Abstract
Abstract: Irreducible symplectic varieties extend the theory of compact hyper-Kähler manifolds to the singular setting, allowing for new constructions beyond the few known smooth families. I will describe one such construction, starting from a K3 surface \(S\) that is a double cover of a del Pezzo surface \(T\). The relative Jacobian associated with the pullback of the anticanonical linear system on \(T\) admits a symplectic compactification carrying a natural fibrewise Prym-type symplectic involution. Taking its quotient and passing to a crepant terminal model produces irreducible symplectic varieties in every even dimension from \(4\) to \(20\). A distinctive feature of this construction is that the involution preserves the entire second cohomology, so the resulting varieties retain comparatively large second Betti numbers relative to the usual finite-quotient constructions.
Abstract: Special biserial algebras are a widely studied class of algebras with a particularly well-behaved and combinatorial representation theory.
In the talk, I will present a geometric model for the module categories of these algebras. Specifically, I will explain how to realise any such algebra as a bilabelled tiling algebra. In this geometric setting, indecomposable modules correspond to curves on the associated surface, morphisms to intersections of curves, and extensions to simultaneous resolutions of crossings.
This is a report on joint work in progress with Raquel Coelho Simões, Johanne Haugland, and Martin Herschend.
Abstract: Saturated fusion systems offer an elegant categorical framework to study local group symmetries. Originally developed in algebraic topology and modular representation theory, they also play a central role in finite group theory—most notably, fusion systems on 2-groups were a key tool in revision of the Classification of Finite Simple Groups.
Given a finite group G and a Sylow p-subgroup S, the fusion system of G on S is the category whose objects are the subgroups of S and whose morphisms are conjugation maps induced by G. But what happens when we abstract this setup beyond groups? Remarkably, there exist exotic fusion systems: structures that mimic the internal symmetries of a finite group without coming from any actual finite group.
In this talk, we will revisit what is currently known about exotic systems and survey ongoing research in the field. We will see how generalizing classical results allows us to frame new families of exotic fusion systems within unified contexts, taking a step further toward a complete classification.
(Joint work with J. Lynd, B. Oliver, C. Parker, J. Semeraro, and M. van Beek).
Abstract: Moduli spaces of polygons are families of symplectic manifolds constructed by symplectic reduction. In this talk I will describe various examples of these moduli spaces both from the symplectic and the algebro-geometric viewpoint. In particular, I will describe how the level of reduction determines their structure by studying their birational geometry under wall-crossing.