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, Raoul, Tu; Loring W. Differential forms in algebraic topology. Graduate Texts in Mathematics, 82. Springer-Verlag, New York - Heidelberg - Berlin, 1982.
Complementary.
Davis, James F.; Kirk, Paul. Lecture notes in algebraic topology. Graduate Studies in Mathematics. 35. American Mathematical Society, Providence, RI, 2001.
Dodson, C.T.J.; Parker, Phillip E. A user's guide to algebraic topology. Mathematics and its Applications 387. Kluwer Academic Publishers, Dordrecht, 1997.
Hatcher, Allen. Algebraic topology. Cambridge University Press, Cambridge, 2002.
Karoubi, M.; Leruste, C. Algebraic topology via differential geometry. London Mathematical Society Lecture Note Series, 99. Cambridge University Press, Cambridge, 1987.
Madsen, Ib; Tornehave, Jørgen. From calculus to cohomology: de Rham cohomology and characteristic classes.
Cambridge University Press, Cambridge, 1997.
Tu, Loring W. Differential geometry. Connections, curvature, and characteristic classes. Graduate Texts in Mathematics, 275. Springer, Cham, 2017.
Tu, Loring W. An introduction to manifolds. Second edition. Universitext. Springer, New York, 2011.
3.1 BASIC AND GENERAL COMPETENCES
GENERAL
CG01 - Introduce students into the research, as an integral part of a deep formation, preparing them for the eventual completion of a doctoral thesis.
CG02 - Acquisition of high level mathematical tools for diverse applications covering the expectations of graduates in mathematics and other basic sciences.
CG03 - Know the broad panorama of current mathematics, both in its lines of research, as well as in methodologies, resources and problems it addresses in various fields.
CG04 - Train for the analysis, formulation and resolution of problems in new or unfamiliar environments, within broader contexts.
CG05 - Prepare for decision making based on abstract considerations, to organize and plan and to solve complex issues.
BASICS
CB6 - Possess and understand knowledge that provides a basis or opportunity to be original in the development and / or application of ideas, often in a research context.
CB7 - That students know how to apply the knowledge acquired and their ability to solve problems in new or unfamiliar environments within broader (or multidisciplinary) contexts related to their area of study.
CB8 - That students are able to integrate knowledge and face the complexity of making judgments based on information that, being incomplete or limited, includes reflections on social and ethical responsibilities linked to the application of their knowledge and judgments.
CB9 - That students know how to communicate their conclusions and the knowledge and ultimate reasons that sustain them to specialized and non-specialized audiences in a clear and unambiguous way.
CB10 - That students have the learning skills that allow them to continue studying in a way that will be largely self-directed or autonomous.
3.2 TRANSVERSAL COMPETENCES
CT01 - Use bibliography and search tools for general and specific bibliographic resources of Mathematics, including Internet access.
CT02 - Optimally manage work time and organize available resources, establishing priorities, alternative paths and identifying logical errors in decision making.
CT03 - Enhance capacity for work in cooperative and multidisciplinary environments.
3.3 SPECIFIC COMPETENCES
CE01 - Train for the study and research in mathematical theories in development.
CE02 - Apply the tools of mathematics in various fields of science, technology and social sciences.
CE03 - Develop the necessary skills for the transmission of mathematics, oral and written, both in regard to formal correction, as well as in terms of communicative effectiveness, emphasizing the use of appropriate ICT.
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
El tema central es la cohomología de De Rham y sus aplicaciones geométricas, lo que supone un conocimiento elemental de la teoría de variedades diferenciables (Geometría y topología de variedades, máster, primer cuatrimestre).
Es recomendable, aunque no imprescindible, haber cursado Topología algebraica (grado).
Jesús Antonio Álvarez López
Coordinador/a- Department
- Mathematics
- Area
- Geometry and Topology
- Phone
- 881813149
- jesus.alvarez [at] usc.es
- Category
- Professor: University Professor
| Monday | |||
|---|---|---|---|
| 10:00-11:00 | Grupo /CLE_01 | Galician | Classroom 10 |
| Tuesday | |||
| 10:00-11:00 | Grupo /CLIL_01 | Galician | Classroom 10 |
| 06.10.2024 10:00-14:00 | Grupo /CLE_01 | Classroom 10 |
| 06.12.2024 10:00-14:00 | Grupo /CLE_01 | Classroom 10 |