Recorded 06 March 2025. Federico Ardila of San Francisco State University presents "Intersection theory and combinatorics: variations on a theme" at IPAM's LatMath 2025 Workshop. Abstract: My talk will discuss some beautiful objects at the intersection of combinatorics, geometry, and algebra called “Chow rings of toric varieties”. I will discuss three ways of thinking about them: they can be approached using algebra, polyhedral geometry, or numerical analysis. I will then explain how combinatorialists have recently used these rings to prove several conjectures from the 1970s and 80s about the colorings of a map. My talk will not assume that you know anything about these objects. I will talk about the work of many people, including my joint work with Carly Klivans, Graham Denham, and June Huh. Learn more online at: https://www.ipam.ucla.edu/programs/special-events-and-conferences/latmath-2025/?tab=overview