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, Vietoris–Rips, 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 Delaunay–Cech and chromatic Delaunay–Rips filtrations as computationally efficient alternatives to the chromatic alpha filtration. We use generalized discrete Morse theory to demonstrate that the Cech, chromatic Delaunay–Cech, and chromatic alpha filtrations are interconnected through simplicial collapses, extending Bauer and Edelsbrunner’s results from non-chromatic to chromatic contexts. Furthermore, we establish that the chromatic Delaunay–Rips filtration is locally stable under perturbations of the underlying point cloud.
Our findings offer theoretical support for the application of chromatic Delaunay–Cech and chromatic Delaunay–Rips 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. 3741–3762. 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

Ponente: Nabil Laiche, Lecturer In Mathematics and Computer Sciences Department, University of Oum El Bouaghi, Argelia.

Título: Estimating Coefficients of a Stochastic Differential Equation Using a Bilinear Time Series Model: An Illustration with Simulations Based on Empirical Moments

Resumen: Our contribution focuses on estimating a sample of stochastic differential equations through discretization into a bilinear time series model. This approach enables us to e§ectively estimate the coefficients using the method of empirical moments. This work addresses key challenges in connecting stochastic di§erential equations with nonlinear time series models, providing valuable insights into their relationships. By leveraging simulations and numerical computations, we illustrate the asymptotic behavior of the estimators derived from the empirical moments method. Our Öndings not only enhance the understanding of the dynamics inherent in these models but also offer practical solutions for applied researchers dealing with complex data structures. Overall, this study contributes to bridging the gap between theoretical frameworks and empirical applications in time series analysis.

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:
 Presencial: Seminario del Dpto. Matemática Aplicada I (B2.85), Escuela Tca. Sup. de Ingeniería Informática
 Online: Enlace de Teams

Ponente: Carlos Andrés Toro, estudiante predoctoral del IMPA

Título: Non-existence of free boundary minimal Möbius bands in the unit three-ball

Resumen: We review the main constructions of free boundary minimal surfaces in the Euclidean unit ball $\mathbb{B}^3$ for compact orientable topologies. In the non-orientable setting we prove the non-existence of free boundary minimal Möbius bands in $\mathbb{B}^3$. This answers in the negative a question proposed by I. Fernández, L. Hauswirth and P. Mira.

Financiación: 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