Workshop I — Statistical Mechanics and Discrete Geometry

Geometric structures associated with bipartite graphs on surfaces have recently emerged and turn out to be crucial for studying a variety of problems. On the one hand, probabilists working in statistical mechanics are interested in finding appropriate embeddings of planar graphs that lead to discrete complex analysis theories, which are well suited for observing conformally invariant objects in the scaling limit. In the course of doing so, they established deep connections between specific immersions of the underlying graphs, integrability of the models, and Harnack curves. On the other hand, several spaces of geometric objects have recently been found to be parametrized by weighted bipartite graphs on surfaces, relating them to cluster algebras and integrable systems. These include objects from discrete differential geometry (e.g. Q-nets, Darboux maps), positive Grassmannians, and higher Teichmüller spaces. Moreover, connections between knot theory and the dimer model have started to emerge and beg for a better understanding.

Topics: Part of the Long Program Geometry, Statistical Mechanics, and Integrability