Alexandros Eskenazis：Functional inequalities in Metric Geometry——I

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https://www.youtube.com/watch?v=kCPCceeSi7M
Hausdorff Center for Mathematics
Alexandros Eskenazis: Functional inequalities in Metric Geometry I

I will present a collection of results in which discrete functional inequalities play the role of invariants to bi-Lipschitz embeddings of finite graphs into Banach and metric spaces. Examples of such invariants include the nonlinear versions of type and cotype, Markov convexity, diamond convexity, the nonlinear spectral gap inequality, and others. From these, one can in turn deduce nonembeddability results for the Hamming cube, l∞-grids, trees, diamond graphs and expanders.

Alexandros Eskenazis——“Some geometric applications of the discrete heat flow”：https://doi.org/10.48550/arXiv.2310.01868

We present two geometric applications of heat flow methods on the discrete hypercube {−1,1}n. First, we prove that if X is a finite-dimensional normed space, then the bi-Lipschitz distortion required to embed {−1,1}n equipped with the Hamming metric into X satisfies cX({−1,1}n)≳supp∈[1,2]nTp(X)min{n,dim(X)}1/p, where Tp(X) is the Rademacher type p constant of X. This estimate yields a mutual refinement of distortion lower bounds which follow from works of Oleszkiewicz (1996) and Ivanisvili, van Handel and Volberg (2020) for low-dimensional spaces X. The proof relies on an extension of an important inequality of Pisier (1986) on the biased hypercube combined with an application of the Borsuk-Ulam theorem from algebraic topology. Secondly, we introduce a new metric invariant called metric stable type as a functional inequality on the discrete hypercube and prove that it coincides with the classical linear notion of stable type for normed spaces. We also show that metric stable type yields bi-Lipschitz nonembeddability estimates for weighted hypercubes.

Alexandros Eskenazis, Yair Shenfeld——“Intrinsic dimensional functional inequalities on model spaces”：https://doi.org/10.1016/j.jfa.2024.110338

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