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. Important examples are Fukaya-Seidel categories and other classes of partially wrapped Fukaya category. In representation theory, examples include 3-Calabi-Yau categories associated with surfaces (arising from Ginzburg dg algebras), variants of Brauer graph algebras, and additive categorifications of cluster algebras arising from higher Teichmüller theory.
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. The article B.2 and upcoming work concerns an extension of these results to the so-called higher rank cluster categories associated with surfaces. These categorify higher rank cluster algebras describing coordinates on the higher Teichmüller spaces of Fock-Gonchraov.
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.