Some possible topics for Further Topics in Differential Geometry, 2 (Spring 2016) are listed below. Only a small number of these could be covered, and students’ interests will influence the choice of topics. Many of the topics are interrelated, and portions of some of them may be covered in the fall semester (Fall 2015).

• Connections on vector bundles. Subtopics include curvature, parallel transport, and holonomy.
• Chern-Weil theory and characteristic classes
• Lie groups (aspects not covered last year in MTG 6256-7)
• Homogeneous spaces
• Principal fiber bundles. Subtopics include:
  • Reduction and enlargement of structure group
  • Associated vector bundles
  • Connections on principal bundles, with sub-subtopics:
    • curvature, parallel transport, and holonomy, and
      the induced structures on associated vector bundles

    • holonomy groups
• Laplacian on differential forms; Hodge Theorem (relating de Rham cohomology and harmonic differential forms)
• Other topics in geometric PDEs (i.e. other than the Hodge Theorem), such as:
  • Heat equation and heat kernel on a Riemannian manifold
  • Gauge theory and the Yang-Mills equations
  • Dirac operators and the Dirac equation
  • Various curvature-related equations on Riemannian manifolds
• Spin and Spin^c groups, structures, and bundles
• Jet bundles, differential operators, symbol of a linear differential operator
• Introduction to complex manifolds and Kaehler manifolds
• Morse Theory (relating the topology of a manifold to the critical points of real-valued functions on it)
• Selected topics in differential topology, such as:
  • intersection numbers
  • degrees of maps
  • Poincare-Hopf Index Theorem (relating Euler characteristic of a manifold to zeroes of vector fields)
  • Lefschetz Fixed-Point Theorem
  • “Easy” Whitney embedding theorem (every smooth compact n-manifold embeds in R^{2n+1})