Babes-Bolyai University of Cluj-Napoca
Faculty of Mathematics and Computer Science
Study Cycle: Master
SUBJECT

Code	Subject	
MMG1005	Cohomology of Differential Forms	

Section	Sem.	Hours: C+S+L	Category	Type	
Mathematics	4	2+1+0	speciality	optional	

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	
Bibliography	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%)