Course unit title: GEOMETRÍA DE RIEMANN /
RIEMANNIAN GEOMETRY
Credit Level: 4/5
ECTS Credits: 7.5
Semester: 2S
Unit code: 05A7
Type of unit: elective
Hours/Week: 5
Language of instruction: Spanish
Lecturer/Organizer
José Antonio pastor González
Unit home page
None
Course contents
Metric tensors; bilinear forms; metric; distance; local and global isometries;
Riemannian manifolds; Levi-Civita connection; compatible connection; torsionfree connection; Riemannian connection; Levi-Civita theorem; geodesics;
minimizing properties of geodesics; Symmetry Lemma; Gauss Lemma; convex
neighborhoods; curvature tensor; sectional curvature; spaces of constant
curvature; Ricci and scalar curvature; Jacobi fields; conjugate points;
exponential map; Riemannian submanifolds; tangent and normal vector fields;
Gauss and Weingarten formulae; Gauss equation; hypersurfaces; complete
manifolds; Hopf-Rinow theorem; Hadamard theorem; Schur Lemma; Cartan
theorem; the hyperbolic space; the space forms; variations of the energy
functional; length and energy of a curve; first variation; second variation;
Bonnet-Myers theorem; Synge-Weinstein theorem; the Index lemma; Rauch
theorem; distance between conjugate points; cut points along a geodesic; cut
locus.
Prerequisites
Linear and Multilinear Algebra; geometry of affine and Euclidean spaces;
Mathematical Analysis (differential and integral calculus of several variables);
General Topology and Differentiable Manifolds; Differential Equations.
Teaching methods
Lecture and problem classes. Oral presentations. Some specific exercises from
problem sheets are set for handing in.
Assessment method
Assessment on the basis of oral presentations and handing in solutions to
problems/exercises.
Course unit title: ÁLGEBRA / ALGEBRA
Credit Level: 4
ECTS Credits: 9
Semester: full year
Unit code: 01A5
Type of unit: compulsory
Hours/Week: 3
Language of instruction: Spanish
Lecturer/Organizer
Juan Jacobo Simón Pinero
Unit home page
http://www.um.es/docencia/depmat/docencia.html (Spanish)
Course contents
Preliminaries: rings, subrings, ideals and quotient rings, rings homomorphisms,
isomorphisms theorems.
Basic Concepts: domains, fields and matrices, modules, submodules and module
homomorphisms, exact sequences, the language of categories.
Artinian rings and algebras: chain conditions, semi-simple artinian rings, Wedderburn
Theorem, Jacobson radical of a ring, the Krull-Schmidt Theorem.
Noetherian rings and algebras: polynomial rings, the euclidean algorithm, factorization,
Principal Ideal Domains (PID), modules over PID, Algebraic integers.
Ring constructions: direct product, axiom of choice and Zorn’s lemma, tensor product,
modules over general rings, projective and injective modules, invariant basis number
and projective-free rings.
General rings: rings of fractions, skew polynomial rings, free algebras, FIR rings
Prerequisites
Basic concepts of set theory and elementary algebraic structures (groups, rings, fields
and polynomial rings). Elementary linear algebra: vector spaces, basis, dimension,
linear transformations, matrices and systems of linear equations
Teaching methods
Lecture, problem classes, assignments to be worked out during practical sessions and
monthly individual tutorial. Some specific exercises from problem sheets are set for
handing in.
Assessment method
Closed book written examinations and ongoing evaluation of participation.
Course unit title: ANÁLISIS FUNCIONAL /
FUNCTIONAL ANALYSIS
Credit Level: 4
ECTS Credits: 6
Semester: 1S
Unit code: 01A6
Type of unit: compulsory
Hours/Week: 4
Language of instruction: Spanish
Lecturer/Organizer
José Orihuela Calatayud
Unit home page
None
Course contents
Part I: Hilbert Spaces.
Riesz Lemma and Theorem: characterization of finite dimensional Banach spaces.
Jordan Von Newman theorem: characterization of norms associated to scalar products.
Projection and Riesz representation theorems: dual of a Hilbert space.Bases in Hilbert
spaces: orthogonal and trigonometric polynomials. Operators in Hilbert spaces:
compact, self-adjoint and normal operators. Existence of eigenvalues: spectral theorem
for compact normal operators in Hilbert spaces. Applications to Sturm-Lioville
Problems.
