syllabus
1. Information regarding the programme
1.1 Higher education institution 	Babes-Bolyai University Cluj-Napoca	
1.2 Faculty	Faculty of Mathematics and Computer Science	
1.3 Department	Department of Mathematics	
1.4 Field of study	Mathematics	
1.5 Study cycle	Master	
1.6 Study programme / Qualification 	Mathematics	

2. Information regarding the discipline 
2.1 Name of the discipline	Algebraic and Differential Topology	
2.2 Course coordinator 	Lect. Dr. Liana Topan	
2.3 Seminar coordinator	Lect. Dr. Liana Topan	
2.4. Year of study	II	2.5 Semester	4	2.6. Type of evaluation	E	2.7 Type of discipline	Elective Course	

3. Total estimated time (hours/semester of didactic activities) 
3.1 Hours per week        
4		Of which: 3.2 course
2		3.3
seminar/laboratory
2		
3.4 Total hours in the curriculum 
56		Of which: 3.5 course
28		3.6 seminar/laboratory
28		
Time allotment:	hours	
Learning using manual, course support, bibliography, course notes	39	
Additional documentation (in libraries, on electronic platforms, field documentation) 	15	
Preparation for seminars/labs, homework, papers, portfolios and essays	30	
Tutorship	15	
Evaluations	20	
Other activities: ..................	-	
3.7 Total individual study hours 	119	
3.8 Total hours per semester	175	
3.9 Number of ECTS credits	7	

4. Prerequisites (if necessary)
4.1. curriculum	•		
4.2. competencies	•	Some backgrounds in differential geometry and mathematical analysis, as well as theory of manifolds	

5. Conditions (if necessary)
5.1. for the course	•		
5.2.  for the seminar /lab activities		

6. Specific competencies acquired 
Professional competencies	Ability to understand and manipulate advanced
concepts of fundamental mathematical structures.
Ability to permanently learn, understand and apply
the most recent scientific results.
Ability to work independently and/or in a team in
order to solve problems in various professional
contexts.
Ability in verbal and written communication of
ideas and knowledge.

Transversal competencies	Ability to transmit and value the studied knowledge
and methods.
Ability to analyze, understand, approach and
modelling problems of mathematical nature from
other areas.

7. Objectives of the discipline (outcome of the acquired competencies)
7.1 General objective of the discipline
	Ability to understand and approach problems from this field and to apply particular tools of algebraic and differential topology	
7.2 Specific objective of the discipline
	The purpose of the course is to present an introduction in algebraic and differential topology and applications.	

8. Content
8.1 Course	Teaching methods	Remarks	
1.	Smooth manifolds: local Euclidean spaces, topological manifolds, differential manifolds	Exposure: description, explanation, examples, proofs		
2.	Tangent spaces. Tangent maps	Exposure: description, explanation, examples, proofs		
3.	The tangent bundle of a smooth manifold	Exposure: description, explanation, examples, proofs		
4.	Partitions of unity	Exposure: description, explanation, examples, proofs		
5.	Submanifolds	Exposure: description, explanation, examples, proofs		
6.	Manifolds with boundary	Exposure: description, explanation, examples, proofs		
7.	Submanifolds of manifolds with boundary	Exposure: description, explanation, examples, proofs		
8.	The degree modulo two of a map	Exposure: description, explanation, examples, proofs		
9.	Elements of Riemannian geometry	Exposure: description, explanation, examples, proofs		
10.	Quotient topologies. Cells attaching	Exposure: description, explanation, examples, proofs		
11.	The groups π_n(X, x_0). The fundamental group	Exposure: description, explanation, examples, proofs		
12.	The homotopic invariance of homotopy groups	Exposure: description, explanation, examples, proofs		
13.	Relative homotopy groups	Exposure: description, explanation, examples, proofs		
14.	Fibrations and covering spaces	Exposure: description, explanation, examples, proofs		
Bibliography
1.	Do Carmo, M.P., Riemannian Geometry, Birkhäuser, 
2.	Dundas, B.I., Differential Topology, course material, 2007 (http://www.uib.no/People/nmabd/pp/070814dt.pdf)
3.	Hatcher, A., Algebraic Topology, Cambridge University Press, 2002 (http://www.math.cornell.edu/~hatcher/AT/AT.pdf)
4.	Nicolaescu, L.I., Lectures on the Geometry of Manifolds, World Scientific, 1996
Additional references
1.	Conlon, L., Differentiable Manifolds, Birkhäuser, 2001
2.	Craioveanu, M., Introducere în geometria diferenţială, Editura Mirton, 2004
3.	May, J.P., A Concise Course in Algebraic Topology, Chicago Lectures in Mathematics, 1999
4.	Postnikov, M., Leçons de géometrie. Varietés différentiables., MIR, 1990
5.	Sandovici, P., Ţarină, M., Geometrie Diferenţială, Litografia UBB, Cluj-Napoca, 1974
            Sharpe, R.W., Differential Geometry, Springer, 1996
	
8.2 Seminar / laboratory	Teaching methods	Remarks	
1.	Examples of smooth manifolds	explanation, examples, proofs		
2.	Equivalent definitions for the tangent space of a smooth manifold	explanation, examples, proofs		
3.	Vector bundles. Examples	explanation, examples, proofs		
4.	Operations with vector bundles	explanation, examples, proofs		
5.	The rank theorem. The local inversion theorem	explanation, examples, proofs		
6.	Sard’s theorem	explanation, examples, proofs		
7.	Whitney’s theorem (I)	explanation, examples, proofs		
8.	Whitney’s theorem (II)	explanation, examples, proofs		
9.	Existence of Riemann metrics on a manifold	explanation, examples, proofs		
10.	Example of quotient spaces	explanation, examples, proofs		
11.	Change of base point	explanation, examples, proofs		
12.	Computation of the fundamental group of n-dimensional sphere and n-dimensional torus. Brower’s Fixed point theorem. Fundamental theorem of algebra.	explanation, examples, proofs		
13.	Exact sequence of homotopy groups of a topological pair	explanation, examples, proofs		
14.	Fibrations and covering spaces	explanation, examples, proofs		
Bibliography
The same as for courses section

	

9. Corroborating the content of the discipline with the expectations of the epistemic community, professional associations and representative employers within the field of the program

•	The course respects the IEEE and ACM Curriculla Recommendations for Computer Science studies;
•	The course exists in the studying program of all major universities in Romania and abroad;

10. Evaluation
Type of activity	10.1 Evaluation criteria	10.2 Evaluation methods	10.3 Share in the grade (%)	
10.4 Course	- know the basic results of the domain;
- apply the course concepts and presented tools
	Final written exam	60%	
10.5 Seminar/lab activities	-be able to elaborate and defend a small proceeding 	-oral expositions
-continuous observations	40%	
10.6 Minimum performance standards	
•	At least grade 5 (from a scale of 1 to 10) after the final exam	

Date				Signature of course coordinator 	Signature of seminar coordinator
..........................		Lect. Dr. Liana Topan	            Lect. Dr. Liana Topan

Date of approval  					  Signature of the head of department	
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