Topology is considered one of the most prominent research fields in mathematics. It is concerned with the properties of a space that are preserved under continuous deformations and provides abstract representations of the space and functions defined on the space. The modern field of topological data analysis (TDA) plays an essential role in connecting mathematical theories to practice. It uses stable topological descriptors as summaries of data, separating features from noise in a robust way. This Dagstuhl Seminar aims to bring together researchers from mathematics, computer science, and application domains (e.g., materials science, neuroscience, and biology) to accelerate emerging research directions and inspire new ones in the field of TDA.

The research topics, listed below, reflect highly active and emerging areas in TDA. They are chosen to span topics in theory, algorithms, and applications. The invitees are chosen based on their background in mathematics, computer science, and application domains. While theoretical talks are welcome, the attendees are strongly encouraged to prepare a talk that is rooted in applications.

Multivariate data analysis. Topics include theoretical studies of multivariate topological descriptors (including multiparameter persistence), efficient algorithms for computing and comparing them, formal guarantees for data analysis based on such comparisons, and the development of practical tools based on such analysis. Combining topological analysis together with statistical learning-based methods will also be of interest.

Geometry and topology of metric spaces. A cornerstone of TDA is the study of metric and geometric data sets by means of filtrations of geometric complexes, formed by connecting subsets of the data points according to some proximity parameter. The study of such filtrations using homology leads to a multi-scale descriptor of the data that combines geometric and topological aspects of its shape. Besides their use in TDA, geometric complexes also play an important role in geometric group theory and metric geometry. The results and insights from both areas carry great promise for mutual interactions, leading to a unified view on computational and theoretical aspects.

Applications. TDA is an emerging area in exploratory data analysis and has received growing interest and notable successes with an expanding research community. The application of topological techniques to large and complex data has opened new opportunities in science, engineering, and business intelligence. This seminar will focus on a few key application areas, including material sciences, neuroscience, and biology.

Parallel and distributed computation. The computational challenges in TDA call for the use of advanced techniques of high-performance computing, including parallel, distributed, and GPU-based software. Many of the core methods of TDA, including persistent homology, mapper, merge trees, and contour trees, have received implementations beyond serial computing, and the interest in utilizing modern state-of-the-art techniques continues unabated. The task of optimizing algorithms in TDA is not only a question of engineering. Many of the key insights leading to breakthrough improvements are based on a careful utilization of theoretical properties and insights.

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Ulrich Bauer, Vijay Natarajan, and Bei Wang