Mathematics Colloquia and Seminars

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Description

Consider a manifold M that is either embedded in Euclidean space or a Riemannian manifold. We sample points X_1,\dots,X_n from an unknown probability measure \mu on M. We observe only a single random graph G on {1,\dots,n}, where edges {i,j} appear independently with probability p(|X_i-X_j|) for a  monotone decreasing connection function p.   This setting asks a basic inverse question: how much of the underlying geometry and sampling measure can be recovered from connectivity alone?    In this talk I will describe the reconstruction results showing that, under natural regularity conditions, the combinatorial structure of G encodes substantial geometric information.    Joint work with Pakawut Jiradilok and Elchanan Mossel.