ECTS credits ECTS credits: 3
ECTS Hours Rules/Memories Student's work ECTS: 51 Hours of tutorials: 3 Expository Class: 9 Interactive Classroom: 12 Total: 75
Use languages Spanish, Galician
Type: Ordinary subject Master’s Degree RD 1393/2007 - 822/2021
Departments: Mathematics
Areas: Geometry and Topology
Center Faculty of Mathematics
Call: Second Semester
Teaching: With teaching
Enrolment: Enrollable | 1st year (Yes)
This course is an introduction to the cohomological methods in the theory of differentiable manifolds.
We aim that students have a deeper knowledge of the algebraic techniques in Geometry and Topology, and be able to apply them to concrete problems in order to appreciate their power and sophistication. To adquire a computation competence with those tools is an objective too.
The adquired knowledge will allow to approach several research lines which are being developed in the Areas of Geometry and Topology, and of Algebra. The course can also be of interest for applications in theorical Physics.
1. De Rham Cohomology (2 expositive hours)
1.1. Cochain complexes and cohomology.
1.2. Differential forms.
1.3. De Rham cohomology of a differentiable manifold.
1.4. De Rham cohomology with compact support.
1.5. Orientation. Integration on manifolds. Stokes theorem.
1.6. Homotopy. Lemma of Poincaré.
2. Computation methods (2 expositive hours)
2.1. Mayer-Vietoris sequence.
2.2. Computation in examples.
2.3. Finite dimension.
2.4. Duality of Poincaré.
2.5. Künneth's formula and Leray-Hirsch's theorem.
2.6. Thom isomorphism.
3. Geometric applications (2 expositive hours)
3.1. Degree of a map.
3.2. Euler characteristic.
3.3. Hopf's theorem.
3.4. Leftschetz's formula.
4. Characteristic classes (3 expositive hours)
4.1. Sphere fiber bundles and vector bundles.
4.2. Double complex of Cech-de Rham.
4.3. Euler class of a sphere fiber bundle.
4.4. Chern classes of complex vector bundles.
4.5. Splitting principle and varieties of flags.
4.6. Pontrjagin classes of real vector bundles.
4.7. Grasmannian and classification of vector bundles.
Basic.
Bott, R., Tu, L.W.: Differential forms in algebraic topology. Springer-Verlag (1982) (Book catalog numbers 1210 73, 57 191). URL https://link-springer-com.ezbusc.usc.gal/book/10.1007/978-1-4757-3951-0
Complementary.
Davis, J.F., Kirk, P.: Lecture notes in algebraic topology. American Mathematical Society (2001) (Book catalog number 55 270).
Dodson, C.T.J., Parker, P.E.: A user's guide to algebraic topology. Kluwer Academic Publishers (1997) (Book catalog number 55 238).
Hatcher, A.: Algebraic topology. Cambridge University Press (2002) (Book catalog number 1210 83, 55 279). URL https://pi.math.cornell.edu/~hatcher/AT/AT.pdf
Karoubi, M., Leruste, C.: Algebraic topology via differential geometry. Cambridge University Press (1987) (Book catalog number 55 170).
Madsen, I., Tornehave, J.: From calculus to cohomology: de Rham cohomology and characteristic classes. Cambridge University Press (1997) (Book catalog number 55 245).
Tu, L.W.: Differential geometry. Connections, curvature, and characteristic classes. Springer (2017). URL https://link-springer-com.ezbusc.usc.gal/book/10.1007/978-3-319-55084-8
Tu, L.W.: An introduction to manifolds. Springer (2011). URL https://link-springer-com.ezbusc.usc.gal/book/10.1007/978-1-4419-7400-6
GENERAL COMPETENCES
CG01 - Acquisition of high-level mathematical tools for various applications meeting the expectations of graduates in mathematics and other basic sciences.
CG02 - Get to know the broad panorama of current mathematics, both in its lines of research, and in methodologies, resources and problems it addresses in various areas.
CG03 - Train for the analysis, formulation and resolution of problems in new or little-known environments, within wider contexts.
CG04 - Prepare to make decisions based on abstract considerations, to organize and plan and to solve complex issues.
SPECIFIC COMPETENCES
CE01 - Train for study and research in developing mathematical theories.
CE02 - Apply the tools of mathematics in various fields of science, technology and social sciences.
CE03 - Develop the skills necessary for the transmission of mathematics, oral and written, both in terms of formal correctness and in terms of communicative effectiveness, emphasizing the use of appropriate ICT.
TRANSVERSAL COMPETENCES
CT01 - Use bibliography and search tools for general and specific Mathematics bibliographic resources, including Internet access.
CT02 - Manage working time optimally and organize available resources, establishing priorities, alternative paths and identifying logical errors in decision-making.
CT03 - Enhance the ability to work in cooperative and multidisciplinary environments.
The development of the subject will consist of expositions of the general lines, the main results of the topìc, and the main ideas of the proofs. The personal work of the students and their participation in class is encouraged. The students will solve problems, and they will have to present some of the topics themselves, delivering the notes they prepare for the presentation.
Each student must solve the proposed problems and make a presentation of some part of the program. The evaluation will take into account the active participation in the classes, the resolution of problems, and, above all, the presentation they make of a topic, as well as the notes prepared to make it. In this scenario, the final grade will be the sum of 30%
of the continuous evaluation grade and 70% of the grade of the exhibition and the work presented.
At the second opportunity, the students will have the same conditions of evaluation and the grade of the continuous evaluation of the first opportunity.
In cases of fraudulent performance of exercises or tests, the provisions of the Regulations for evaluating student academic performance and reviewing grades.
CLASSROOM WORK (Hours)
Blackboard courses 21 (9 expositive hours end 12 lab hours)
Courses with computer/ in lab
Tutorial hours 3
Total classroom hours 24
HOMEWORK (Horas)
Autonomous/ in group study 33
Exercises solving /writing, reports, other works 15
Programming / experiments, works with computer / in lab 3
Total homework hours / each student 51
The central topic is De Rham's cohomology and geometric applications, which assumes an elementary knowledge of the theory of differentiable varieties (Geometry and topology of varieties, master's degree, first semester).
Although not essential, studying Algebraic Topology (graduate studies) is recommended.
Jesús Antonio Álvarez López
- Department
- Mathematics
- Area
- Geometry and Topology
- Phone
- 881813149
- jesus.alvarez [at] usc.es
- Category
- Professor: University Professor
| Tuesday | |||
|---|---|---|---|
| 10:00-11:00 | Grupo /CLIL_01 | Galician | Classroom 10 |
| 06.04.2024 10:00-14:00 | Grupo /CLE_01 | Classroom 10 |
| 07.14.2024 10:00-14:00 | Grupo /CLE_01 | Classroom 10 |