other defined in terms of geometry, are in fact the same. Every such category encodes certain numbers, and the original coincidence boils down to the simple observation that two identical categories encode precisely the same numbers! This, then, is one of the primary goals of a pure mathematician: to investigate whether apparent coincidences can be explained in a natural way, and it is precisely this search for the "natural" notion that makes pure mathematics important to so many fields of science and engineering.

The current proposal aims to do precisely this. As with the McKay correspondence described above, it has been known for some time that many such correspondences (called equivalences of categories) do exist even for rather different types of geometry which encode the same kind of numbers, and some of these have been described very elegantly by the work of several mathematicians over the last fifteen years or so. Even now, the general picture eludes us, but the following question has been posed by mathematicians Bondal and Orlov: "If we have two types of geometry that, while being different are nevertheless similar in a controlled way, does there exist a correspondence as above to explain the similarity?".

Here we aim to lay the foundation for a new, geometric approach to this problem by introducing an abstract generalisation of a particular map - a kind of "machine" - that the PI has studied in depth. Crucially, we believe that we understand precisely the right level of abstraction to shed light on the correct path: too little abstraction may illuminate nothing at all; while too much abstraction may be so blinding as to provide no help whatsoever. While we do not have the full picture, we do believe that we have found the correct foundation for the problem, and to provide a "proof of concept" for our approach we will demonstrate that it works for an interesting class of examples. The results that will come from this proposal will, we believe, provide solutions to several interesting problems that, taken together, provide an important, geometric step towards our understanding of the celebrated conjecture of Bondal and Orlov.

The potential beneficiaries of the research described in this proposal are principally mathematicians and theoretical physicists with an interest in derived categories of coherent sheaves, D-branes, and Calabi-Yau manifolds. The current proposal aims to establish not only a firm foundation on which to explore one of the most important conjectures concerning such manifolds, namely the Bondal-Orlov Conjecture, but it also formulates the Derived Moveable Cone Conjecture that we believe will generate considerable interest in the longer term future. Thus, we anticipate considerable interest within academia from algebraic geometers and string theorists, in addition to symplectic geometers studying the Fukaya category and Mirror Symmetry. We believe that this interest will come from three directions:

- first, from those who wish to use our derived linearisation map from Objective (3) to tackle open questions linking derived categories and birational geometry such as the Bondal-Orlov Conjecture;

- second, from those excited by our Derived Moveable Cone Conjecture from Objective (4); and

- third, from those interested in the classical question to describe the set of Chern classes of slope-stable torsion-free sheaves on del Pezzo surfaces, which we aim to resolve in Objective (2).

The research output arising from this proposal will be posted online and published in world-leading journals. The PI, his collaborator Bayer, and the PDRA, will all travel widely to disseminate the results at international conferences and universities across the world. The track records and research profile of the PI and Bayer (and, we anticipate, of the PDRA) will ensure that our results are considered seriously by the academic community.