Central objects in my research are so called perverse schobers, which categorify perverse sheaves. The word 'schober' is german and means (hay) stack, perverse schobers are perverse sheaves of stable ∞-categories. This notion was proposed by Kapranov-Schechtman in 2014. While the general theory of perverse schobers remains conjectural, there exist working definitions for perverse schobers on some classes of stratified spaces. These include surfaces with boundary and hyperplane arrangements.

Global sections of perverse schober describe interesting classes of stable ∞-categories and provide new local-to-global arguments for their study. There are many examples coming from Fukaya(-Seidel) categories and representation theory. Examples of the latter type include the derived categories of (skew-) gentle algebras, (graded, relative) Brauer graph algebras, (relative) Ginzburg dg-algebras of triangulated surfaces and, last but not least, cluster categories of surfaces.

The usual definition of perverse sheaves is as objects in the heart of the perverse t-structure on the derived category of constructible sheaves. It is currently unclear how to categorify this definition, as it is not clear what the categorified derived category should be. One can however already speak about chain complexes of stable ∞-categories (see for instance my joint articles A.3, A.4 with T. Dyckerhoff and T. Walde). It's less clear what a good notion of quasi-isomorphism is in this context. Eventually answering such questions is the goal of the theory of categorified homological algebra.

My work explores the theory of perverse schobers on topological surfaces with boundary (article A.2) and on C^n with the coordinate hyperplace stratification (article A.3). In the context of surfaces with boundary, one can efficiently describe perverse sheaves and schobers in terms of constructible sheaves on a choice of spanning ribbon graph of the surface. Exploring the theory of perverse schobers in higher dimensions will be the subject of future work.

Preprint: arXiv:2311.16597.

Preprint: arxiv:2402.12251.

Preprint: arxiv:2301.02606.

Cluster algebras are a class of commutative algebras which come equipped with combinatorial data that allows to relate certain finite subsets called clusters with each other via a process called mutation. Cluster algebras have been categorified in terms of remarkable categorical structures. One kind of such categorifications of cluster algebras are called cluster categories. In my article B.1, I show that the cluster categories associated with marked surfaces can be described in terms of perverse schobers. More precisely, I describe a perverse schober whose global sections describe the so-called Higgs category, which is a relative version of the standard cluster category. Furthermore, this Higgs category turns out to be equivalent to the topological Fukaya category of the surface itself, with 1-periodic coefficients. Work in progress concerns extending these results to the so-called higher rank cluster categories associated with surfaces. These categorify higher rank cluster algebras in the sense of Fock-Gonchraov.

Preprint: arXiv:2209.06595.

Perverse schobers allow to study their stable ∞-categories of global sections with powerful and flexible local-to-global arguments. In the article C.1, I explore how one can construct objects and describe derived Hom's via gluing. The result is that to every suitable decorated curve in the surface, one can associate a global section, which defines an object in the derived category of a Ginzburg algebra. Further, I relate the derived Hom's with certain counts of intersections of the curves. This can also be seen as an algebraic analog of the matching sphere construction from symplectic geometry.

In the articles C.2 and C.3, we use similar constructions of objects from curves to describe the tilting theory of t-structures with a finite length heart on the global sections of suitable perverse schobers. This then allows to relate the spaces of Bridgeland stability conditions with spaces of quadratic differentials on the surface.

(with Fabian Haiden and Yu Qiu)

Preprint: arxiv:2407.00154.

Preprint: arxiv:2303.18249.

Preprint: arXiv:2107.10091.

2023. Based on the papers A.2, A.5, B.1, C.1.