Lecture Schedule: Difference between revisions
Created page with "====== 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 * diffeomorphism..." |
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=== Week 1 === | === Week 1 === | ||
Topics: Review of differential geometry | Topics: Review of differential geometry | ||
* Topological and smooth manifolds, | * Topological and smooth manifolds, | ||
* Vector and tensor fields, | * Vector and tensor fields, | ||
| Line 11: | Line 12: | ||
Reading: | Reading: | ||
* Taubes Ch 1 | * Taubes Ch 1 | ||
* Frenkel Sections 1-3 | * Frenkel Sections 1-3 | ||
| Line 18: | Line 20: | ||
=== Week 2 === | === Week 2 === | ||
Topics: Review of differential geometry | Topics: Review of differential geometry | ||
* vector fields | * vector fields | ||
* Lie brackets | * Lie brackets | ||
| Line 31: | Line 34: | ||
Reading: | Reading: | ||
* Rudolph-Schmidt (Differential Geometry and Mathematical Physics. Part I) Section 5 | * Rudolph-Schmidt (Differential Geometry and Mathematical Physics. Part I) Section 5 | ||
=== Week 3 === | === Week 3 === | ||
Topics: | Topics: | ||
* Further details on smooth actions, | * Further details on smooth actions, | ||
* Symplectic structure on cotangent bundles | * Symplectic structure on cotangent bundles | ||
| Line 47: | Line 52: | ||
=== Week 4 === | === Week 4 === | ||
Topics: | Topics: | ||
* Associated fibre bundles: vector bundles, adjoint bundle. | * Associated fibre bundles: vector bundles, adjoint bundle. | ||
* Reduction and extension of structure group. | * Reduction and extension of structure group. | ||
| Line 56: | Line 62: | ||
=== Week 5 ==== | === Week 5 ==== | ||
Topics: | Topics: | ||
* Connections: Koszul connections, principal connections, Ehresmann connections. | * Connections: Koszul connections, principal connections, Ehresmann connections. | ||
* Equivalences between them. | * Equivalences between them. | ||
Revision as of 13:06, 18 January 2026
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: