Schenck H. Algebraic Foundations for Applied Topology and Data Analysis

Cham: Springer, 2022. — 231 p.How do we reveal, characterize, and exploit the structure in data? Meeting this central challenge of modern data science requires the development of new mathematical approaches to data analysis, going beyond traditional statistical methods.Fruitful mathematical methods can originate in geometry, topology, algebra, analysis, stochastics, combinatorics, or virtually any field of mathematics.Confronting the challenge of structure in data is already leading to productive new interactions among mathematics, statistics, and computer science, notably in machine learning. We invite novel contributions (research monographs, advanced textbooks, and lecture notes) presenting substantial mathematics that is relevant to data science. Since the methods required to understand data depend on the source and type of the data, we very much welcome contributions comprising significant discussions of the problems presented by particular applications. We also encourage the use of online resources for exercises, software, and data sets.Contributions from all mathematical communities that analyze structures in data are welcome. Examples of potential topics include optimization, topological data analysis, compressed sensing, algebraic statistics, information geometry, manifold learning, tensor decomposition, support vector machines, neural networks, and many more.