Babes-Bolyai University of Cluj-Napoca
Faculty of Mathematics and Computer Science
Study Cycle: Master
SUBJECT
| MMG1005 | Cohomology of Differential Forms |
| Teaching Staff in Charge |
Assoc.Prof. BLAGA Paul Aurel, Ph.D., pablaga cs.ubbcluj.ro |
| Aims |
|
The course of the cohomology of differential forms is a natural continuation of the courses of smooth manifolds, calculus on manifolds and riemannian geometry. The goal of the course is to familiarize the students with the basic notions of the de Rham cohomology, a theory lying at the interface between the differential topology, differential geometry and algebraic topology. We shall discuss, also, the connections between the de Rham cohomology, Hochschild homology and cyclic homology. |
| Content |
|
1. Differential forms, exterior differential, integrations on manifolds, Stokes@ theorem
2. de Rham cohomology 3. Chain complexes and homology in a category 4. Basic properties of the de Rham cohomology 5. Applications: Brouwer fixed point theorem, vector fields on spheres 6. Riemannian manifolds, Hodge theory 7. Degree of mappings, linking numbers, index of vector fields 8. The POincare-Hopf theorem 9. Computation of the de Rham cohomology for selected manifolds 10. Currents, de Rham@s theorem 11. Vector bundles and connections 12. Characteristic classes and the classifications of vector bundles 13. The Hochschild homology of smooth functions 14. The connection between the de Rham cohomology and cyclic homology for smooth manifolds |
| References |
|
1.Bott, R., Tu, L.: Differential Forms in Algebraic Topology, Springer, 1982
2.Connes, A.: Noncommutative Geometry, Academic Press, 1994 3.Flanders, H.: Differential Forms with Applications to the Physical Sciences, Dover, 1989 4.Lafontaine, J.: Introduction aux varietes differentielles, EDP Sciences, 1996 5.Loday, J.L.: Cyclic Homology, Springer, 1992 6.I. Madsen, J. Tornehave - From Calculus to Cohomology, Cambridge University Press, 1997 7.S. Morita - Geometry of Differential Forms, AMS, 2001 8.Warner, F.: Foundations of Differentiable Manifolds and Lie Groups, Springer, 1983 9.Weibel, Ch.: An Introduction to Homological Algebra, Cambridge University Press, 1994 10.von Westenholtz, C.: Differential Forms in Mathematical Physics, North Holland, 1978 |
| Assessment |
|
Final exam (70%), presentations (30%) |
| Links: | Syllabus for all subjects Romanian version for this subject Rtf format for this subject |