PART II: Banach spaces.
Hahn-Banach theorem: separation of convex sets and their applications. The principle of
Uniform Boundedness: applications to holomorphic vector valued functions. Closed
Graph and Open Mapping theorems: applications to Schauder bases.
Prerequisites
Linear algebra, elementary set-theoretic topology, calculus including integration and
differential equations techniques.
Teaching methods
Lecture. Problem classes based on periodical problem sheets. Some specific exercises
from problem sheets are set for handing in.
Assessment method
Closed book written examinations and ongoing evaluation on the basis of handing in
solutions to problems from problem sheets.
Course unit title: ANÁLISIS COMPLEJO /
COMPLEX ANALYSIS
Credit Level: 4
ECTS Credits: 6
Semester: 1S
Unit code: 01A7
Type of unit: compulsory
Hours/Week: 4
Language of instruction: Spanish
Lecturer/Organizer
Bernardo Cascales
Unit home page
None
Course contents
Mittag-Leffler’s Theorem. Infinite products and the Weiertrass factorization theorem.
The Gamma function. The Riemann Zeta function. Open mapping Theorem and
conformal mapping. The maximum Modulus Principle and the Schwartz’Lemma. Space
of analytic functions and Montel’s Theorem. The Riemann Mapping Theorem.
Harmonic functions. The Dirichlet Problem and the Poisson Integral. Jensen’s formula.
Prerequisites
Elementary properties of analytic functions: power series representation; complex
integration and Cauchy’s Theorem; singularities and residues.
Teaching methods
Lecture. Problem classes based on monthly problem sheets. Some specific exercises
from problem sheets are set for handing in.
Assessment method
Closed book written examinations and ongoing evaluation on the basis of participation
in problem classes and handing in solutions to problems from problem sheets.
Course unit title: ECUACIONES EN DERIVADAS PARCIALES /
PARTIAL DIFFERENTIAL EQUATIONS
Credit Level: 4
ECTS Credits: 6
Semester: 2S
Unit:01A8
Type of unit: compulsory
Hours/Week: 4
Language of instruction: Spanish
Lecturer/Organizer
Francisco Balibrea Gallego
Unit home page
None
Course contents
Examples of partial differential equations on Mathematical Physics. Cuasi linear and
linear equations of first order. Cauchy theory. Characteristics method. Nonlinear first
order equations. Cauchy theory. Lagrange-Charpit method. Linear equations of second
order with variable coefficient. Propagation of singularities. Classification of equations.
Cauchy-Kowalwvsky theorem. Laplace equation and Dirichlet problem. Green
functions. Heat equation. Maximum problems. Wave linear equation. Boundary
problem and separation of variables.
Prerequisites
Ordinary differential equations, several variables real analysis.
Teaching methods
Lecture, problem classes, assignments to be worked out during practical sessions and
monthly individual tutorial. Some specific exercises as homework weekly. Also some
relations of exercises and problems to be solved.
Assessment method
Closed book written examinations and ongoing evaluation of participation
Course unit title: GEOMETRÍA Y TOPOLOGÍA
GEOMETRY AND TOPOLOGY
Credit Level: 4
ECTS Credits: 9
Semester: full year
Unit code: 01A9
Type of unit: compulsory
Hours/Week: 3
Language of instruction: Spanish
Lecturer/Organizer
Ángel Ferrández Izquierdo
Unit home page
None
Course contents
Differential manifolds, diffeomorphisms, tangent vectors, submanifolds, vector fields,
integral curves, Lie bracket, tensor fields, tensor derivatives, exterior forms, Poincaré
lemma, integration and Stokes theorem.
Prerequisites
General topology, linear algebra, elementary two variables analysis, and differential
geometry of curves and surfaces.
Teaching methods
Lecture, problem classes, assignments to be worked out during practical sessions and
monthly individual tutorial. Some specific exercises from problem sheets are set for
handing in.
Assessment method
Closed book written examinations and ongoing evaluation of participation
Course unit title: CÁLCULO NUMÉRICO /
NUMERICAL ANALYSIS
Credit Level: 4
ECTS Credits: 9
Semester: full year
Unit code: 02A0
Type of unit: compulsory
Hours/Week: 3
Language of instruction: Spanish
Lecturer/Organizer
Francisco Esquembre and Antonio Linero
Unit home page
None
Course contents
Lagrange and Hermite polynomial interpolation, Numerical differentiation,
Richardson’s extrapolation, methods of interpolatory quadrature, Newton-Côtes
formulas, Peano’s error representation, Gauss quadrature, Euler-MacLaurin formula,
integrating by extrapolation, Romberg’s method , adaptative quadrature.
Ordinary differential equations, Euler’s method, Taylor’s method, Runge-Kutta’s
method, multistep methods (convergence, consistency, stability), predictor-corrector
method, introduction to two-points boundary value problems, shoot method.
Programming, JAVA language.
Prerequisites
Real Mathematical Analysis (one and several variables), ordinary differential equations,
linear difference equations, basics of JAVA language.
Teaching methods
Lecture and practical sessions in the computer room.
Assessment method
Closed book written examinations, practical (JAVA language) and ongoing evaluation
of practical computer work.
Course unit title: ANÁLISIS MULTIVARIANTE /
MULTIVARIANT ANALYSIS
Credit Level: 4/5
ECTS Credits: 7.5
Semester: 2S
Unit code: 02A3
Type of unit: elective
Hours/Week: 5
Language of instruction: Spanish
Lecturer/Organizer
Jorge Navarro
Unit home page
None
Course contents
Random vectors. Multivariate Data Analysis. Graphical representations. Principal
Component Analysis (PCA). Factor Analysis (FA). Canonical Correlation Analysis.
Discriminant Analysis. Cluster Analys. Practical sessions using Minitab and SPSS.
Prerequisites
None
Teaching methods
Lecture, problem classes, assignments to be worked out during practical sessions. Some
specific exercises from problem sheets are set for handing in.
Assessment method
Closed book written examinations (50%). Open book practical examinations (50%).
Course unit title: MODELOS LINEALES /
LINEAR MODELS
Credit Level: 4/5
ECTS Credits: 7.5
Semester: 1
Unit code: 02A4
Type of unit: elective
Hours/Week: 5
Language of instruction: Spanish
Lecturer/Organizer
Juan Antonio Cano and Manuel Franco
Unit home page
None
Course contents
Simple linear regression, full rank models, related designs, less than full rank models,
analysis of variance models
Prerequisites
Matrix algebra, vector calculus, probability and statistical methods
Teaching methods
Lecture, problem classes, practical sessions in the computer room.
Assessment method
Written examination and assessment on the basis of handing in solutions to
problems/practical exercises.
Course unit title: MODELOS DE INVESTIGACIÓN
Unit code: 02A5
OPERATIVA / MODELS OF OPERATIONAL
RESEARCH
Credit Level: 4/5
ECTS Credits: 6
Semester: 1S
Type of unit: elective
Hours/Week: 4
Language of instruction: Spanish
Lecturer/Organizer
Blas Pelegrín
Unit home page
All the information during the course, would be downloaded from the SUMA virtual
environment, at the URL http://suma.um.es
Course contents
Multicriteria decision making: Weighted methods, Compromise method and Goal
programming. Network models: Arcs and nodes routing and Project control analysis.
Locational analysis: Center and Median models. Game Theory: Two person zero-sum
games, Two person general games, and N-person cooperative games.
Prerequisites
Optimisation methods.
Teaching methods
Lecture, classes on problem modelling, assignments to be worked out during practical
sessions in the computer room. Some specific exercises from problem sheets are set for
handing in.
Assessment method
Written examinations on theoretical questions and exercises. Handing a collection of
optimisation problems solved by standard software.
Course unit title: TÉCNICAS DE MUESTREO Y CONTROL
DE CALIDAD / SAMPLING TECHNIQUES
AND QUALITY CONTROL
Credit Level: 4/5
ECTS Credits: 6
Semester: 1S
Unit code: 02A7
Type of unit: elective
Hours/Week: 4
Language of instruction: Spanish
Lecturer/Organizer
Félix Belzunce
Unit home page
None
Course contents
Quality control charts for attributes and for variable and attribute data, process
capability indices, acceptance sampling plans, reliability measures, coherent systems,
finite population sampling, Horvitz-Thompson estimators, simple random sampling,
stratified sampling, cluster sampling, systematic sampling
Prerequisites
Probability theory and basic statistical inference notions.
