Speaker: Professor Nathan Albin, Kansas State University Title: Modulus, Duality, and Families of Objects on Graphs Abstract:

Given a discrete graph and a family of objects (walks, spanning trees, edge covers, etc.) on the graph, p-modulus provides a mathematical way to quantify the "richness" or "robustness" of that family. Acting as a tunable metric, p-modulus generalizes classical graph metrics—such as shortest path, effective resistance, and minimum cut—to provide a multifaceted view of the graph's topology and geometry. Through the lens of modulus, we can explore a variety of structural properties of the graph. This talk will introduce p-modulus, describe its basic properties, connect it to well-known graph-theoretic quantities, and explore the powerful theory of Fulkerson blocking duality, which connects each family of objects to a natural dual family that provides deep insights into the graph's structural properties.

Speaker: Professor Pietro Poggi-Corradini, Kansas State University Title: Discrete p-Modulus and Orthodiagonal Maps Abstract:

This project is joint work with Nathan Albin, Joan Lind and Pekka Pankka. Our goal is to approximate planar p-capacity (or continuous p-modulus) in topological rectangles using discrete p-modulus defined on an approximating orthodiagonal map. To that end, I will first introduce the planar p-capacity problem we are interested in and then I will give an overview of the theory of p-modulus on finite graphs, describing various notions of duality, and establishing its relation to the discrete p-Laplacian and to non-linear flows.

Speaker: Mr. Andrew Jensen, Kansas State University Title: Modulus of Families of Lipschitz Chains with Arbitrary Dimension and Codimension Abstract:

Recently, Lohvansuu (2023) introduced the p-modulus for families of k-dimensional Lipschitz chains and their dual families of (n-k)-dimensional chains. While he established an upper bound for the duality of these families on Lipschitz cubes, the corresponding lower bound remained an open question. Subsequently, Kangasniemi and Prywes (2025) developed dMod, a related notion of modulus based on differential forms, and successfully established a full duality result. In this talk, I will explore the implications of these developments and discuss related open problems.