Ponente: Abhinav Natarajan, investigador de la Universidad
de Oxford, Reino Unido.
Título: Morse Theory for Chromatic Delaunay Triangulations
Resumen: This talk is focused on new techniques for the topological
data analysis (TDA) of labelled point cloud data.
Well-established filtrations in TDA for a point cloud data include
the Cech, VietorisRips, and alpha filtrations. Bauer and Edelsbrunner
[BE16] demonstrated that the Cech filtration can be simplicially
collapsed onto the alpha filtration, showing that they are homotopy
equivalent.
Recent techniques in data collection across fields like cancer biology,
geospatial analysis and ecology have produced chromatic (labeled)
data that express interactions among points of different colors.
In such cases, it is crucial to understand not only the overall
spatial structure of the data but also the spatial relationships
among subsets defined by their labels. The chromatic alpha filtration
[Mon+24] is a generalization of the alpha filtration that captures
these relationships, making it particularly useful for multi-species
data in TDA.
In this talk we introduce the chromatic DelaunayCech and chromatic
DelaunayRips filtrations as computationally efficient alternatives
to the chromatic alpha filtration. We use generalized discrete Morse
theory to demonstrate that the Cech, chromatic DelaunayCech,
and chromatic alpha filtrations are interconnected through simplicial
collapses, extending Bauer and Edelsbrunners results from
non-chromatic to chromatic contexts. Furthermore, we establish that
the chromatic DelaunayRips filtration is locally stable under
perturbations of the underlying point cloud.
Our findings offer theoretical support for the application of chromatic
DelaunayCech and chromatic DelaunayRips filtrations,
and we illustrate their computational advantages through numerical
experiments.
This is joint work with T. Chaplin and A. Brown from University
of Oxford and M.J. Jimenez from Universidad de Sevilla (arXiv:2405.19303).
[BE16] Ulrich Bauer and Herbert Edelsbrunner. The Morse theory
of Cech and Delaunay complexes. In: Transactions of the American
Mathematical Society 369.5 (Dec. 27, 2016), pp. 37413762.
DOI:10.1090/tran/6991.
[Mon+24] Sebastiano Cultrera di Montesano et al. Chromatic Alpha
Complexes. Feb. 7, 2024. arXiv: 2212.03128
Financiación:
- Ayuda VIIPPIT-2024-III.3 A Estancias breves. PROGRAMA DE INVESTIGADORES
DE OTROS CENTROS NACIONALES Y EXTRANJEROS EN DEPARTAMENTOS E INSTITUTOS
DE INVESTIGACIÓN DE LA US. Modalidad A. Estancias breves en la US
de profesores e investigadores de reconocido prestigio de otras
Universidades o Centros de Investigación.
- Ayudas para estancias breves del Departamento. VII PLAN PROPIO
DE INVESTIGACIÓN Y TRANSFERENCIA DE LA UNIVERSIDAD DE SEVILLA. LÍNEA
ESTRATÉGICA 5. ACCIONES ESTRATÉGICAS DE INVESTIGACIÓN Y TRANSFERENCIA.
V.1 PROGRAMA AL ESTÍMULO DE ÁREAS CON NECESIDADES INVESTIGADORAS
Y DE ACTIVIDAD INVESTIGADORA EMERGENTE. Modalidad A1. Ayudas para
áreas de conocimiento con necesidades investigadoras y con alto
potencial.
Lugar:
Seminario del Dpto. Matemática Aplicada I (B2.85), Escuela
Tca. Sup. de Ingeniería Informática
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