Given the data of a variety, an algebra, or more generally a triangulated category, Bridgeland stability produces a complex manifold (the space of stability conditions). What does the geometry of this manifold tell us about the starting data? In this talk we'll investigate this question by looking for so-called "geometric" stability conditions on surfaces that arise as free quotients by finite groups. This is joint work with Edmund Heng and Anthony Licata, based on arxiv:2307.00815 and arxiv:2311.06857. Time permitting, we will also discuss work in progress in the case of bielliptic surfaces, which is joint with Gebhard Martin.

We give a conceptual construction of a twisted Chern character from the K-theoretical Hall algebra of a symmetric quiver to its cohomological Hall algebra. The more conceptual approach allows us to carry our work over to K-theoretical Hall algebras of quivers with potential and their critical cohomological Hall algebras. This is joint work with Š. Špenko

In the classical setting of modules over a commutative ring, depth, projective dimension and global dimension are important invariants. For a triangulated category one can think of the generation time of $X$ from another object $G$ as a generalization of projective dimension. Loosely speaking generation time is the maximal number of cones that are necessary to obtain $X$ from $G$. I will present a notion of depth for objects in a triangulated category and its connection to generation time. This is joint work with Antonia Kekkou and Marc Stephan.

Toric Fano varieties build an important example class in algebraic geometry as their high symmetry allows to describe them in a purely combinatorial manner via the so-called Fano polytopes. The anticanonical complex is a combinatorial tool that was invented to extend the features of the Fano polytope to wider classes of varieties. In this talk we give an outline of the construction of anticanonical complexes and show how they can be used to read off the Gorenstein index of Fano varieties with torus action of higher complexity in full analogy to the toric Fano polytope.

TBA

A classical approach to certifying the non-negativity of a real polynomial is to express it as a sum of squares. However, the cone of polynomials that can be written as a sum of squares is, in general, strictly contained within the cone of nonnegative polynomials. The fundamental cases where these two cones coincide were characterized by Hilbert. An alternative approach for certifying non-negativity is to express the polynomial as a Sum of Non-negative Circuits (SONC). In this talk, I will discuss necessary and sufficient combinatorial conditions under which the cone of SONC polynomials coincides with the cone of nonnegative polynomials. Our approach builds on extending Viro’s patchworking to singular hypersurfaces and using properties of the positive A-discriminant. This talk is based on joint work with Timo de Wolff.

In this talk, I will report about work in progress on questions related to (non-)finite generation of quasiaffine algebras. This in particular relates to certain Cox rings, reductive invariant rings of quasiaffine algebras, and singularities of such rings. Results include abstract statements as well as computational methods.

Quantized enveloping algebras of Kac-Moody algebras and their representation theory have played a significant role in mathematics and physics over the past decades. In this talk, I will discuss the first attempt to quantize a class of equivariant map algebras that realize parabolic subalgebras of affine Kac-Moody algebras. After presenting some structural results, I will introduce the classification of finite-dimensional irreducible representations over fields of characteristic zero, assuming the deformation parameter is not a root of unity. The classification is formulated in terms of Drinfeld polynomials, revealing new phenomena—for instance, for maximal parabolic subalgebras, certain divisibility conditions will appear.