Explore the interconnections between three distinct mathematical problems through the lens of Chern classes in this 48-minute conference talk. Discover how the Tverberg-Vrećica conjecture from combinatorial geometry, the existence of linearly independent complex line fields on manifolds from algebraic topology, and the calculation of persistent equivariant cohomology of the circle from applied topology are unified by common mathematical techniques. Learn how Chern classes emerge as crucial tools in resolving these problems and examine the symmetries that connect these seemingly disparate areas of mathematics. Gain insights into collaborative research spanning multiple mathematical domains and consider intriguing questions for future investigation in the intersection of combinatorial geometry, algebraic topology, and applied topology.