Join the GW Data Science Program for a seminar with Vitaliy Kurlin, professor of algorithms and computing systems at the University of Liverpool. Professor Vitaliy Kurlin is a mathematician by training and leads the Data Science Theory and Applications group in the Materials Innovation Factory at the University of Liverpool, UK.
 
Dr. Kurlin's talk, The geo-mapping problem in Geometric Data Science, is based on the papers [1,2,3] extending Topological Data Analysis (TDA) to a wider area of Geometric Data Science, which aims to continuously parametrise moduli spaces of real objects under practical equivalences. The key example is a cloud A of unordered points under isometry in ℝ. Standard filtrations (Vietoris-Rips, Cech, Delaunay) of complexes on A are invariant under isometry (any distance-preserving transformation). Hence, persistent homology of these filtrations can be considered a partial solution to the following geo-mapping problem: design an invariant I of clouds of m unordered points satisfying these conditions: 

• Completeness: any clouds A,B in ℝn are related by rigid motion if and only if I(A)=I(B);
• Realisability: the invariant space {I(A) for all clouds A in ℝn} is explicitly parameterised so that any sampled value I(A) can be realised by a cloud A, uniquely under motion in ℝn; 
• 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; 
• Computability: the invariant I, the metric d, and a reconstruction of A in ℝn 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, which remains open for other data (embedded graphs, meshes, or complexes) and relations (dilation, affine, or projective maps).

 
[1] V.Kurlin. Complete and continuous invariants of 1-periodic sequences in polynomial time. SIAM J Mathematics of Data Science, v.7, p.1643-1663 (2025).
 
[2] P.Smith, V.Kurlin. Generic families of finite metric spaces with identical or trivial 1-dimensional persistence. J Applied and Computational Topology, v.8, p.839–855 (2024).
 
[3] D.Widdowson, V.Kurlin. Recognizing rigid patterns of unlabeled point clouds by complete and continuous isometry invariants with no false negatives and no false positives. Proceedings of CVPR 2023, p.1275-1284.