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Vector bundles over differentiable manifolds is a mathematical setup for physical theories. In this series of lectures, we describe the calculation of the Chern and Pontrjagin classes of a vector bundle (E) over a differentiable manifold (M), in terms of the invariant polynomials of the matrix of the curvature 2-form of the bundle (F). Action integrals expressed in terms of the curvature 2-form of the bundle have topological lower bounds given by the characteristic classes of the bundle. We illustrate these situations for specific examples. The lectures are based on the book “Characteristic Classes” by Milnor and Stasheff.

Ayse Bilge Hümeyra