ECTS credits ECTS credits: 6
ECTS Hours Rules/Memories Student's work ECTS: 102 Hours of tutorials: 6 Expository Class: 18 Interactive Classroom: 24 Total: 150
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: First Semester
Teaching: With teaching
Enrolment: Enrollable | 1st year (Yes)
- To know the fundamental notions and the basic tools of Lie theory and homogeneous spaces.
- To use differential methods for obtaining classical results and calculation of numerical invariants.
- To manage approximation and tubular neighbourhoods techniques.
- To study critical points of real valued functions.
Differential topology
Topological and differentiable manifolds. Manifolds with boundary. (1h)
Submanifolds. Rank theorem. Frobenius theorem. (2h)
Embedding theorems. Morse-Sard theorem. Consequences. Morse functions. (3h)
Transversality. Differentiable homotopies. Parametric transversality theorem. Existence of tubular neighbourhoods. (3h)
Lie groups and Lie algebras
Lie groups. Homomorphisms. Topological properties. (4h)
Lie algebras. The Lie algebra of a Lie group. Exponential map. (7h)
Classical linear groups. (4h)
Lie subgroups and Lie subalgebras. Cartan’s theorem. (4h)
Lie transformation groups. Homogeneous spaces. (8h)
Representations of Lie groups and Lie algebras. (8h)
Basic bibliography
J. M. LEE, Introduction to smooth manifolds. Second edition. Graduate Texts in Mathematics, 218.
F. W. WARNER, Foundations of Differentiable Manifolds and Lie Groups, Scott, Foresman and Company, Illinois, 1971.
W. ZILLER, Lie groups. Representation theorey and symmetric spaces. Disponible en https://www.math.upenn.edu/~wziller/math650/LieGroupsReps.pdf (último acceso 26/05/2021)
Complementary bibliography
L. CONLON, Differentiable Manifolds. A first Course. Birkhäuser, Boston, 1993.
V. GUILLEMIN, A. POLLACK, Differential topology. Reprint of the 1974 original. AMS Chelsea Publishing, Providence, RI, 2010.
M. W. HIRSCH, Differential topology. Corrected reprint of the 1976 original. Graduate Texts in Mathematics, 33. Springer-Verlag, New York, 1994.
A. W. KNAPP, Lie groups beyond an introduction. Second edition. Progress in Mathematics, 140. Birkhäuser Boston, Inc., Boston, MA, 2002.
I. MADSEN, J. TORNEHAVE, From calculus to cohomology. de Rham cohomology and characteristic classes. Cambridge University Press, Cambridge, 1997.
Y. MATSUSHIMA, Differentiable Manifolds. Marcel Dekker, New York, 1972.
J. MILNOR, Morse Theory, Princeton University Press, 1963.
E. OUTERELO, J. M. RUIZ, Topología diferencial. Addison-Wesley Iberoamericana España S.A., Madrid, 1998.
CB6, CB7, CB8, CB9, CB10, CG01, CG02, CG03, CG04, CG05, CT01, CT02, CT03, CE01, CE02, CE03.
We will follow the general directives established in the Master in Mathematics of the University of Santiago de Compostela (USC).
A key aspect in teaching at any educational level is the motivation of the concepts that are introduced. Thus, in Mathematics it is necessary to adopt a methodological approach that first introduces the notions and results that will be studied by means of examples. In this initial phase, the new concepts must be connected in a natural way with previous knowledge, in order to contribute to a unifying image of Mathematics. After this first stage, the properties, results and methods associated with the concepts will be developed. Finally, such contents will be reinforced through more examples, exercises and problems of different difficulty and nature. In addition, in accordance with the spirit of the European Higher Education Area, where students become an active subject and motor of their own learning, many of these exercises and problems must be carried out by the students, in order to consolidate and assimilate contents.
Among the teaching methodologies presented in the syllabus, we will use, above all:
M1 Teacher expositions
M2 Presentations by the students
M3 Resolution of exercises
M4 Reading and study by students
M5 Discussions in class
M9 Summaries and proposed works
M10 Complementary readings
The assessment system will follow USC regulations, located at
https://www.xunta.gal/dog/Publicados/2011/20110721/AnuncioG2018-190711-…
For the calculation of the final grade, the continuous assessment and the final exam will be considered. In any case, the final mark will never be lower than that of the final examination.
Continuous assessment (40%). The continuous assessment will be carried out through the completion of proposed exercises, both individually or in groups, and through the participation of students.
These activities will be used to assess the learning outcomes, the ability to work in teams, and self-study.
Final examination (60%). Each student will sit a written final exam to verify the acquisition of concepts and results, and their applicability to concrete cases.
In case of dishonest execution of exercises or assessments the following regulation will be applied:
https://www.xunta.gal/dog/Publicados/2011/20110721/AnuncioG2018-190711-….
Classroom or telematic work
Blackboard or telematic lectures 44
Tutoring in groups (face-to-face or telematic) 4
Classroom or telematic work load 48
Personal work
Study 74
Exercises and other work 22
Work with computers 6
Total personal work load 102
Total work load 150
The student should have taken some courses on topics related to Geometry and Topology.
Jose Carlos Diaz Ramos
Coordinador/a- Department
- Mathematics
- Area
- Geometry and Topology
- Phone
- 881813363
- josecarlos.diaz [at] usc.es
- Category
- Professor: University Professor
Victor Sanmartin Lopez
- Department
- Mathematics
- Area
- Geometry and Topology
- Category
- Professor: LOU (Organic Law for Universities) PhD Assistant Professor
| Tuesday | |||
|---|---|---|---|
| 10:00-11:00 | Grupo /CLE_01 | Galician, Spanish | Classroom 10 |
| 11:00-12:00 | Grupo /CLE_01 | Galician, Spanish | Classroom 10 |
| Wednesday | |||
| 11:00-12:00 | Grupo /CLIL_01 | Spanish, Galician | Classroom 10 |
| 01.17.2025 10:00-14:00 | Grupo /CLE_01 | Classroom 10 |
| 06.16.2025 10:00-14:00 | Grupo /CLE_01 | Classroom 10 |