Course unit title: GEOMETRÍA DE RIEMANN / Unit code: 05A7 RIEMANNIAN GEOMETRY

Credit Level: 4/5 Type of unit: elective ECTS Credits: 7.5 Hours/Week: 5 Semester: 2S 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; torsion-free 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 Unit code: 01A5

Credit Level: 4 Type of unit: compulsory ECTS Credits: 9 Hours/Week: 3 Semester: full year 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 / Unit code: 01A6 FUNCTIONAL ANALYSIS

Credit Level: 4 Type of unit: compulsory ECTS Credits: 6 Hours/Week: 4 Semester: 1S 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 / Unit code: 01A7 COMPLEX ANALYSIS

Credit Level: 4 Type of unit: compulsory ECTS Credits: 6 Hours/Week: 4 Semester: 1S 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 / Unit:01A8 PARTIAL DIFFERENTIAL EQUATIONS

Credit Level: 4 Type of unit: compulsory ECTS Credits: 6 Hours/Week: 4 Semester: 2S 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 Unit code: 01A9

GEOMETRY AND TOPOLOGY

Credit Level: 4 Type of unit: compulsory ECTS Credits: 9 Hours/Week: 3 Semester: full year 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 / Unit code: 02A0 NUMERICAL ANALYSIS

Credit Level: 4 Type of unit: compulsory ECTS Credits: 9 Hours/Week: 3 Semester: full year 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 / Unit code: 02A3 MULTIVARIANT ANALYSIS

Credit Level: 4/5 Type of unit: elective ECTS Credits: 7.5 Hours/Week: 5 Semester: 2S 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 / Unit code: 02A4

LINEAR MODELS

Credit Level: 4/5 Type of unit: elective ECTS Credits: 7.5 Hours/Week: 5 Semester: 1 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 Type of unit: elective ECTS Credits: 6 Hours/Week: 4 Semester: 1S 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 Unit code: 02A7 DE CALIDAD / SAMPLING TECHNIQUES AND QUALITY CONTROL

Credit Level: 4/5 Type of unit: elective ECTS Credits: 6 Hours/Week: 4 Semester: 1S 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 / Unit code: 02A8 NON-LINEAR OPTIMIZATION

Credit Level: 4/5 Type of unit: elective ECTS Credits: 6 Hours/Week: 4 Semester: 2S 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 Unit code: 02A9 PROBABILITY THEORY

Credit Level: 4/5 Type of unit: elective ECTS Credits: 6 Hours/Week: 4 Semester: 1S 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 Unit code: 03A0 INVESTIGACIÓN OPERATIVA / EXTENSION OF OPERATIONAL RESEARCH MODELS

Credit Level: 4/5 Type of unit: elective ECTS Credits: 6 Hours/Week: 4 Semester: 2S 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 Type of unit: elective ECTS Credits: 4.5 Hours/Week: 3 Semester: 2S 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 Unit code: 03A7 COMPUTATIONAL ALGEBRA

Credit Level: 4/5 Type of unit: elective ECTS Credits: 7.5 Hours/Week: 5 Semester: 1S 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 Type of unit: elective ECTS Credits: 6 Hours/Week: 4 Semester: 1S 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 Type of unit: elective ECTS Credits: 6 Hours/Week: 4 Semester: 1S 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 Unit code: 04A0 LA MECÁNICA / MATHEMATICAL METHODS IN MECHANICS

Credit Level: 4/5 Type of unit: elective ECTS Credits: 6 Hours/Week: 4 Semester: 2S 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 / Unit code: 04A5 ALGEBRAIC NUMBER THEORY

Credit Level: 4/5 Type of unit: elective ECTS Credits: 6 Hours/Week: 4 Semester: 2S 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 / Unit code: 04A6 ALGEBRAIC GEOMETRY

Credit Level: 4/5 Type of unit: elective ECTS Credits: 7.5 Hours/Week: 5 Semester: 2S 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 / Unit code: 04A7 REPRESENTATIONS OF GROUPS

Credit Level: 4/5 Type of unit: elective ECTS Credits: 6 Hours/Week: 4 Semester: 2S 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 / Unit code: 04A8 HOMOLOGICAL ALGEBRA

Credit Level: 4/5 Type of unit: elective ECTS Credits: 6 Hours/Week: 4 Semester: 1S 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 Type of unit: elective ECTS Credits: 6 Hours/Week: 4 Semester: 2S 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 / nit code: 05A4

ALGEBRAIC TOPOLOGY

Credit Level: 4/5 Type of unit: elective ECTS Credits: 7.5 Hours/Week: 5 Semester: 1S 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 Jordan-Brouwer. 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 Unit code: 05A5 ADVANCED DIFFERENTIAL GEOMETRY

Credit Level: 4/5 Type of unit: elective ECTS Credits: 6 Hours/Week: 4 Semester: 2S 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, energy-momentum, 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 / Unit code: 05A6 GEOMETRY OF SUBMANIFOLDS

Credit Level: 5 Type of unit: elective ECTS Credits: 6 Hours/Week: 4 Semester: 1S 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; non-positive 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 Unit code: 05A1 CONMUTATIVA / EXTENSION OF COMMUTATIVE ALGEBRA

Credit Level: 4/5 Type of unit: elective ECTS Credits: 6 Hours/Week: 4 Semester: 2S 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.