Lectures

Tentative lecture plan is as follows:

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

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

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:

Topics:

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

Reading:

Topics:

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

Reading: