The enumerative content of Gromov-Witten theory

*Abstract:* I will discuss old and new ways of answering questions in enumerative geometry. New methods have many advantages and one major drawback. In this talk I will discuss this drawback. I will introduce Gromov--Witten invariants and I will give evidence that they do not give correct curve counts. I will introduce new enumerative invariants from curves with cusps and I will argue that cuspidal invariants have a better enumerative meaning. In the end, I will highlight one application. This is based on work in collaboration with L Battistella, F Carocci and T Coates.

*Abstract:*
I will survey the existence of hidden recursive structures in the Gromov--Witten (GW) theory of a complex projective variety. I will discuss a characterisation of recursions for genus zero invariants in terms of associative deformations of the cup product in cohomology, and some alternative presentations of the deformed cup product motivated by singularity theory (the celebrated "mirror symmetry"). In some happy (and central) instances mirror symmetry is often a tool powerful enough to determine recursively all GW invariants starting from minimal input data. I will consider one application of these ideas to low-dimensional topology, which is partly joint work with Gaetan Borot, relating a class of smooth invariants of 3-manifolds to recursions for GW invariants.

Stability conditions of the Kronecker quiver

*Abstract:* To a quiver Q, we can associate a sequence of Calabi--Yau-n triangulated categories. The spaces of stability conditions of these categories can then be computed. I will give a description of these stability manifolds, and discuss the relationship between them and the Frobenius structure of the quantum cohomology of the projective line.

Simultaneous deformations of Hall algebras

*Abstract:* In this talk, we discuss how Ringel-Hall algebras, an algebra associated to suitably finite Abelian categories, can be viewed in certain cases as simultaneously deforming two simpler algebras. One of these algebras is the universal enveloping algebra of a Lie algebra, while the other is a Poisson algebra. Time permitting we also discuss an analogous deformation picture for a generalization of Ringel-Hall algebras due to Bridgeland.

Grothendieck groups and singularity categories of quotient singularities

*Abstract:* We study the K-theory of the Buchweitz-Orlov singularity category for quasi-projective algebraic schemes. Particularly, we show for isolated quotient singularities with abelian isotropy groups that the Grothendieck group of the singularity category is finite torsion and that rational Poincare duality is satisfied on the level of Grothendieck groups. We consider also consequences for the resolution of singularities of such quotient singularities and study dual properties in this setting, more concretely we prove a conjecture of Bondal and Orlov in the case of quotient singularities.

On categorical entropy

*Abstract:* The categorical entropy of triangulated endofunctors was defined by Dimitrov-Haiden-Katzarkov-Kontsevich motivated by classical dynamical theory. In this talk, I'll explain relations to classical entropy theory, basics on categorical entropy, many examples and future directions.