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 geometricallydefined 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 categores) 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.
