Fundamentals of Differential K-Theory and Geometric Cohomology

Differential K-theory combines topology and geometry to refine classical topological invariants with geometric data such as connections and curvature. This text-based course offers an accessible path into understanding how smooth vector bundles, differential forms, and characteristic classes come together in modern geometric topology. You will start from core definitions and progressively build up the theoretical framework, learning how geometric structures refine ordinary K-theory and how differential characters operate in modern mathematical physics and differential geometry. What you'll learn: - Understand the foundational topology of vector bundles and connections on smooth manifolds - Apply Chern-Weil theory and explore Chern-Simons forms as secondary invariants - Learn the construction of differential K-theory groups from geometric data - Analyze differential characters and their relations to differential cohomology theories - Explore modern connections to gauge theory and mathematical physics models The course begins by clarifying essential definitions and basic algebraic topology concepts before guiding you step-by-step through differential refinements and their practical applications. This course is designed for mathematics and physics students or curious learners seeking a clear, accessible entry point into differential cohomology without unneeded complexity. Start reading today to deepen your understanding of modern geometric topology.