BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:David Hume (Bristol) DTSTART:20230124T140000Z DTEND:20230124T160000Z DTSTAMP:20260423T023043Z UID:WienGAGT/32 DESCRIPTION:Title: Thick embeddings of graphs into symmetric spaces\nby David Hume (Bri stol) as part of Vienna Geometry and Analysis on Groups Seminar\n\n\nAbstr act\nInspired by the work of Kolmogorov-Barzdin in the 60’s and more rec ently by Gromov-Guth on thick embeddings into Euclidean spaces\, we consid er thick embeddings of graphs into more general symmetric spaces. Roughly\ , a thick embedding is a topological embedding of a graph where disjoint p airs of edges and vertices are at least a uniformly controlled distance ap art (consistent with applications where vertices and edges are considered as having volume). The goal is to find thick embeddings with minimal “vo lume”.\n\nWe prove a dichotomy depending upon the rank of the non-compac t factor of the symmetric space. For rank at least 2\, there are thick emb eddings of \\(N\\)-vertex graphs with volume \\(\\leq C N\\log(N)\\) where \\(C\\) depends on the maximal degree of the graph. By contrast\, for ran k at most 1\, thick embeddings of expander graphs have volume \\(\\geq c N ^{1+a}\\) for some \\(a\\geq 0\\).\n\nThe key tool required for these resu lts is the notion of a coarse wiring\, which is a continuous embedding of one graph inside another satisfying some additional properties. We prove t hat the minimal “volume” of a coarse wiring into a symmetric space is equivalent to the minimal volume of a thick embedding. We obtain lower bou nds on the volume of coarse wirings by comparing the relative connectivity (as measured by the separation profile) of the domain and target\, and up per bounds by direct construction.\n\nThis is joint work with Benjamin Bar rett.\n LOCATION:https://researchseminars.org/talk/WienGAGT/32/ END:VEVENT END:VCALENDAR