Stability conditions on triangulated categories have been introduced by Bridgeland in 2007, generalizing the notion of slope stability for sheaves on curves. Since then, the theory has developed very fast leading to many applications in algebraic geometry. The space parametrizing stability conditions has the structure of a complex manifold. Since it is not compact, an interesting problem is the construction of a compactification. 

The aim of these talks is to survey the recent progresses by Bapat, Deopurkar and Licata, who propose a construction in analogy to the Thurston compactification of the Teichmuller space for Riemann surfaces. We will study the case of the bounded derived category of a smooth projective curve investigated by Kikuta, Koseki and Ouchi. Then we will focus on non commutative curves and K3 surfaces of Picard rank 1, which is part of a work in progress joint with C. Dare, B. Farman, E. Macrì, L. Marquand, T. Peng, X. Qin, N. Rekuski, F. Rota.

I will illustrate the most recent results on how to enhance triangulated categories (and exact functors) of geometric nature. We will then move to the problem of lifting equivalences between various triangulated categories and illustrate the new interplay between the theory of weakly approximable triangulated categories and the existing results about the uniqueness of enhancements. Applications to a generalization of a classical result by Rickard and to derived invariants of schemes will be discussed. The new results are joint work (partly in progress) with Alberto Canonaco and Amnon Neeman.

I will discuss the basic ideas and properties of F-bundles and non-commutative Hodge structures, as well as applications to birational geometry. Joint work with Katzarkov, Kontsevich and Pantev.

The integral Hodge conjecture—though long known to be false in general—holds for some special classes of algebraic varieties, notably Calabi–Yau threefolds. Recently, Perry formulated a categorical version of the conjecture, where triangulated categories play the role of algebraic varieties. 

I will discuss the conjecture in the case of twisted derived categories, which turns out to be closely connected to a set of classical problems in algebra and topology. Finally, I will explain some recent results for twisted Calabi–Yau threefolds. Includes joint work with Alex Perry. 

 Atomic sheaves on hyper-Kähler manifolds have well-behaved Hodge-theoretic properties, and are the natural analogs of coherent sheaves on K3 surfaces. Despiting their nice properties, only a few concrete examples are known on general projective hyper-Kähler manifolds. In this talk, I will first review the theory of atomic sheaves, then I'll explain two methods to construct atomic sheaves supported on Lagrangians in the moduli spaces of stable objects in the K3 categories of Gushel-Mukai fourfolds. If time permits, I'd like to talk about an explicit example of atomic Lagrangians in a hyper-Kähler fourfold. This is based on the joint work with Hanfei Guo.

I will discuss recent joint work with Daniel Halpern-Leistner on constructing a partial compactification of (a quotient of) the space of Bridgeland stability conditions. I will give an overview of previous results motivating the partial compactification, before sketching some features of its construction.