2026 Analysis on Metric Space Seminar: Nonlinear Neumann Eigenvalues in Outward Cuspidal Domains With Weighted Measure" by Professor Alexander Menovschikov, HSE University

Speaker: Professor Alexander Menovschikov, HSE University

Title: Nonlinear Neumann Eigenvalues in Outward Cuspidal Domains With Weighted Measure

Abstract:

We consider the nonlinear Neumann eigenvalue problem in outward cuspidal domains with a weighted measure. Using composition operators on Sobolev spaces, we establish embeddings of Sobolev spaces into weighted Lebesgue spaces. These embeddings give the solvability of the Neumann spectral problem in this setting and provide estimates for the corresponding weighted Neumann eigenvalues

2026 Analysis on Metric Space Seminar"Base Modulus for Matroid Truncation, Strength, and Fractional Arboricity" by Huy Truong, Kansas State University

Speaker: Huy Truong, Kansas State University

Title: Base Modulus for Matroid Truncation, Strength, and Fractional Arboricity

Abstract: 

In previous work, we studied the -modulus of the family of all bases of a matroid and showed that it recovers several classical concepts in matroid theory, including strength, fractional arboricity, and principal partitions. These results generalize corresponding concepts for spanning trees in graphs. Due to computational constraints, one may impose a bound on the number of elements sampled from a base. For instance, when exploring a tree, we may stop at forests with edges. Such objects are captured by matroid truncations. In this paper, we study the modulus of matroid truncations and determine the universal density for every truncation of a given matroid. As a consequence, we show that the truncation modulus serves as an approximation of the original matroid modulus.

2026 Analysis on Metric Space Seminar: "On Mappings Generating Embedding Operators in Sobolev Classes on Metric Measure Spaces" by Prof.Alexander Menovschikov, HSE University

Speaker: Professor Alexander Menovschikov, HSE University

Title: On Mappings Generating Embedding Operators in Sobolev Classes on Metric
Measure Spaces

Abstract:

Let X =(X,ρ,µ) be a doubling metric measure space which supports the weak p-Poincar´e inequality. We consider bi-measurable homeomorphisms φ : Ω → Ω, of bounded domains Ω, Ω ⊂ X, which generate bounded composition operators on Newtonian–Sobolev spaces N1,p(Ω) → N1,q(Ω),1 < q ≤ p < ∞. We prove the Luzin N−1-property of such mappings with respect to capacities and obtain necessary and sufficient conditions on bi-measurable homeomorphisms that generate bounded composition operators on Newtonian–Sobolev spaces. We prove this by using special test functions generated by distance functions. On the base of the composition operators we consider Sobolev type embedding theorems in weak (p,q)-quasiconformal α-regular domains

2026 Analysis on Metric Space Seminar: "The p-elastic flows of inextensible planar closed curves for p ∈ (1, ∞)" by Prof.Chun-Chi Lin, National Taiwan Normal University

Speaker: Professor Chun-Chi Lin, National Taiwan Normal University

Title: The p-elastic flows of inextensible planar closed curves for p ∈ (1, ∞)

Abstract:

Building on the work of Blatt, Hopper, and Vorderobermeier (2022) on the existence of geometric flows for the p-elastic energy with p∈ [2,∞), we investigate how to extend their results to the case p∈(1,2). In this talk, we focus on the class of inextensible planar closed curves. We establish the existence of global weak solutions to the L 2 -gradient flow of the p-elastic energy for all p ∈ (1, ∞). Further directions, including extensions to open curves and applications to variational problems of curves in sub-Riemannian geometry, will also be discussed. The results presented are based on joint work with Ying-Hsian Tsai.

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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.