• Hausdorff School 2015/2016
• Derived Categories: dimensions, Stability conditions, and Enhancements

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I will give an introduction to Bridgeland's notion of a space of stability conditions, along with a survey of applications. In particular, I will focus on the relation to groups of autoequivalences of derived categories.

The classical notion of generators for a category does not work in triangulated categories. Nevertheless it is useful to consider analogs, and over the decades people have come up with several. We will review the various notions and their applications, and discuss some open problems.

Let be a generator. Another idea, whose usefulness only became apparent relatively recently, is to measure how many steps it takes to get from the generator to another object in the category. This leads to various notions of dimension. We will review what is known, some applications, and open problems.

The interplay between triangulated and dg categories is subject to a growing interest with several nice applications to algebraic geometry. In these talks we review some recent developments concerning the quest for a (possibly unique) lift of exact functors and triangulated categories. To this end, we first present some recent advancements about Fourier--Mukai functors. Then we discuss the general belief, formally conjectured by Bondal, Larsen and Lunts, that the dg enhancement of the bounded derived category of coherent sheaves or the category of perfect complexes on a (quasi-)projective scheme is unique. This was proved by Lunts and Orlov in a seminal paper and we explain how to extend Lunts-Orlov's results to several interesting geometric contexts. The new results presented in this talks are joint work with A. Canonaco.