This Research Training Group (RTG) project at USC will train early career researchers at the undergraduate, graduate, and postdoctoral levels working in the areas of geometry, topology, and representation theory. These mathematical fields address fundamental questions related to shape, symmetry, and motion. They have close connections to many other areas of science, including high-energy physics, quantum computation, and orbital dynamics. This RTG project will fund potentially transformative research in these areas. It will also support workshops on cutting-edge research developments, undergraduate research programs, innovative course design projects, and activities connecting researchers with K-12 math teachers. These activities will foster lasting improvements in mathematics research and STEM training at USC and beyond. This RTG project will support student researchers and postdoctoral scholars supervised by senior personnel working in symplectic geometry, representation theory, homotopy theory, and low-dimensional topology. The project will leverage emerging interactions between these areas to develop new tools and structures, with applications to mirror symmetry, 4-manifold topology, combinatorics and algebraic geometry. The project will also fund a closely integrated set of training activities. An annual Current Developments Mini-Workshop will be conducted, bringing together experts on rapidly developing topics for an intensive period of research and collaboration wi