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Boundary and Curvature on Graphs

Stefan Steinerberger, University of Washington
Tuesday, March 8, 2022 - 1:30pm to 3:30pm
PDL C-401

Abstract:  I will discuss new definitions of boundary and curvature on combinatorial graphs. The boundary will include an earlier definition of Chartrand-Erwin-Johns-Zhang and satisfy an isoperimetric principle: graphs with many vertices have a large boundary (or gigantic diameter: the path graph only has 2 boundary vertices independently of length).  There are multiple definitions of curvature on graphs (combinatorial, via the behavior of the Laplacian, via Optimal Transport). We propose a new one based on potential theory: this curvature can be computed by solving a linear system and has a large number of desirable properties: it satisfies a Bonnet-Myers theorem (graphs with curvature bounded from below have diameter bounded from above) as well as an inverse Bonnet-Myers and a Lichnerowicz theorem. The von Neumann Minimax Theorem from Game Theory plays a crucial role in the proofs. There will be many pictures and many open problems.

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