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

Code	Subject	
MG266	Cohomology of Differential Forms	

Section	Semester	Hours: C+S+L	Category	Type	
Algebra and Geometry - in English	2	2+2+0		compulsory	

Teaching Staff in Charge	Prof. ANDRICA Dorin, Ph.D., dandrica@math.ubbcluj.ro	
Aims	1.The study of some important notions and results in Analysis on Manifolds, a new and modern direction in Mathematics.
2.The study of techniques and methods  in this discipline and using them in connection to other fields.
3.The connection to other mathematical fields and to some research problems in Mechanics and Physics. 
	
Content	Course1.Review on smooth manifolds.Opening new perspectives of study.
Seminar.Examples of manifolds.

Course 2.Algebra of real  smooth mappings of a manifold.
Seminar.Example of smooth mappings.Smooth partition of unity.

Course 3.The tangent bundle of a manifold.Tangent vector fields.
Seminar.Locally and globally flows on a manifold.

Course 4.Integrability of vector fields.The problem of completeness.
Seminar.Lie algebra of vector fields.

Course 5.Tensor algebra of a vector space.
Seminar.Equivalent definitions for the tensor product.

Course 6. Exterior algebra of a vector space.
Seminar.Properties of the exterior product.

Course 7.Determinants, volumes and the Hodge&#8217;s operator.
Seminar.Properties of Hodge&#8217;s operator.

Course 8.Differential forms on a manifold.
Seminar.Working paper.

Course 9.Exterior operator.De Rham differential complex.
Seminar.The module structure over the ring of real smooth mappings.

Course 10.Orientability and the volume element.
Seminar.Existence and uniquiness of the exterior operator.

Course 11.The integration of the m-differential forms.
Seminar.Curve integrals on a manifold.

Course 12.The Stokes theorem.
Seminar.Applications of the Stokes theorem.

Course 13.Classical theorems of Green and Gauss.
Seminar.Problems and applications in the integration of forms.

Course 14.The Poincare&#8217;s lemma.
Seminar.Equivalent formulations for Poincare&#8217;s lemma.
	
Bibliography	1.Abraham,R.,s.a.,Manifolds.Tensor Analysis and Applications,Springer Verlag,1988.
2.Andrica,D.,Critical Point Theory and Some Applications,Cluj University Press,2005.
3.Andrica,D.,Pintea,C.,Elemente de teoria omotopiei cu aplicatii la studiul punctelor critice,Editura Mirton,Timisoara,2002.
4.Bredon,G.Topology and Geometry,Springer Verlag,1993.
5.Conlon,L.,Differentiable Manifolds.A First Course,Birkhauser,2001.
6.Godbillion,C.,Elements de topologie algebrique,Hermann,Paris,1971.
7.Jost,J.,Riemannian Geometry and Geometric Analysis,Springer Verlag,1995.
8.Jurchescu,M.,Introducere in analiza pe varietati,Universitatea din Bucuresti,1980.
9.Pop,I.,Topologie algebrica,Ed.Stiintifica,Bucuresti,1990.
10.Bott,R.,Tu,L.W.,Differential Forms in Algebraic Topology,Springer Verlag,1982.

Assessment	The final evaluation is given as follows:
-	the working paper during the semester 20%
-	the evaluation of the report during the semester 10%
-	the final examination 70%