Advanced Functional Analysis

Algebra
GROUPS: Groups, subgroups, normal subgroups, quotient groups, homomorphism, cyclic groups, symmetric groups, direct products, the homomorphism theorems, finite abelian groups and finite groups.
RINGS: Rings, subrings, ideals, quotient rings, homomorphisms, extensions and quotients of rings, rings of polynomials, principal ideal domains, Euclidean domains, unique factorization domains, zeros of polynomials.
FIELDS: Prime fields, field extensions, splitting fields, finite fields.

Algebraic Topology
The course deals with two main topics: Fundamental Group and Covering Spaces. "A First Course in Algebraic Topology" by C. Kosniowski, C.U.P., 1980, gives a good idea of what the course is about.

Algorithms and Data Structures
Structured Programming. Formal Algorithm Verification: Hoare's Axiomatic Recursive Algorithms. Searching and Sorting Algorithms Computational Efficiency and Complexity Analysis. Dynamic Data Structures. Abstract Data Types: Stacks, Queues, Trees and their Applications. Graphs and Networks. Hashing. External Storage Techniques.

Applications of Mathematics

Category Theory
Categories, functors and natural transformations. Epimorphisms and monomorphisms. Principle of Categorical Duality. Yoneda Lemma. Limits and colimits. Adjoint functors. Cartesian closed categories. Bibliography: F. Borceux, Handbook of Categorical Algebra , vol.1, Cambridge University Press, 1994.

Combinatorial Theory

Commutative Algebra
Prerequisites: elementary concepts of linear algebra and of second year algebra: commutative rings, prime and irreducible elements, unique factorization domains, principal ideal domains. 1. Prime and maximal ideals; consequences of Zorn's Lemma; nilradical and Jacobson radical; the spectrum of a ring; Zariski's topology. 2. Rings of fractions and localization. 3. Introduction to module theory; Noether isomorphism theorems; quotient modules; sums and products; free modules and presentations; torsion and cyclic modules. 4. Chain conditions, Noetherian rings and modules, and the Basis Theorem of Hilbert and Noether. 5. Algebraic varieties; Zariski topology; irreducible varieties and irreducible decomposition of a variety; statement of Hilbert's Nullstellensatz; applications. Bibliography: M. Atiyah & I. MacDonald, Introduction to Commutative Algebra, Add. Wesley 1969 (the most recommended), O. Zariski & P. Samuel, Commutative Algebra, vols I & II, GTM, Springer 1986, E. Kunz, Introd. to Commutative Algebra and Algebraic Geometry, Birkauser 1985

Compilers
Introduction: formal definition of a Language; Interpreters and Compilers. Lexical Analysis: Regular Expressions and Finite Automata. Syntax Analysis: top-down parsing and bottom-up shift/reduce parsing (LR,LALR). Semantic Analysis: syntax-oriented translation. Tools: Lex and Yacc.

Complex Analysis I

Complex Analysis II
Introduction to function theory; Applications of Cauchy theorem: Theorems of Residue, Rouché and Lagrange; Theorems of Weierstrass and Mittag-Leffler; Elliptic Functions Theory; Normal families; Riemann mapping theorem; Theory of harmonic functions; Dirichlet problem.
References: L. Ahlfors, Complex Analysis (3rd ed.), Mc-Graw-Hill Book Co., 1979; W. Rudin, Real and Complex Analysis, Mc-Graw-Hill International Editions, 3 ed., 1987; E.T. Wittaker and G.N. Watson, A Course of Modern Analysis (4th ed.), Cambridge Unversity Press, 1927.
Remark: The students should have the basic notions given in the course Complex Analysis I.

Computer Graphics I

Computer Graphics II

Computer Graphics III
I - Colour Study. II - Rendering. III - Advanced Topics.
References: J.D. Foley, A. van Dam, S.K. Feiner, J.F. Hughes, "Interactive Computer Graphics - Principles and Practice", Addison-Wesley, 1990. D. Hearn, M.P. Baker, "Computer Graphics", 2nd ed., Prentice-Hall International, 1994.

