Living systems, from cancer cells to microbial colonies and human populations, are inherently spatial, meaning that individuals interact within complex networks of relationships. However, current theory attempting to understand and predict the future evolution of these systems largely overlooks this spatial complexity. This project will build new, more realistic frameworks to understand how the intricate "shape" of biological populations, such as their patterns of interaction and reproduction, influences their future evolution. This includes studying how quickly mutations spread, how pathogenic populations expand and how organisms adapt to new treatments and environments. By combining cutting-edge mathematics, computational tools, and data from real biological systems, the project will create open-source software and visualization tools to help researchers interpret emerging, highly detailed spatial datasets. These tools will provide a foundation for studying evolutionary dynamics in spatially structured populations across diverse areas, including cancer progression, aging and microbial ecosystems. The project also emphasizes education and public engagement, incorporating interactive visualizations and undergraduate and graduate research experiences to make complex evolutionary concepts accessible and inspiring to a broad audience and train the future US scientific and industrial workforce. At a technical level, the project develops novel mathematical and computational mo