Z. Wang - Surface models of graded gentle algebras 2

This lecture series explores the connections between the representation theory of graded gentle algebras and surface topology, providing a comprehensive framework for understanding derived equivalences. Diverging from traditional topological definitions of line fields, the approach utilizes a purely combinatorial geometric model. The main geometric objects are compact, oriented smooth surfaces equipped with boundary stops and dissections. By assigning integer labels to boundary paths, one can dictate the degrees of arrows in the quivers and explicitly construct the associated graded gentle algebras. The primary focus of these lectures is the application of combinatorial winding numbers to classify derived equivalence between these algebras. For genus zero surfaces, this equivalence is determined entirely by boundary stop data and boundary winding numbers. In higher genera, additional invariants are necessary. The framework introduces a greatest common divisor (gcd) invariant derived from non-separating simple curves for genus one surfaces, while higher-genus surfaces require richer extractions like parity data and Arf invariants to complete the classification. Finally, we will explain why these invariants work by transitioning to a strictly algebraic perspective using the mechanism of 𝐴∞ algebras and categories. Specifically, we will demonstrate how a cyclic higher product allows an object to be replaced by an iterated twisted complex without altering the underlying perfect derived category. This algebraic mechanism of generator replacement naturally connects to the topological partially wrapped Fukaya category of a graded surface with stops. https://if-summer-2026.sciencesconf.org/