On the Gonality of Metric Graphs

Abstract

The last decades have seen an extremely fruitful interplay between Riemann surfaces and graphs with a metric. A deformation process called tropicalisation transforms the former into the latter. Under this process, additional structure on the Riemann surfaces yields additional structure on the metric graphs. For instance, meromorphic functions on Riemann surfaces yield piecewise linear functions on metric graphs. In this manner, theorems in algebraic geometry have deep combinatorial consequences; and conversely, combinatorial arguments can be used to prove theorems in algebraic geometry.

Discrete and metric divisorial gonality can be different

“…3.12] using algebraic geometry. A purely combinatorial proof of this result was recently found by Draisma and Vargas [14], with many promising avenues still to be explored [15]. However, for discrete graphs, Conjecture 1.1 is still wide open.…”

Laplacians on Infinite Graphs

“…It is impossible to give even a very brief account on these matters. The key features of Laplacians on metric graphs, which are also widely known as quantum graphs, include their use as simplified models of complicated quantum systems and the appearance of metric graphs in tropical and algebraic geometry, where they can be seen as non-Archimedean analogs of Riemann surfaces (we only refer to a very brief selection of recent monographs and collected works [11,24,25,62,67,69,182]). The subject of discrete Laplacians on graphs is even wider and has been intensively studied from several perspectives (a partial overview of the immense literature can be found in [12,43,44,91,136,212]).…”

Laplacians on infinite graphs: discrete vs continuous

“…Laplacian operators on graphs have a long history and enjoy deep connections to several branches of mathematics and mathematical physics. There are two different notions of Laplacians appearing in this context: the key features of (continuous) Laplacians on metric graphs, which are also known as quantum graphs, include their use as simplified models of complicated quantum systems (see, e.g., [4], [19], [21], [56]) and the appearance of metric graphs in tropical and algebraic geometry, where they serve as non-Archimedean analogues of Riemann surfaces (see, e.g., [1], [17]). The subject of discrete Laplacians on graphs is even wider and a partial overview of the immense literature can be found in [2], [9], [10], [43], [70].…”