Applied derived categories

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Derived categories are abstract algebraic objects that package geometric information. The way they do this is inspired by topology -- a more flexible type of geometry which allows more deformations. As such they endow the original geometry with more flexibility and symmetries. They also filter out a little of the geometric information, so two different geometries might lead to the same derived category. The way in which they do this is very interesting, both in mathematics and physics, where derived categories describe topological D-branes .It has become clear in recent years that derived categories are not quite as abstract, mysterious or fearsome as often thought. Extracting the geometry (and invariants of the geometry) turns out to be quite natural in many situations, and the different geometries that can come from the same derive category give new and important points of view that solve previously intractable problems. Their extra symmetries and flexibility make them more useful in many applications.Derived categories bring a different philosophy to problems, suggesting new approaches to them. We propose to bring this new way of thinking to areas of broad areas of geometry, linking many which have not been touched derived categories before. We hope to solve problems and develop new areas of mathematics, helping to make derived categories into standard mathematical tools used all over the subject.