2026 Analysis on Metric Space Seminar: "Modulus of Families of Lipschitz Chains with Arbitrary" by Dr.Andrew Jensen, Kansas State University

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.

2026 Analysis on Metric Space Seminar: "Discrete p-Modulus and Orthodiagonal Maps" by Prof.Pietro Poggi-Corradini, Kansas State University

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.

2026 Analysis on Metric Space Seminar: "Modulus, Duality, and Families of Objects on Graphs" by Prof.Nathan Albin, Kansas State University

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.

2026 Analysis on Metric Space Seminar "Ollivier-Ricci curvature in non-smooth Lorentzian geometry and causal set theory" by Dr.Samuël Borza, University of Vienna

Speaker: Dr. Samuël Borza, University of Vienna

Title: Ollivier-Ricci curvature in non-smooth Lorentzian geometry and causal set theory

Abstract:

This talk will explore some aspects of non-smooth Lorentzian geometry, the mathematical framework underlying Einstein’s general relativity, which is currently being developed. Just as metric length spaces provide a synthetic generalisation of smooth Riemannian manifolds, the time-separation function plays the role of a “distance” in Lorentzian geometry. The need for a non-smooth Lorentzian framework appeared early on, most famously with Penrose’s singularity theorems. After introducing the basic concepts and some initial results in this synthetic setting, we will turn to causal set theory, a radical approach to quantum gravity in which spacetime is modelled as a discrete causal graph. I will formulate a new notion of curvature, inspired by Ollivier-Ricci curvature on metric graphs, using optimal transport between causal diamonds. We will see that it does recover Ricci curvature on smooth Lorentzian manifolds, and numerical examples will be presented.

2026 Analysis on Metric Space Seminar: "P-Dirichlet spaces and the resolution of the resistance and energy image density conjectures" by Prof.Sylvester Eriksson-Bique, University of Jyv¨askyl¨a

Speaker: Sylvester Eriksson-Bique, University of Jyv¨askyl¨a

Title: P-Dirichlet spaces and the resolution of the resistance and energy image density conjectures

Abstract: I will describe the resolution of two conjectures related to Dirichlet forms. In both cases a conceptually simple solution arises by stepping away from the p=2 regime. This leads to a new definition of a p-Dirichlet space, which unifies three quite different areas: Dirichlet form theory, Analysis on fractals and Analysis on metric spaces. The talk includes joint work with Mathav Murugan

2026 Analysis on Metric Space Seminar: "The Trace Theorem for Sobolev Homeomorphisms" by Dr.Aleksis Koski, Aalto University

Speaker: Aleksis Koski, Aalto University

Title:The Trace Theorem for Sobolev Homeomorphisms

Abstract: Classical Sobolev trace theory tells us when a boundary map can be extended as a Sobolev function inside a given domain in R^n. For the purposes of minimization problems in Nonlinear Elasticity, it is natural to rephrase this question in the context of extending a given embedding of the boundary as a homeomorphic Sobolev map. In this talk, I will explain what is known about this problem, ending with a full trace theory for Sobolev homeomorphisms in 2D.

2026 Analysis on Metric Space Seminar: "Dynamic inverse problems regularized with Wasserstein-1 transport" by Dr.Julius Lohmann, Institute of Science Tokyo

Speaker:Julius Lohmann JSPS International Research Fellow, Institute of Science Tokyo

Title: Dynamic inverse problems regularized with Wasserstein-1 transport Julius Lohmann JSPS International Research Fellow, Institute of Science Tokyo

Abstract: The (classical, balanced) Wasserstein-p distance can be used as a measure of how close a source and sink mass distribution (with equal mass) are. In recent years, the Wasserstein-2 distance has been employed in the temporal regularization of dynamic inverse problems. The so-called Benamou–Brenier formula states that it can be written as the square root of the performed physical work through the transport from the source to the sink. In my talk, I will instead focus on dynamic inverse problems regularized with Wasserstein-1 transport. The Wasserstein-1 distance can be interpreted as the optimal transport cost with respect to the Euclidean distance: it equals infπ R |x−y|dπ(x, y), where measure element dπ(x, y) indicates the (infinitesimal) amount of mass moving from location x to y. I will explain a novel dynamic inverse problem on time-parameterized curves in the induced Wasserstein-1 (metric) space. It is a natural extension of static sparse optimization problems such as lasso or TV regularization. One essential difference to classical regularization with Wasserstein-2 transport is that it allows for discontinuous decision variables (realized as BV curves). Despite this weak regularity requirement and the non-differentiability of the cost function (x, y) 7→ |x − y|, it is possible to prove the existence of a sparse solution and its characterization. I will present this result. Further, I will detail an adaption of the fully-corrective generalized conditional gradient method to the problem and highlight a natural discretization approach. Finally, I will show some numerical examples. Joint work with: Marcello Carioni

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2025 Analysis on Metric Space Seminar: "Almost splitting and quantitative stratification for super Ricci flow" by Professor Yohei Sakurai, Saitama University

Speaker: Professor Yohei Sakurai, Saitama University

Title: Almost splitting and quantitative stratification for super Ricci flow

Abstract:

I will discuss almost rigidity properties of super Ricci flow whose Muller quantity is non-negative.
I will present almost splitting and quantitative stratification theorems that have been established by Bamler for Ricci flow. This talk is based on the joint work with Keita Kunikawa (Tokushima university).

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2025 Analysis on Metric Space Seminar: " Fluid structure interaction and Sobolev spaces on changing domains" by Professor Malte Kampschulte, Charles University

Speaker: Professor ​Malte Kampschulte, Charles University in Prague.

Title: Fluid structure interaction and Sobolev spaces on changing domains

Abstract:

Dynamic interaction between fluids and solids occurs everywhere, from a fish swimming in the water to an airplane wing bending in the wind. When studying them, one of the key issues is that finding the domain on which the fluid equations are posed itself is part of the problem. Thus before we can apply all the standard machinery from PDE and the calculus of variations, we must first translate it to work on changing domains. The aim of this talk will be to explore precisely this. On one hand it will be an introduction into fluid structure interaction highlighting some recent results. On the other hand this will be used as an opportunity to illustrate and discuss several old and new methods on how to deal with Sobolev spaces when the domain itself is not fixed. This talk is based on results obtained with S.Schwarzacher and B.Benešová as well as work with N.Evseev and A.Menovschikov.

Analysis on Metric Spaces Seminar 2025 | Topological regularity of Busemann spaces, Professor Tadashi Fujioka, Kyoto University

Title: Topological regularity of Busemann spaces, Professor Tadashi Fujioka, Kyoto University

Abstract:

We discuss the topological regularity theorem for Busemann spaces of nonpositive curvature, while reviewing the corresponding results for Alexandrov spaces and CAT spaces. All of these are metric spaces with upper or lower curvature bounds in some synthetic senses, and we address the question of when such spaces are topological manifolds. This is joint work with Shijie Gu (Northeastern University, China). Preprint available at arXiv:2504.14455.

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