Universitatea Babeş-Bolyai Cluj-Napoca
Facultatea de Matematică şi Informatică
Ciclul de studii: Masterat
Domeniul: Matematica
Programul de studii: Algebră şi Geometrie (în limba engleză)
Limba de predare: Engleză

SYLLABUS

I. General data
Code	Subject	
MG266	Cohomology of Differential Forms	

Semester	Hours: C+S+L	Category	Status	
2	2+2+0		compulsory	

II. Full status faculty members
Name and	Scientific	Didactic	Chair	Type of activity	
surname	title	title		C	S	L	
ANDRICA Dorin	Ph.D.	Prof.	Geometry	*	*		

Associated faculty members
Name and	Scientific	Institution	Type of	Type of activity	
surname	title		position	C	S	L	

III. Course objectives
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. 	

IV. Course contents
1.Differentiable manifolds.
2.Tensor algebra and exterior algebra of a smooth manifold.
3.Differential forms. De Rham differential complex.
4.Integration on manifolds.
5.Applications.	

V. 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.Conlon,C.,Elements de topologie algebrique,Hermann,Paris,1971.	

VI. Thematic of didactic activities per weeks
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’s operator.
Seminar.Properties of Hodge’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’s lemma.
Seminar.Equivalent formulations for Poincare’s lemma.	

VII. Didactic methods used
Presentations, dialogs, proofs, problem-solving, reports.	

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

IX. Additional references
1.Burghelea,D.,s.a.,Introducere in topologia diferentiala,Ed.Stiintifica si enciclopedica,Bucuresti,1972.
2.Berger,M.,Gostieaux,B.,Geometrie Differentiale,Armand Colin,1972.
3.Gauld,D.,Differential Topology.An Introduction,Marcel Dekker,1982.
4.Guillemin,V.,Pollak,A.,Differential Topology,Prentice Hall,1974.
5Jost,J.,Riemannian Geometry and Geometric Analysis,Springer Verlag,1995.
6.Jurchescu,M.,Introducere in analiza pe varietati,Universitatea din Bucuresti,1980.
7.Pop,I.,Topologie algebrica,Ed.Stiintifica,Bucuresti,1990.
8.Bott,R.,Tu,L.W.,Differential Forms in Algebraic Topology,Springer Verlag,1982.
9.Hirsch,M.,Differential Topology,Springer Verlag,1976.	

	Dean,							Chair head,

	Date, October 2/2008				     Course responsible,
								Prof. ANDRICA Dorin, Ph.D.