When
12:30 – 1:30 p.m., March 19, 2026
Where
Speaker: Geoff Vasil, The University of Edinburgh, Scotland
Title: From the discrete to continuum and back on metric graphs
Abstract: A metric graph is what most people first think about when picturing a network. Euler first created the idea of an algebraic graph with the Seven Bridges of Königsberg with abstract relations between the intersections. Ironically, the actual streets automatically form a metric graph, with the notion of length between nodes. Metric graphs are ubiquitous: optical fibre networks, electrical power grids, spider webs, vasculature, river systems, neuronal networks, textiles, fishing nets, or British Rail. Importantly, slender ``wires'' (edges) support wave or diffusion equations subject to Kirchhoff boundary conditions at the nodes.
This talk will discuss a continuum-limit framework that replaces discrete vertex-based equations with a coarse-grained partial differential equation (PDE) defined on the continuous space occupied by the network, calculating all macroscopic parameters from first principles via a systematic discrete-to-continuous local homogenization, finding an anomalous effective embedding dimension resulting from a homogenized diffusivity. The talk will also discuss some applications to biological transport networks, like leaf venation of blood flow. Finally, these applications lead to deep connections to complex analysis from a fully discrete vantage point.