Stability conditions in families, and on Kuznetsov components 
-- Arend Bayer, April 26, 2019 

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In this talk, I will describe a recent construction (joint with Marti Lahoz, Howard Nuer, Emanuele Macri, Alex Perry and Paolo Stellari) that gives a construction of stability conditions on the derived categories of varieties in a family. It comes equipped with relative moduli spaces of stable objects. It seems to become particularly powerful when restricted to certain subcategories, callled Kuznetsov components, of derived categories of Fano varieties: it provides a new tool to study a family of such varieties by studying the associated families of moduli spaces. I will illustrate this via examples for Fano threefolds, and for cubic fourfolds.