The last twenty years of rapid development of Topological Data Analysis (TDA) have shown the need of understanding the shape of data to better understand the data. Since an explosion of new ideas in 2000’ including those of Persistent Homology and Mapper Algorithms, the community rushed to solve detailed theoretical questions related to the existing invariants. However, topology and geometry have still much to offer to the data science community. New tools and techniques are within reach, waiting to be brought over the fence to enrich our understanding and potential to analyze data.

At the same time, the fields of Discrete Morse Theory (DMT) and Combinatorial Topology (CT) are developed in parallel with no strong connection to data-intensive TDA or to other statistical pipelines (e.g. machine learning).

Our major goal is to soften this strict separation between TDA, CT, and DMT by creating new cross-community connections. We plan to bring together cross-sections of both communities, including researchers with theoretical, applied, and computational backgrounds. Initial speed up of interpersonal communication should result in a significant boost to all the involved areas and even encourage them to search for connections and common ground to create community in-between – transferring ideas, tools and methods.

This Dagstuhl Seminar aims to bring together a number of experts in Discrete Morse Theory, Combinatorial Topology, Topological Data Analysis, and Statistics to (i) enhance the existing interactions between these fields on the one hand, and to (ii) discuss the possibility of adopting new invariants from algebra, geometry, and topology; in particular inspired by continuous and discrete Morse theory and combinatorial topology; to analyze and better understand the notion of shape of the data.

We will start our discussion by understanding interactions of DMT and CT with the current tools of TDA and Statistics, i.e. we will start by examining how to input Mappers into statistical pipelines, study how to define averages and study the limit behavior. Building on this, new possible techniques will be explored and integrated into proper statistical frameworks. We will also discuss how to adopt new invariants from DMT and CT to the setting of TDA and how to input them into Statistics. Possible invariants from DMT and CT to be examined include invariant sets of discrete vector fields and algebraic invariants of (multi-)filtered complexes, respectively. Their stability, robustness and computational complexity will be thoroughly discussed. Lastly, if time permits, we want to test them on a number of sample datasets and compare their efficacy to the already existing tools.

We hope that our seminar in Dagstuhl will provide an interactive environment for the discrete Morse theory, combinatorial topology, and statistics communities that will result in new joint projects and research collaborations in applied topology and data science.

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Pawel Dlotko, Dmitry Feichtner-Kozlov, Anastasios Stefanou, and Yusu Wang