Dimers, networks, and integrable systems

I will review two combinatorial constructions of integrable systems: Goncharov-Kenyon construction based on counting perfect matchings in bipartite graphs, and Gekhtman-Shapiro-Tabachnikov-Vainshtein construction based on counting paths in networks.  After that I will outline my proof of equivalence of those constructions.  The talk is based on my recent preprint arXiv:2108.04975.