Persistent Homology of Chromatic Alpha Complexes Sebastiano Cultrera di Montesano, Ondˇ rej Draganov, Herbert Edelsbrunner, and Morteza Saghafian ISTA (Institute of Science and Technology Austria), Klosterneuburg, Austria Abstract Motivated by applications in medical sciences, we study finite chromatic sets in Euclidean space from a topological perspective. Based on persistent homology for images, kernels and cokernels, we design provably stable homological quantifiers that describe the geometric micro- and macro- structure of how the color classes mingle. These can be efficiently computed using chromatic variants of Delaunay mosaics and Alpha complexes. Funding. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme, grant no. 788183, from the Wittgenstein Prize, Austrian Science Fund (FWF), grant no. Z 342-N31, and from the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry and Dynamics’, Austrian Science Fund (FWF), grant no. I 02979-N35. 1 Introduction This paper takes a topological approach to characterize the mingling of a small number of point sets. The aim is the development of a mathematical language to answer questions like: “how and how often do blue points surround groups of red points?”, or “are there cycles made out of blue, red, and green points that make essential use of all three colors?”. We tackle these questions from a homological perspective, with the goal of disentangling patterns such as the ones shown in Figure 1. 1+2 1+1 1+0 2+0 3+0 2+1 Figure 1: The mingling patterns distinguished by the number of colors needed to form the cycle and the number of additional colors needed to fill the cycle. The drawing is a caricature of similar patterns for cycles different from circles and fillings different from disks. One of the motivations for this work is the recent growth of interest in spatial biology, which combines the biological properties of cells with their locations. An example is the tumor immune microenvironment [2] in cancer research, which focuses on the interplay between tumor and immune cells. Can we identify as well as quantify patterns in the interaction between cell types that correlate with clinical outcomes? Another biological process that raises similar mathematical questions is the segregation of cell types 1 arXiv:2212.03128v1 [math.AT] 6 Dec 2022