Stability conditions on Fukaya categories of surfaces: Some new techniques and results

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In this talk I will present some upcoming work on Bridgeland stability conditions on partially wrapped Fukaya categories of topological surfaces. The main result is a proof that the stability conditions defined by Haiden, Katzarkov and Kontsevich using quadratic differentials cover the entire stability space. This proof uses a definition of the new concept of relative stability conditions, which is a relative version of Bridgeland's definition, with functorial behavior analogous to compactly supported cohomology. This definition is exclusive to the setting of these categories, and I will discuss problems and possibilities regarding generalization to other types of categories.