Teaching methods
Lecture, problem classes, assignments to be worked out during practical sessions and
monthly individual tutorial. Some specific exercises from problem sheets are set for
handing in.
Assessment method
Closed book written examinations and on the basis of handing in solutions to problems.
Course unit title: OPTIMIZACIÓN NO LINEAL /
NON-LINEAR OPTIMIZATION
Credit Level: 4/5
ECTS Credits: 6
Semester: 2S
Unit code: 02A8
Type of unit: elective
Hours/Week: 4
Language of instruction: Spanish
Lecturer/Organizer
Blas Pelegrín
Unit home page
During the course, all the information will be downloaded from the SUMA virtual
environment, at http://suma.um.es
Course contents
Convex sets. Convex functions and generalizations. Optimality properties. Global
optima in polyhedral sets. Algorithms for unconstrained optimisation. Optimality
conditions for constrained optimisation. Methods of feasible directions. Penalty and
barrier functions. Fractional and quadratic programming. Dynamic optimisation
techniques.
Prerequisites
Fundaments of Mathematical Analysis.
Teaching methods
Lecture, classes on problem solving, assignments to be worked out during practical
sessions in the computer room. Some specific exercises from problem sheets are set for
handing in.
Assessment method
Written examinations on theoretical questions and exercises. Handing a collection of
optimisation problems solved by standard software.
Course unit title: TEORÍA DE LA PROBABILIDAD
PROBABILITY THEORY
Credit Level: 4/5
ECTS Credits: 6
Semester: 1S
Unit code: 02A9
Type of unit: elective
Hours/Week: 4
Language of instruction: Spanish
Lecturer/Organizer
Noemí Zoroa
Unit home page
None
Course contents
Modes of convergence and their interrelationships. Central limit theorem. Laws of large
numbers. Three series theorem. Infinitely divisible laws. Kolmogorov theorem. Markov
chains. Poisson process. Martingales.
Prerequisites
Basic results from set theory and combinatorics. General knowledge of linear algebra,
mathematical analysis, probability and mathematical statistics .
Teaching methods
Lecture. Problem classes. Some specific exercises from problem sheets are set for
handing in.
Assessment method
Closed book written examinations.
Course unit title: AMPLIACIÓN DE MODELOS DE
INVESTIGACIÓN OPERATIVA /
EXTENSION OF OPERATIONAL
RESEARCH MODELS
Credit Level: 4/5
ECTS Credits: 6
Semester: 2S
Unit code: 03A0
Type of unit: elective
Hours/Week: 4
Language of instruction: Spanish
Lecturer/Organizer
Unit home page
http://www.um.es/or/ampliacion
Course contents
Simulation, queueing theory, inventory control.
Prerequisites
Basic notions of probability, statistics and operational research
Teaching methods
Lecture, problem classes, computer.
Assessment method
Written examination + assessment on the basis of handing in solutions to
problems/exercises with oral presentations.
Course unit title: DIDÁCTICA DE LAS MATEMÁTICAS EN
Unit code: 03A6
LA ENSEÑANZA SECUNDARIA /
DIDACTICS OF MATHEMATICS IN SECONDARY
EDUCATION
Credit Level: 4/5
ECTS Credits: 4.5
Semester: 2S
Type of unit: elective
Hours/Week: 3
Language of instruction: Spanish
Lecturer/Organizer
Dolores Carrillo Gallego
Unit home page
None
Course contents
Mathematics education, mathematics learning, curriculum, didactical situations,
problem solving, errors and obstacles in mathematics learning, history and mathematics,
education, mathematics reasoning, evaluation.
Prerequisites
None
Teaching methods
Lecture, exercises, practical sessions, assignments to be worked out during practical
sessions
Assessment method
Examinations: multiple choice questions. Written report of group work.
