Details of MA5118 (Autumn 2020)
| Level: 5 | Type: Theory | Credits: 4.0 |
| Course Code | Course Name | Instructor(s) |
|---|---|---|
| MA5118 | Principles Bundles & Representation Ring | Somnath Basu |
| Syllabus |
|---|
| Part I: Principal Bundles
G-spaces, orbit category, principal $G$-bundles, properties, Vector bundles, properties, associated bundle constructions, classifying spaces, construction and existence, May's bar construction, Elmendorf's theorem. Part II: Representation Ring Review of (compact) Lie groups and Cartan subgroups, Weyl group, Riemannian geometry of Lie groups, Cartans theorem, representation ring of (compact) Lie groups, augmentation ideal. |
| Prerequisite |
|---|
| Consent of the Instructor. |
| References |
|---|
| Suggested Texts:
1. Bott, R. Homogeneous Vector Bundles; Annals of Mathematics, 2nd Ser., Vol. 66, No. 2 (1957). 2. Elmendorf, A. Systems of Fixed Point Sets; Transactions of the American Mathematical Society, Vol. 277, No. 1 (1983). 3. Hsiang, W.Y. Lectures on Lie Groups; Singapore: World Scientific (2000). 4. Kirillov, A. Jr. An Introduction to Lie Groups and Lie Algebras; Cambridge Studies in Advanced Mathematics, Vol. 113 (2008). 5. May et al. Equivariant Homotopy and Cohomology Theory; CBMS Vol. 91, American Mathematical Society (1996). 6. Milnor, J. Construction of Universal Bundles I; Annals of Mathematics, Vol. 63, No. 2 (1956). 7. Milnor, J. Construction of Universal Bundles II; Annals of Mathematics, Vol. 63, No. 3 (1956). 8. Milnor, J. Curvatures of Left Invariant Metrics on Lie Groups; Advances in Mathematics, Vol. 21 (1976). 9. Husemoller et al. Basic Bundle Theory and $K$-Cohomology Invariants, Lecture Notes in Physics book series (LNP, Vol. 726) Springer (2008). 10. Segal, G. The Representation-ring of a Compact Lie Group; Publ. IHES, tome 34 (1968). 11. Steenrod, N. The Topology of Fibre Bundles; Princeton University Press (1999). |
Course Credit Options
| Sl. No. | Programme | Semester No | Course Choice |
|---|---|---|---|
| 1 | IP | 1 | Not Allowed |
| 2 | IP | 3 | Not Allowed |
| 3 | IP | 5 | Not Allowed |
| 4 | MR | 1 | Not Allowed |
| 5 | MR | 3 | Not Allowed |
| 6 | MS | 5 | Not Allowed |
| 7 | MS | 7 | Not Allowed |
| 8 | MS | 9 | Elective |
| 9 | RS | 1 | Not Allowed |
| 10 | RS | 2 | Not Allowed |