Nathan Ilten (Simon Fraser University): Deformation theory, computations, and toric geometry

Cristina Manolache (University of Sheffield): Reduced invariants

Francesco Meazzini (Università degli Studi di Roma "La Sapienza"): An introduction to deformations of sheaves

Michael Wemyss (University of Glasgow): Noncommutative Deformations and Classification Problems

• Abstract. This mini-course will be an introduction to noncommutative deformation theory, and will also outline some of its applications to algebraic geometry. The main difference when we add the adjective "noncommutative" to deformation theory is that we make the category of test objects larger, compared to the classical case. By doing this, we get more information. The first lecture will outline how to formulate noncommutative deformation theory using naive functors, and will give some general results. The second lecture will outline how to do this using DGAs. There are two key points: (a) we really do need a DGA to be able to noncommutatively deform; a DGLA does not suffice, and (b) deforming multiple objects is now also possible. The third and fourth lectures will explain how this works when we apply the above theory to curves within 3-folds, where noncommutative deformations turn out to be the classifying object.

Simon Felten (Columbia University): Curved Lie algebras in logarithmic deformation theory

• Abstract. Every classical deformation problem in characteristic 0 is controlled by a dg Lie algebra. This is no longer true in logarithmic geometry, where the infinitesimal deformation functor of a log smooth variety cannot always be controlled by a dg Lie algebra. In this talk, we will see how allowing curvature in the dg Lie algebra remedies this situation.

Alessandro Lehmann (University of Antwerp and SISSA): Curved deformations of differential graded algebras

• Abstract. It is a classical fact that the Hochschild complex of an algebra governs its deformations as an associative algebra. Surprisingly this does not immediately generalize to differential graded algebras (and by extension, categories), since in general deformations given by Hochschild cocycles can introduce curvature. In this talk I’ll explain how to make sense of these curved deformations via a novel notion of a type of derived category; I will also explain how this construction suggests a new notion of deformation of a triangulated category, based on categorical square zero extensions. This is based on joint work with Wendy Lowen.