Course unit title: ÁLGEBRA COMPUTACIONAL
COMPUTATIONAL ALGEBRA
Credit Level: 4/5
ECTS Credits: 7.5
Semester: 1S
Unit code: 03A7
Type of unit: elective
Hours/Week: 5
Language of instruction: Spanish
Lecturer/Organizer
José Luis García Hernández
Unit home page
None
Course contents
Algorithms: corrections and efficiency. Computational complexity. Cryptosystems:
public key (RSA, ElGamal, digital signatures) and private key (Rijndael). Square roots
and discrete logarithms. Primality: probabilistic tests, deterministic tests, prime
certificates. Integer factorization: old algorithms and sieve methods. Elliptic curves and
their applications in cryptography and number theory algorithms.
Prerequisites
Some knowledge of abstract algebra, including field extensions and Galois theory up to
a basic level.
Teaching methods
Lecture and assignments to be worked out during practical sessions.
Assessment method
Implementation of a computer program with MATHEMATICA which will efficiently
develop at least one of the algorithms of the course.
Course unit title: AMPLIACIÓN DE ECUACIONES EN
Unit code. 03A8
DERIVADAS PARCIALES / EXTENSION OF
PARTIAL DIFFERENTIAL EQUATIONS
Credit Level: 4
ECTS Credits: 6
Semester: 1S
Type of unit: elective
Hours/Week: 4
Language of instruction: Spanish
Lecturer/Organizer
Francisco Balibrea Gallego
Unit home page
None
Course contents
Nonlinear wave equations. Solitons. Special functions and boundary problems of more
than one variable. The Helmholtz equation. Integral transforms. Hopf-Cole, Hodograf
and Legendre transforms. Weak solutions of partial differential equations. Distributions
theory. Operations. Convolution. Solutions as distributions. Fundamental solutions.
Malgrange-Ehrenpreis theorem. Applications.
Prerequisites
None
Teaching methods
Lecture, problem classes, assignments to be worked out during practical sessions and
monthly individual tutorial. Some specific exercises as homework weekly. Also some
relations of exercises and problems to be solved.
Assessment method
Closed book written examinations and ongoing evaluation of participation
Course unit title: ANÁLISIS NUMÉRICO DE LAS
Unit code: 03A9
ECUACIONES EN DERIVADAS
PARCIALES / NUMERICAL ANALYSIS
OF PARTIAL DIFFERENTIAL EQUATIONS
Credit Level: 4/5
ECTS Credits: 6
Semester: 1S
Type of unit: elective
Hours/Week: 4
Language of instruction: Spanish
Lecturer/Organizer
Víctor Jiménez
Unit home page
Detailed information on the main programming tool we use (Easy Java Simulations)
can be found at http://fem.um.es/Ejs/Ejs_en/index.html.
Course contents
Numerical methods for solving partial differential equations. Finite difference methods:
parabolic equations in one and two space variables; hyperbolic equations in one space
variable. Finite elements method: two-dimensional elliptic equations.
Prerequisites
A basic course on partial differential equations, some basic knowledge in Java
programming, and some basic knowledge in numerical methods for solving systems of
linear equations.
Teaching methods
Lecture, problem classes, practical sessions in the computer room, practical computer
work.
Assessment method
Closed book written examination and assessment on the basis of handing in solutions to
practical problems.
Course unit title: MÉTODOS MATEMÁTICOS PARA
LA MECÁNICA / MATHEMATICAL
METHODS IN MECHANICS
Credit Level: 4/5
ECTS Credits: 6
Semester: 2S
Unit code: 04A0
Type of unit: elective
Hours/Week: 4
Language of instruction: Spanish (English, if agreed)
Lecturer/Organizer
Francisco Esquembre
Unit home page
None
Course contents
1.- Basic concepts of Mechanics at an introductory Physics level.
2.- Newtonian mechanics.
3.- Lagrangian mechanics.
4.- Hamiltonian machanics.
Prerequisites
Matrix algebra. Vector calculus. Ordinary differential equations.
Teaching methods
Selected plenary lectures. Students should prepare in advance so that classroom time
can be devoted to discussions on the theory and problems. Homework will be assigned
in the form of problems to be worked out at home and discussed in the classroom.
Assessment method
Closed book written examinations of practical nature and ongoing evaluation of
participation and homework problems.
