Lie groups and bundles

Marcel Nicolau (UAB) and Joan Porti (UAB)
 Mon-Wed 15h-17h, from Sept. 22, 2014
Aula A2, Centre de Recerca Matemàtica, Bellaterra Campus

Both Lie groups and line bundles are transversal tools that are used in many mathematical fields. We start from a geometrical pointview, but the course provides basic tools for students interested in algebraic and differential topology,
algebraic geometry and mathematical physics.

  1. Lie Groups. Matrix groups and Lie groups. The Lie algebra of a Lie group. The exponential map. Lie subgroups and homogeneous spaces. Symmetric spaces. Linear representations of Lie groups. An overview of the classification of Lie groups and of Lie algebras.
  2. Fiber Bundles. Vector bundles. The de Rham cohomology of a differentiable manifold. Linear connections and curvature. Principal bundles. The Weil homomorphism. Characterístic classes. The Euler class. Chern and Pontrjagin classes.
  3. Sheafs theory. Sheaves and presheaves. Axiomatic sheaf cohomology. Cech cohomology. The de Rham theorem. Cohomology of vector bundles.
  4. Hodge theory. Elliptic differential operators and elliptic complexes. The Hodge decomposition theorem on compact Riemannian manifolds.
  • Warner, Frank W. Foundations of differentiable manifolds and Lie groups.  Springer-Verlag, 1983.
  • Greub, Werner; Halperin, Stephen; Vanstone, Ray. Connections, curvature, and cohomology. Vol. I and II. Academic Press, 1972
  • Helgason, Sigurdur. Differential geometry, Lie groups, and symmetric spaces.  American Mathematical Society, 2001.
  • Bröcker, Theodor; tom Dieck, Tammo. Representations of compact Lie groups. Springer-Verlag, 1995.
  • Milnor, John W.; Stasheff, James D. Characteristic classes. Princeton University Press, 1974.