Central objects in my research are so called perverse schobers, meaning categorified perverse sheaves. The general notion of perverse schobers is conjectural, but there exists a working theory for perverse schobers on surfaces with boundary.
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 or representation theory, the latter including 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.
Perverse schobers further form natural objects of study in the emerging theory of categorified homological algebra.