Lecture Schedule

Lectures

Tentative lecture plan is as follows:

Week 1

Topics: Review of differential geometry

• Topological and smooth manifolds,
• Vector and tensor fields,
• Algebra of differential forms,
• De Rham Cohomology

Reading:

 * Taubes Ch 1
 * Frenkel Sections 1-3
 * Nakahara Sections 5.1, 5.2, 5.4. 6.
 * Sontz Chapters 2, 3, 5 

Week 2

Topics: Review of differential geometry

 * vector fields
 * Lie brackets
 * diffeomorphisms
 * Lie derivatives
 * Lie groups and Lie algebras.
 * Exponential maps
 * adjoint representation
 * Basics of Symplectic Geometry
 * Hamiltonian vector fields
 * Hamiltonian group actions
 * Moment maps and co-moment maps.

Reading:

 * Rudolph-Schmidt (Differential Geometry and Mathematical Physics. Part I) Section 5

Week 3

Topics:

 * Further details on smooth actions,
 * Symplectic structure on cotangent bundles
 * Quotients of free, proper, smooth actions
 * Principal bundles.
 * Frame bundles.
 * Associated fibre bundles.

Reading:

Week 4

Topics:

 * Associated fibre bundles: vector bundles, adjoint bundle.
 * Reduction and extension of structure group. 
 * Group of gauge transformations. 
 * Linear connections.

Reading:

Week 5 =

Topics:

 * Connections: Koszul connections, principal connections, Ehresmann connections. 
 * Equivalences between them. 
 * Atiyah sequence and differential operators.

Reading: