Research

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.

A: Perverse schobers and categorified homological algebra

To define perverse schobers on a given stratified space, one typically uses some explicit linear algebra-type description of the category of perverse sheaves, which one then directly categorifies. For instance, a perverse sheaf on the complex line C with a singularity at 0 can be described in terms of its vector spaces of vanishing cycles and nearby cycles, as well as two arising linear maps between these two (which satisfy a linear relation). This description is categorified by a spherical adjunction.
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.

  • A.5 Relative Calabi-Yau structures and perverse schobers on surfaces
    Preprint: arXiv:2311.16597.
  • A.4 Lax Addivity (with Tobias Dyckerhoff and Tashi Walde)
    Preprint: arxiv:2402.12251.
  • A.3 Complexes of stable ∞-categories (with Tobias Dyckerhoff and Tashi Walde)
    Preprint: arxiv:2301.02606.
  • A.2 Ginzburg algebras of triangulated surfaces and perverse schobers
    Forum Math. Sigma, 10:e8, 2022, doi:10.1017/fms.2022.1. arXiv:2101.01939.
  • A.1 Spherical monadic adjunctions of stable infinity categories
    Int. Math. Res. Not. IMRN, 2023(15):13153–13213, 2022, doi:10.1093/imrn/rnac187. arXiv:2010.05294.

    • B: Categorifications of cluster algebras

    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.

  • B.1 Cluster theory of topological Fukaya categories
    Preprint: arXiv:2209.06595.

    • C: Representation theory of triangulated categories arising from surfaces

    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 article C.2, 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.

  • C.2 Perverse schobers, stability conditions and quadratic differentials (with Fabian Haiden and Yu Qiu)
    Preprint: arxiv:2303.18249.
  • C.1 Geometric models for the derived categories of Ginzburg algebras of n-angulated surfaces via local-to-global principles
    Preprint: arXiv:2107.10091.

    • Other writings

  • Thesis: Perverse schobers and cluster categories.
    2023. Based on the papers A.2, A.5, B.1, C.1.
  • Lecture notes on perverse schobers in representation theory.