Exact Structures in Representation Theory

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The aim of this project is to develop a theory of stability conditions in a general setting and thus advance the knowledge in this field. The objectives of this proposal are to: 1) study stability conditions in the new context of exact categories, and 2) interpret the matrix reduction technique as a change of exact structure. The approach is, in both cases, to work with exact structures that have been defined by Quillen in his fundamental work on K-theory. They provide a framework that allows the use of methods from homological algebra, but is strictly more general than the setting of abelian categories. Stability conditions have been studied systematically in geometry and theoretical physics. The motivation stems from a phenomenon called mirror symmetry, relating two seemingly unrelated geometrical objects relevant to theoretical physics. The matrix reduction technique is an elementary tool that has been used by the Kiev school to obtain fundamental results in representation theory of algebras. The formal setup for matrix reduction has been given using a certain setup called called BOCSes. The disadvantage of this approach is that iterated reduction of matrices, per se not a complicated technique, produces more and more complicated descriptions of objects in the categories to be studied. We propose here a fundamentally novel approach: instead of changing the objects, we suggest to keep the same category, but to change its exact structure. We plan to apply this technique of reductions of exact structures to solve questions related to representation theory and mirror symmetry.