Abstract:

Finite graphs and metric graphs provide useful combinatorial analogs of algebraic curves. Since the work of Cools-Draisma-Payne Robeva, it has been known that chains of loops are particularly useful graphs in this regard. I will describe a new perspective on these chains of loops, by describing a version of Brill-Noether theory for curves or graphs with two marked points. Divisors on twice-marked graphs are associated with permutations; the extent to which a divisor is special is measured by the inversions of the permutation. This twice-marked Brill-Noether theory is well-suited to inductive arguments; when two twice-marked graphs are glued together, permutations are combined by an operation called the Demazure product. I will describe how this framework provides a short proof of some of the results of Cools-Draisma-Payne Robeva, as well as more recent results on tropical Hurwitz-Brill-Noether theory, and may provide a route to identifying new classes of Brill-Noether-General graphs.