The geo-mapping problem in Geometric Data Science.

Meeting Details

For more information about this meeting, contact Rachel Weaver, John Harlim, Pierre-Emmanuel Jabin, Toan T. Nguyen, Wenrui Hao.

Speaker: Vitaliy Kurlin, University of Liverpool

Abstract: The talk introduces the central problem in the emerging area of Geometric Data Science [1], which aims to continuously parametrise moduli spaces of all real objects under practical equivalences. The key example is a cloud A of unordered points under an isometry in R^n. Standard filtrations (Vietoris-Rips, Cech, Delaunay) of complexes on A are invariant under isometry (any distance-preserving transformation). Hence, persistent homology of these filtrations [2] can be considered a partial solution to the following geo-mapping problem: design an invariant I of clouds of m unordered points satisfying the conditions below. (a) Completeness: any clouds A, B in R^n are related by rigid motion if and only if I(A)=I(B); (b) Realisability: the invariant space {I(A) for all clouds A in R^n} is explicitly parameterised so that any sampled value I(A) can be realised by a cloud A, uniquely under motion in R^n; (c) Bi-continuity: the bijection from the space of clouds to the space of complete invariants is Lipschitz continuous in both directions in a suitable metric d on the invariant space; (d) Computability: the invariant I, the metric d, and a reconstruction of A in R^n from I(A) can be obtained in polynomial time in the size of A, for a fixed dimension n. The talk will outline a full solution to this problem [3], which is open for other data (graphs, complexes) and relations (affine or projective maps)

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Room Reservation Information

Room Number: 114 McAllister

Date: 01/12/2026

Time: 12:00pm - 1:30pm