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

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.

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