Abstract: The Laplace operator is a fundamental tool for studying the geometry of manifolds. Motivated by electric networks, Laplacians on graphs are defined using edge weights that play the role of conductance. When the weights are constant, the graph Laplacian reduces to the combinatorial Laplacian, which captures rich combinatorial information about the graph.
For a graph embedded on a surface, it is natural to consider a geometric Laplacian whose edge weights reflect the underlying metric. On Euclidean surfaces, the 1-skeleton of a geodesic triangulation satisfies the well-known cotangent formula, which expresses these weights in terms of the Euclidean geometry. In this talk, we present the hyperbolic analogue of this construction and discuss its applications to discrete harmonic maps into hyperbolic surfaces, which realize the weighted graph as the 1-skeleton of an equivariant convex polyhedral surface.
This is joint work with Ivan Izmestiev.

Abstract: Identifiability shows up everywhere, from rigidity questions in discrete geometry to tensor completion problems and secant varieties in algebraic geometry. At its core lies a simple question: when do a collection of algebraic relations determine the underlying points uniquely?
I will present a new framework that tackles this question for systems where algebraic constraints interact with combinatorial structure. The key idea is to understand how the combinatorics of a graph or hypergraph control local and global identifiability. Starting from the classical setting of graph rigidity, where relations come from Euclidean distances, we extend the theory to hypergraphs equipped with general algebraic measurements. This perspective allows us to obtain several necessary or sufficient conditions for identifiability, using tools from both rigidity theory and the geometry of secant varieties.
The results come from several joint works: (a) with James Cruickshank, Anthony Nixon, and Shin-ichi Tanigawa; (b) with Sean Dewar, Georg Grasegger, Kaie Kubjas, and Anthony Nixon; and (c) with Louis Theran and Jessica Sidman.