Computers in Mathematics Education
The impact and uses of computers in the curriculum of Mathematics. Exploration of software tools for the teaching of Mathematics. Evaluation and documentation of educational software. The LOGO language. Internet and the teaching of Mathematics.
References: Papert, Seymour - Logo: computadores e educação, Ed. Brasiliense, São Paulo, 1988; NCTM - Computers in Mathematics Education, NCTM Yearbook 1984. New curriculum for Mathematics in Secondary School, Min. Education of Portugal; NCTM - Normas para o curriculo e a avaliação em Matemática, ed. APM/IIE, 1991.

Data Bases

Decision Theory and Statistics
1 - Basic Principles of Statistical Decision Theory (bayes decision rules, minimax decision rules, unbiased decision rules, invariante decision rules, asymptotic principles) 2 - Neyman-Pearson Theory of Testing of Hypotheses 3 - Some Chi-Square Tests (goodness-of-fit, independence and homogeneity tests)

Differential Equations
Introduction to differential equations. First-order ordinary differential equations. Linear differential equations of order n. First-order systems of differential equations. Qualitative theory of differential equations

Differential Geometry
I - Local Curve Theory in R^3.
1- Preliminaries. 1.1- The euclidean space R^n. 1.2- A brief review of vectorial-valued functions of real variable. 2- Regular curves. 3- Arc length and parameterization by arc length. 4- Curvature and torsion. Frenet-Serret frame. Frenet-Serret formulas. 5- Frenet-Serret frame for curves not parameterized by arc length. 6- Cylindrical helices. 7- Involutes and evolutes. 8- Congruent curves. 9- Fundamental Theorem of Curves.
II- Local Surface Theory in R^3.
1- Preliminaries. 1.1- The metric space R^n. 1.2- A brief review of continuity and differenciability in R^n. 2- Surfaces in R^3: basic definitions and examples. 3- Tangent space and tangent plane. 4- First fundamental form. 5- Orientable surfaces. 6- The geometry of the Gauss map: the second fundamental form; normal curvature, geodesic curvature and Gauss's formulas; the Dupin indicatrix.

Differential Manifolds I
Differentiable manifolds. Tangent and cotangent bundles. Submanifolds. Vector fields. First order differential equations. Distributions and Frobenius' theorem. First notions of Lie groups.

Differential Manifolds II
Tensor Bundles. Exterior derivative. Lie derivative. De Rham Cohomology. Poincaré' Lemma. Integration on manifolds and Stokes' theorem. Applications: Hamiltonian Mechanics.

Educational Psychology

Elementary Topics in Mathematics
Chapter I: Basic Concepts in Logic 1.1 Propositions. 1.2 The propositional calculus (logic operations with propositions). 1.3 The predicate calculus (quantification). Validity of an expression. 1.4 Tautologies. Contradictions. Analysis of arguments for logic validity.
Chapter II: Methods of Mathematical Proof 2.1 Introduction 2.2 Proving set inclusion 2.3 Proving set equality 2.4 Proving involving the empty set 2.5 Conclusions involving *, but not *, *, or * 2.6 Conclusions involving *, and * but not * 2.7 Proof by specialization and division into cases 2.8 Proof by mathematical induction 2.9 Conclusions involving * followed by * 2.10 Indirect proofs: derivation of conclusions involving disjunction, proof by contra-positive, proof by contradiction
Chapter III: Methods of Mathematical Proof 3.1 Basic definitions and notation 3.2 Relations between sets 3.3 Power set 3.4 Operations on sets 3.5 Relations 3.6 Mappings 3.7 Restriction and extension of a mapping 3.8 Inverse mappings 3.9 Families
Chapter IV: Finite, countable and uncountable sets 4.1 An axiomatization for the set N of the natural numbers 4.2 Well ordered principle and mathematical induction 4.3 Finite sets and infinite sets. Cardinal number of a set. 4.4 Countable sets 4.5 Uncountable sets
Chapter V: Construction of number systems 5.1 Fields 5.2 Ordered fields 5.3 Development of the ring, Z, of the integers. 5.4 Development of the (ordered) field, Q, of the rational numbers. 5.5 Development of the (complete ordered) field, R, of the real numbers. 5.6 Development of the field, C, of the complex numbers.

