# Schedule of the Hausdorff School: Derived Categories: dimensions, Stability conditions, and Enhancements

## Tuesday, March 29

 9:30 - 10:30 (Self-)Registration and Coffee 10:30 - 12:15 Amnon Neeman: Generators and dimension in triangulated categories 12:15 - 14:15 Lunch Break 14:15 - 16:00 Paolo Stellari: Uniqueness of dg enhancements 16:00 - 16:30 Afternoon Break 16:30 - 18:15 Short Talks

## Wednesday, March 30

 9:00 - 10:45 Paolo Stellari: Uniqueness of dg enhancements 10:45 - 11:15 Group Photo and Coffee Break 11:15 - 13:00 Amnon Neeman: Generators and dimension in triangulated categories 13:00 - 14:30 Lunch Break 14:30 - 16:15 Arend Bayer: Applications of stability conditions 16:15 - 16:45 Afternoon Break 16:45 - 18:15 Questions and Answers and Discussions 19:30 Dinner

## Thursday, March 31 (program has changed)

 9:00 - 10:45 Arend Bayer: Applications of stability conditions 10:45 - 11:15 Coffee Break 11:15 - 13:00 Amnon Neeman: Generators and dimension in triangulated categories 13:00 - 14:30 Lunch Break 14:30 - 16:15 Paolo Stellari: Uniqueness of dg enhancements 16:15 - 16:45 Afternoon Break 16:45 - 18:15 Questions and Answers and Discussions

## Friday, April 1 (program has changed)

 9:00 - 10:45 Paolo Stellari: Uniqueness of dg enhancements 10:45 - 11:15 Coffee Break 11:15 - 13:00 Arend Bayer: Applications of stability conditions 13:00 - 13:45 Lunch Break 13:45 - 14:30 Coffee Break 14:30 Siebengebirge Hike or Free Afternoon

## Saturday, April 2

 9:00 - 10:45 Amnon Neeman: Generators and dimension in triangulated categories 10:45 - 11:15 Coffee Break 11:15 - 13:00 Arend Bayer: Applications of stability conditions 13:00 End of Hausdorff School

## Arend Bayer: Applications of stability conditions

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.

## Amnon Neeman: Generators and dimension in triangulated 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 $G$ 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 $G$ to another object in the category. This leads to various notions of dimension. We will review what is known, some applications, and open problems.

## Paolo Stellari: Uniqueness of dg enhancements

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.