ATLAS: Algebraic thoughts looking at space

I will describe constructions of noncommutative cubic 4-folds, using suitable 6-dimensional Artin-Schelter regular algebras, and what happens to the link between cubic 4-folds and K3 categories when the cubic 4-folds become noncommutative. We will see that several of Michel’s important contributions to noncommutative algebraic geometry play an essential role in these constructions.

We consider moduli spaces parameterising bundles on an elliptic curve C equipped with two filtrations whose factors are semistable of ascending slope. Such moduli spaces occur as Betti moduli spaces for q-difference equations, and are also very natural from the point-of-view of derived categories. The main result gives a description of a class of such moduli spaces in terms of universal families of anti-canonical embeddings of C into del Pezzo surfaces. This is joint work with Pierrick Bousseau and Luca Giovenzana.

In 2006, Yang Han conjectured that a finite-dimensional algebra has finite global dimension if its Hochschild homologyvanishes in all sufficiently high degrees. The analogous statement forfinitely generated commutative algebras follows from work of BACH and of Avramov--Vigué-Poirier. The conjecture itself was shown to hold for many classes of algebras by several authors including Bergh, Cibils, Erdmann, Lanzilotta, Marcos, Redondo, Solotar, Suarez-Alvarez, Vivas, Ren Wang, Xiaoxiao Xu, Yunge Xu, Jinbi Zhang, and Guodong Zhou. In late August 2026, Bochao Kong, Yeqin Liu and Yu Shen posted a preprint where they construct a counter-example starting from Krah's universal phantom on the blow-up of the projective plane in 10 general points. We will sketch their construction, focusing on the passage from a proper dg algebra to a finite-dimensional algebra concentrated in degree zero.

I will describe a complete and cocomplete pretriangulated dg category over the ring of integers of a non-archimedean field, built from microlocal sheaf theory and almost mathematics. It can be viewed both as a large integral Fukaya category of a symplectic torus and as a kind of category of analytic quasi-coherent sheaves with norms on a power of a Tate elliptic curve. The symplectic geometry is not imposed but recovered as categorical symmetries.

I will talk about two examples of smooth projective varieties whose spaces of Bridgeland stability conditions are disconnected (in fact, have infinitely many connected components). In addition, for one of these varieties, the braid group action on the set of full exceptional collections is not transitive (in fact, has infinitely many orbits). Although the idea behind the construction is quite simple and the examples are not at all exotic, to the best of my knowledge neither of these disconnectedness phenomena had been observed before on smooth projective varieties.