Finite Mathematics

Fluid Mechanics

Formal Languages
Semigroups. Generative grammar. The Chomsky hierarchy of languages. Context-free languages. Linear grammars and regular languages. Greibach normal form. Regular expressions. Context-sensitive languages. Phrase-structure languages. Automata and their languages. Decidability.

Foundations and Teaching of Algebra
1-Elementary Logic; Proposicional Calculus, Predicate Calculus; Methods of mathematical proof.
2-Intuitive set theory and axiomatic set theory (the Zermelo-Fraenkel set theory with choice).
3-Relations,Functions, Mappings, and Orderings.

Foundations and Teaching of Analysis

Foundations and Teaching of Geometry
1. Axiomatics of plane euclidean geometry: the classical material in more than usual detail. 2. Plane Trigonometry: sine, cosine from perspective of geometry and analysis, the familiar theorems: sine theorem, cosine theorem, addition theorems, some applications. 3. Theorems on triangles, quadrangles, and circles: Ceva and consequences, Menelaos, Euler, Steiner-Lehmus, pedal triangles, medial triangle, Appollonius circle, inscribed and circumscribed quadrilaterals etc. Examples of Olympiad problems. 4. Vector calculus: motivation of vector space axioms from axiomatic euclidean geometry; some applications to geometry. 5. Analytic Geometry in plane and space: coordinate systems, equations of rectilinear and plane figures, conics, distances in plane and space.

Fourier Analysis and Functional Spaces
I- Some concepts from Measure and Integration. II- Fourier series. III- Distributions. IV- Fourier transform. V- Sobolev spaces. References: P. Benoist-Gueutal, M. Courbage, Mathematiques pour la Physique, Tome 2, 2nd ed., Eyrolles, 1995; M.A.Al. Gwaiz, Theory of Distributions, Marcel Dekker, Inc. 1992; D.G. Figueiredo, Analise de Fourier e Equacoes Diferenciais, Projecto Euclides, 1977.

Functional Analysis
Topological linear spaces. Normed spaces, Banach spaces. Bounded linear operators. Compact operators. Dual spaces. The Hahn-Banach theorem. Dual operators. Inner product spaces. Hilbert spaces. The Riesz representation theorem. Adjoint operator. Symmetric normal, unitary and selfadjoint operators. Orthogonal projections. Spectral analysis: an introduction. Weak topology and weak* topology in normed spaces.

Fundamentals of Computer Organization
I - Introduction to Digital Computers (Analogic and Digital information and systems; Binary digital systems, Binary information, Representation of numbers in computer's memory). II - Architectural Aspects of a Computer (CPU, Memory, Buses, Peripherals; Assembler language; Example: the 68000 microprocessor). III - Programming in C Language.

Geometry

Geometric Modelling
Introduction. Polynomial Interpolating Curves. Bernstein-Bézier Curves. B-Spline Curves. Bernstein-Bézier Surfaces. Solid Modelling - A Brief Introduction.

Group Theory

History of Mathematics

Infinitesimal Analysis I
Sequences of real numbers; infinite series. Properties of the real numbers. Topology. Limits of functions. Continuity. Derivatives. Riemann integral. Sequences and infinite series of functions.

Infinitesimal Analysis II

Integer Programming
Formulations; Optimality, Relaxation and Bounds; Well-Solved Problems; Matchings and Assignments; Dynamic Programming; Complexity and Problem Reduction; Branch-and-Bound; Lagrangian Duality; Cutting-Plane Algorithms.
References: L. Wolsey, Integer Programming, Wiley-Interscience, 1998; A. Schrijver, Theory of Linear and Integer Programming, Wiley-Interscience, 1986.

Introduction to Topology
I - Continuous functions versus metric and topological spaces. II - Topological spaces: examples and basic notions. III - Connected topological spaces. IV- Compact topological spaces. V - Metric spaces: convergence, completeness and compactness. VI - Function spaces. Bibliography: J.A. Pereira da Silva, Elementos de Topologia, Textos de Matemática, Coimbra, 1987; W.A. Sutherland, Introduction to Metric and Topological Spaces. Oxford University Press, 1975.