Course unit title: TEORÍA DE NÚMEROS ALGEBRAICOS /
ALGEBRAIC NUMBER THEORY
Credit Level: 4/5
ECTS Credits: 6
Semester: 2S
Unit code: 04A5
Type of unit: elective
Hours/Week: 4
Language of instruction: Spanish
Lecturer/Organizer
José Ramón Caruncho Castro
Unit home page
None
Course contents
Algebraic integers. Quadratic and cyclotomic fields. Dedekind domains. Prime
decomposition in Dedekind rings. Class group and class numbers. Minkowski’s
Theorem. Applications to computation of class numbers. Dirichlet units theorem.
Prerequisites
Field theory, finitely generated abelian groups.
Teaching methods
Lecture, problem classes. Some specific exercises from problem sheets are set for
handing in.
Assessment method
Closed book written examinations and ongoing evaluation of participation.
Course unit title: GEOMETRÍA ALGEBRAICA /
ALGEBRAIC GEOMETRY
Credit Level: 4/5
ECTS Credits: 7.5
Semester: 2S
Unit code: 04A6
Type of unit: elective
Hours/Week: 5
Language of instruction: Spanish
Lecturer/Organizer
Pedro A. Guil Asensio
Unit home page
None
Course contents
Affine and projective algebraic sets. The ring of regular functions. Gröbner bases and
applications. Algebraic varieties and morphisms. Local rings. Birrational equivalence.
Dimension of a variety. Tangent spaces and singular points. Introduction to sheaves and
schemes.
Prerequisites
Some knowledge of abstract algebra and linear geometry, including basic notions of
projective geometry, field extensions and commutative rings.
Teaching methods
Lecture and problem classes.
Assessment method
Closed book written examinations and ongoing evaluation of participation
Course unit title: REPRESENTACIONES DE GRUPOS /
REPRESENTATIONS OF GROUPS
Credit Level: 4/5
ECTS Credits: 6
Semester: 2S
Unit code: 04A7
Type of unit: elective
Hours/Week: 4
Language of instruction: spanish
Lecturer/Organizer
Antonio Álvarez
Unit home page
None
Course contents
Group actions. Sylow’s Theorems. Finite p-groups. Composition series. Solvable
groups. Algebras, modules and representations. Characters. The characters table.
Burnside’s paqb Theorem. Induced characters.
Prerequisites
Wedderburn Theory.
Teaching methods
Lecture, problem classes and individual tutorial. Some specific exercises from problem
sheets are set for handing in.
Assessment method
Closed book written examinations and ongoing evaluation of participation
Course unit title: ÁLGEBRA HOMOLÓGICA /
HOMOLOGICAL ALGEBRA
Credit Level: 4/5
ECTS Credits: 6
Semester: 1S
Unit code: 04A8
Type of unit: elective
Hours/Week: 4
Language of instruction: Spanish
Lecturer/Organizer
Juan Martínez Hernández
Unit home page
None
Course contents
Categories and functors. Natural transformations. Exact sequences, Chain complexes.
Homology and cohomology groups. Homology functors. Homothopy. Derived functors.
The Ext and Tor functors. Universal coefficient Theorem.
Prerequisites
Algebra (01A5).
Teaching methods
Lecture supplemented by small group tutorials, problem sheets
Assessment method
Students may choose between a closed book examination and an ongoing evaluation of
their participation based on the problem sheets and personal expositions supervised by
the lecturer
Course unit title: ALGEBRAS DE BANACH Y TEORIA ESPECTRAL / Unit code: 05A3
BANACH ALGEBRA AND SPECTRAL THEORY
Credit Level: 4
ECTS Credits: 6
Semester: 2S
Type of unit: elective
Hours/Week: 4
Language of instruction: Spanish
Lecturer/Organizer
Stanimir Troyanski
Unit home page
None
Course contents
Elements of Banach Algebras. Spectral theory of self-adjoint operators in Hilbert
spaces. Compact operators in Banach spaces, Lomonosov’s Theorem.
Prerequisites
Basic knowledge on Complex and Functional Analysis (see units ANALISIS
COMPLEJO-COMPLEX ANALYSIS and ANALISIS FUNCIONAL-FUNCTIONAL
ANALYSIS)
Teaching methods
Lecture, problem classes. Some specific exercises from problem sheets are set for
handing in.
