BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:María José Jiménez Rodríguez (Universidad de Sevilla - Spain) DTSTART:20241122T160000Z DTEND:20241122T170000Z DTSTAMP:20260423T022931Z UID:GEOTOP-A/91 DESCRIPTION:Title: Morse Theory for Chromatic Delaunay Triangulations\nby María José Jiménez Rodríguez (Universidad de Sevilla - Spain) as part of GEOTOP-A s eminar\n\n\nAbstract\nThis talk is focused on new techniques for the topol ogical data analysis (TDA) of labelled point cloud data.\nWell-established filtrations in TDA for a point cloud data include the Čech\, Vietoris– Rips\, and alpha filtrations. Bauer and Edelsbrunner [BE16] demonstrated t hat the Čech filtration can be simplicially collapsed onto the alpha filt ration\, showing that they are homotopy equivalent.\nRecent techniques in data collection across fields like cancer biology\, geospatial analysis an d ecology have produced chromatic (labeled) data that express interactions among points of different colors. In such cases\, it is crucial to unders tand not only the overall spatial structure of the data but also the spati al relationships among subsets defined by their labels. The chromatic alph a filtration [Mon+24] is a generalization of the alpha filtration that ca ptures these relationships\, making it particularly useful for multi-speci es data in TDA.\nIn this talk we introduce the chromatic Delaunay–Čech and chromatic Delaunay–Rips filtrations as computationally efficient alt ernatives to the chromatic alpha filtration. We use generalized discrete M orse theory to demonstrate that the Čech\, chromatic Delaunay–Čech\, a nd chromatic alpha filtrations are interconnected through simplicial colla pses\, extending Bauer and Edelsbrunner’s results from non-chromatic to chromatic contexts. \nOur findings offer theoretical support for the appli cation of chromatic Delaunay–Čech and chromatic Delaunay–Rips filtrat ions\, and we illustrate their computational advantages through numerical experiments.\n This is joint work with A. Natarajan\, T. Chaplin and A. Br own from University of Oxford (arXiv:2405.19303).\n[BE16] Ulrich Bauer an d Herbert Edelsbrunner. “The Morse theory of Cech and Delaunay complexes ”. In: Transactions of the American Mathematical Society 369.5 (Dec. 27 \, 2016)\, pp. 3741–3762. DOI:10.1090/tran/6991.\n[Mon+24] Sebastiano Cu ltrera di Montesano et al. Chromatic Alpha Complexes. Feb. 7\, 2024. arXiv : 2212.03128\n LOCATION:https://researchseminars.org/talk/GEOTOP-A/91/ END:VEVENT END:VCALENDAR