Cohomology of product spaces from a categorical viewpoint

Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any:2016, Director: Carles CasacubertaOne of the aims of Algebraic Topology is to study complex problems within Topology by translating them into the more workable world of Algebra. This is usually done by defining invariants such as singular homology and singular cohomology. Both of them help us find properties of spaces. In many cases, homology is not enough and it is desirable to know the cohomology of spaces. For instance, although a product of two Klein Bottles and ($S^1 \vee \mathbb {R}P^2) \times (S^1 \vee \mathbb {R}P^2$) have isomorphic homology modules, their respective cohomologies have different ring structures, as shown in Example 3.15, which can also be found in [8]. All other examples and counterexamples given in this project are due to the author. These invariants are very powerful since they contain much information of the spaces that are being studied. The first goal of the present work is to prove a formula that relates the homology of a product of spaces with the homologies of the factors. This formula was obtained in the first half of the past century after the work of H. Künneth, who found in 1923 a relation between the Betti numbers of the product of two spaces and the Betti numbers of each of its factors; see [3, Chapter II, Section 5]. In this work, we want to study Künneth formulas for both homology and cohomology. Although the homology Künneth formula is a very standard result, the cohomology Künneth formula is harder to be found in modern literature, and this is why its study is interesting

Distributing Persistent Homology via Spectral Sequences

We set up the theory for a distributed algorithm for computing persistent homology. For this purpose we develop linear algebra of persistence modules. We present bases of persistence modules, and give motivation as for the advantages of using them. Our focus is on developing efficient methods for the computation of homology of chains of persistence modules. Later we give a brief, self contained presentation of the Mayer-Vietoris spectral sequence. Then we study the Persistent Mayer-Vietoris spectral sequence and present a solution to the extension problem. Finally, we review PerMaViss, a method implementing these ideas. This procedure distributes simplicial data, while focusing on merging homological information.Comment: Comments: 31 pages, 14 figures, 1 algorithm. Changes to previous version: longer introduction added, some material has been removed due to length constraints, section 5.2 added describing the procedure of computing the persistence Mayer-Vietoris spectral sequence, followed by complexity estimates in section 5.

Distributed Persistent Homology for 2D Alpha Complexes

We introduce a new algorithm to parallelise the computation of persistent homology of 2D alpha complexes. Our algorithm distributes the input point cloud among the cores which then compute a cover based on a rectilinear grid. We show how to compute the persistence Mayer-Vietoris spectral sequence from these covers and how to obtain persistent homology from it. For this, we introduce second-page collapse conditions and explain how to solve the extension problem. Finally, we give an overview of an implementation in C++ using Open MPI and discuss some experimental results.Comment: 32 pages, 15 figure

A Topological Approach to Measuring Training Data Quality

Data quality is crucial for the successful training, generalization and performance of artificial intelligence models. Furthermore, it is known that the leading approaches in artificial intelligence are notoriously data-hungry. In this paper, we propose the use of small training datasets towards faster training. Specifically, we provide a novel topological method based on morphisms between persistence modules to measure the training data quality with respect to the complete dataset. This way, we can provide an explanation of why the chosen training dataset will lead to poor performance

Nomenclatura en cuidados paliativos y de soporte renal: no solo al final de la vida

Resumen: La visión multidimensional de la enfermedad es fundamental en la atención de patologías complejas como la enfermedad renal crónica (ERC). Es oportuno definir y unificar conceptos que permitan que los diferentes profesionales encargados de la atención ofrezcan una atención multidisciplinar, alineados a las necesidades de cada persona.Debido al creciente aumento de la incidencia de ERC en el mundo y teniendo en cuenta que pueden existir diferentes trayectorias en el curso de la enfermedad, es necesario establecer abordajes integrales personalizados para cada paciente y sus familias de manera más temprana. Este planteamiento va más allá del simple control de los síntomas urémicos o de la congestión y consiste en abordar tempranamente los problemas no solo sintomáticos sino también funcionales, sociales y de afrontamiento de la enfermedad, facilitando la toma de decisiones tanto en el escenario de la ERC como en situaciones agudas, potencialmente irreversibles o en intervenciones que no mejoren el pronóstico vital.Para asegurar la excelencia en la atención es relevante evaluar indicadores para la atención paliativa y de soporte renal, como la presencia de la planificación anticipada y compartida de la atención, la inclusión de atención psicosocial, ética, espiritual y la atención al duelo. Esto permite ofrecer una atención integral, humanizada y de calidad para el paciente y sus familiares.Los cuidados paliativos y de soporte renal no se orientan únicamente a los pacientes en los últimos días de vida. Definir, unificar y evaluar los conceptos permitirá aplicarlos de manera oportuna en cada momento específico de la trayectoria de la ERC. Abstract: The multidimensional view of the disease is fundamental in the care of complex diseases such as chronic kidney disease (CKD). It is appropriate to define and unify concepts that allow the different professionals involved in care to provide multidisciplinary care tailored to the needs of each individual.Given the increasing incidence of CKD worldwide and the fact that the disease may progress at different rates, there is a need to establish personalized, comprehensive approaches for each patient and their families at an earlier stage. This approach goes beyond the simple control of uremic symptoms or congestion and consists of addressing not only symptomatic but also functional, social and coping problems at an early stage, facilitating decision making both in the CKD and in acute settings, potentially irreversible situations or interventions that do not improve life expectancy.To ensure excellence in care, it is important to assess indicators of palliative care and renal support, such as the presence of advance and shared care planning, the inclusion of psychosocial, ethical, spiritual and bereavement care. This enables the provision of comprehensive, humanized, and high-quality care for patients and their families.Palliative and renal care is not just about patients in the last days of life. Defining, unifying, and evaluating the concepts will allow them to be applied in a timely manner at each specific moment of the CKD trajectory