Assessment method
Oral presentations and ongoing evaluation of participation.
Course unit title: TOPOLOGÍA ALGEBRAICA /
ALGEBRAIC TOPOLOGY
Credit Level: 4/5
ECTS Credits: 7.5
Semester: 1S
nit code: 05A4
Type of unit: elective
Hours/Week: 5
Language of instruction: Spanish
Lecturer/Organizer
Luis Alías Linares
Unit home page
None
Course contents
Homotopy and relative homotopy. The Seifert-Van Kampen theorem and applications.
Covering spaces and apllications.Singular homology. The Mayer-Vietoris sequence.
Homology of spheres. Some classical theorems: Brouwer fixed point and JordanBrouwer. Degree theory.
Prerequisites
General topology, topology of surfaces.
Teaching methods
Lecture, problem classes, assignments to be worked out during practical sessions and
monthly individual tutorial. Some specific exercises from problem sheets are set for
handing in.
Assessment method
Ongoing evaluation of participation and oral presentations.
Course unit title: GEOMETRÍA DIFERENCIAL AVANZADA
ADVANCED DIFFERENTIAL GEOMETRY
Credit Level: 4/5
ECTS Credits: 6
Semester: 2S
Unit code: 05A5
Type of unit: elective
Hours/Week: 4
Language of instruction: Spanish
Lecturer/Organizer
José Antonio pastor González
Unit home page
None
Course contents
Lorentz geometry, causal character of vectors, time-cones, time-orientation, local
Lorentz geometry, space-times, special relativity, some relativistic effects, energymomentum, general relativity, the Einstein equation, perfect fluids, Robertson-Walker
space-times, redshift, Schwarzschild space-time, perihelion advance, light-like orbits,
Kruskal space-time, black holes.
Prerequisites
Riemannian Geometry.
Teaching methods
Lecture, problem classes, assignments to be worked out during practical sessions.
Assessment method
Ongoing evaluation of work and participation.
Course unit title: GEOMETRÍA DE SUBVARIEDADES /
GEOMETRY OF SUBMANIFOLDS
Credit Level: 5
ECTS Credits: 6
Semester: 1S
Unit code: 05A6
Type of unit: elective
Hours/Week: 4
Language of instruction: Spanish
Lecturer/Organizer
Pascual Lucas
Unit home page
None
Course contents
Basic equations of submanifolds; fundamental theorem of submanifolds;
minimal submanifolds; umbilical submanifolds; r-planes; r-spheres;
hypersurfaces; convex Euclidean hypersurfaces; Einstein hypersurfaces; nonpositive curvature submanifolds; Chern-Kuiper theorem; Jorge-Koutroufiotis;
codimension reduction; parallelism of first normal space; complete submanifolds
of constant sectional curvature; isometric immersions; bilinear forms; rigidity;
local and global rigidity of hypersurfaces; conformally flat hypersurfaces; flat
conformally flat submanifolds; low dimension.
Prerequisites
Basic concepts and main results of Topology, Mathematical Analysis,
Differential Geometry and Differential Equations.
Teaching methods
Lecture and problem classes. Some specific exercises from problem sheets are
set for handing in.
Assessment method
Assessment on the basis of handing in solutions to problems/exercises.
Course unit title:
AMPLIACIÓN DE ÁLGEBRA
CONMUTATIVA / EXTENSION OF
COMMUTATIVE ALGEBRA
Credit Level: 4/5
ECTS Credits: 6
Semester: 2S
Unit code: 05A1
Type of unit: elective
Hours/Week: 4
Language of instruction: Spanish
Lecturer/Organizer
Manuel Saorín Castaño
Unit home page
None
Course contents
Primary decomposition in Noetherian rings. Dimension Theory. Regular local rings.
Complete local rings
Prerequisites
The student should know the basic concepts concerning ideals and modules over
commutative rings, as well as the formation of rings and modules of fractions.
Teaching methods
Lectures, problem classes based on problem sheets, assignments to be worked out
during practical sessions, tutorials.
Assessment method
Open book written examination and ongoing evaluation of participation.