Algebraic constructions related to marked Riemann surfaces

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A marked Riemann surface (S,M) is formed by a Riemann surface S and a finite set M of “marked” points. Fomin, Shapiro and Thurston constructed a cluster algebra A(S,M) for each marked surface, and the work of Amiot, Cerulli Irelli, Keller, Labardini-Fragoso and Plamondon allows to define the corresponding cluster category C(S,M). The main focus of this research proposal is the interplay between these three objects: the geometry of the Riemann surface with its Mapping Class Group, the cluster algebra with its cluster automorphisms, and the triangulated category C(S,M). An important combinatorial invariant of the marked surface is its flip graph which is formed by all triangulations of (S,M) and where edges are given by the flip of an arc. The same graph occurs as the cluster exchange graph of A(S,M) or C(S,M), with respective notions of mutations which correspond to the flip of an arc. The cluster exchange graph can be endowed with an orientation, and maximal paths in this oriented graph are referred to as maximal green sequences. These maximal green sequences are studied in various contexts, as they give rise to quantum dilogarithm identities and non-commutative Donaldson–Thomas invariants. The case of a pentagon (S,M) yields the classical quantum dilogarithm identity in two skew-commuting variables, and the construction has been generalized by Reineke and Keller to many other cases. Moreover, Riemann surfaces are also studied in mathematical physics, in the context of string theory. In particular, the complete spectrum of a BPS (Bogomol’nyi–Prasad–Sommerfield) particle in string theory can be computed using maximal green sequences. The string theory approach is based on quadratic differentials on S, and recent work of Bridgeland and Smith relates this to studying stability conditions on the triangulated category C(S,M). The main objectives of this research proposal are: • To provide a definition of the cluster category C(S,M) which is independent of any triangulation, using Cohen-Macaulay modules over orders. • To study minimal paths in the exchange graph. • To characterize the existence of maximal green sequences in terms of stability conditions on C(S,M).