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Graduate Research Seminar in Applied Topology (GRST)

Monday, April 12, 2021 - 3:00pm

Sara Kalisnik

Wesleyan University


University of Pennsylvania

via Zoom (link in abstract)

A common problem in data science is to determine properties of a space from a sample. For instance, under certain assumptions a subspace of a Euclidean space may be homotopy equivalent to the union of balls around sample points, which is in turn homotopy equivalent to the Čech complex of the sample. This enables us to determine the unknown space up to homotopy type, in particular giving us the homology of the space. A seminal result by Niyogi, Smale and Weinberger states that if a sample of a closed smooth submanifold of a Euclidean space is dense enough (relative to the reach of the manifold), there exists an interval of radii, for which the union of closed balls around sample points deformation retracts to the manifold. A tangent space is a good local approximation of a manifold, so we can expect that an object, elongated in the tangent direction, will better approximate the manifold than a ball. We present the result that the union of ellipsoids of suitable size around sample points deformation retracts to the manifold while requiring much smaller density than in the case of union of balls. The proof requires new techniques, as unlike the case of balls, the normal projection of a union of ellipsoids is in general not a deformation retraction. Zoom link: