Lecturer: Jørgen Tornehave
Content:
The concept of curvature is one of the corner stones in modern geometry, with wide ramifications in other areas of mathematics - pure and applied.
The course introduces curvature for connections on vector bundles over smooth manifolds and constructs characteristic classes in de Rham cohomology from this point of view. These cohomology classes, originally defined by L. Pontrjagin and S. S. Chern in the 1940'ies, have played a fundamental role in mathematics, and, more recently, in mathematical physics. For example, the index theorem for elliptic differential operators is expressed in terms of characteristic classes.
The course will cover the Gauss - Bonnet theorem, generalized to arbitrary dimensions, and the Poincarè - Hopf theorem, which relates the singularities of vector fields to the Euler characteristic of the manifold.
The course will be structured with a mixture of lectures and student seminars.
Prerequisites: Topology 1, chap 1-10 of the book below.
Literature: Ib Madsen and Jørgen Tornehave, From Calculus to Cohomology, Cambridge Univ. Press, 1997.