Linear Algebra and Analytic Geometry
I - Complex Numbers; II - Linear Spaces; III - Linear Tranformations; IV - Matrices and Systems of Equations; V - Determinants and Systems of Equations; VI - Space Analytic Geometry; VII - Multilinear Transformations; VIII - Inner Product Spaces.
References: J. Vitoria and T.P.Lima, Álgebra Linear, Universidade Aberta, Lisboa, 1998; F. R. Dias Agudo, Introdução à Álgebra Linear e Geometria Analítica, Lisboa, 1989.

Linear Programming
Definition of linear program. Some samples of problems as a linear program. The primal simplex method. A first feasible basic solution. The convergence of the algorithm.
Theory of duality. The dual simplex algorithm and the primal-dual algorithm. Analisys of sensibility.

Mathematical Logic
I. Propositional Calculus (PC), II. Computability.
I. 1. Well formed formulae, truth tables, disjunctive Normal Form. 2. Typical applications: the resolution theorem in PROLOG, construction of classical circuitry, mention of NP-completeness of the simplification problem for Boolean formulae. 3. Compactness theorem of propositional calculus. 4. Formal deduction in PC. 5. Completeness theorem of PC.
II. 1. Turing Machines. 2. Computability of certain elementary functions. 3. General theorems concerning computability of functions defined via recursion or composition or minimum operator. 4. Corollaries deduced from (3). 5. Existence of noncomputable functions and nondecidable problems. 6. A concrete example: the halting problem. 7. Enumerability. 8. Definability in Arithmetics. 9. Tarski's theorem on non-arithmeticity of truth. 10. Gödel's incompleteness theorem as a corollary.

Measure and Integration
0 - Riemann integral and Jordan measure 1 - Sets and classes 2 - Measures and outer measures 3 - Extension of measures 4 - Measurable functions 5 - Integration 6 - Product measures 7 - Transformation of measures 8 - L^p spaces 9 - Signed measures

Mechanics
Introduction: Some considerations about the axiomatic concepts of rigid body, particle, space, time and about motion and coordinate systems. Kinematics of a point: Description of particle's motion, velocity and acceleration: cartesian, cylindrical, spherical and intrinsic (normal and tangential) components; some particular motions: rectilinear and plane motions (radial and transverse components of velocity and acceleration; central motions), 3D motions. Kinematics of a rigid body: degrees of freedom of a moving rigid body, vector fields of velocities, instantaneous axis of rotation, vector fields of the acelerations, translational motions, rotation about a fixed axis, plane, screw and tangent motions; moving coordinate systems; motion about a fixed point (Euler angles) and the general motion of a rigid body as a composition of simpler motions.
Kinetics:
- Kinetics of systems of particles: some considerations about mass and mass-systems; the concepts and properties of the mass-center, the linear momentum, the kinetics momentum and the kinetics energy; Koenigs Theorems.
- Kinetics of a rigid body: moments and products of inertia, inertia matrix, principal axis, kinetic energy of a rigid body in a general motion.
Dynamics:
- Some considerations about forces and their mathematical representations.
- Dynamics law for rigid bodies; forces work, energy and power; conservative fields; conservation of energy.
- Study of some particular particle motions: projectile motions under gravity, constrained particle motions, motion of a particle under a central force.
- The general three-dimensional rigid body motions: Euler equations. Some constrained rigid body motions.
References: Brousse - Cours de Mécanique - Armand Colin, Paris, 1973, A.S.Alves - Mecânica Geral, INIC/CMUC, Coimbra, 1989, D. Euvrard - Cours de Mécanique Générale, policopiado de Paris VI, 1985

Methodology of Mathematics

Methods and Techniques of Education

Network Programming
Minimal Spanning Tree Problem - Algorithms of Kruskal, Prim and Cheng.
Optimal Path Problems - The Optimality Principle; label setting and label correcting algorithms; computational complexity and practical efficiency.
Maximal Flow Problem - min cut-max flow theorem; incremental chains algorithm; computational complexity and practical efficiency.
Minimal Cost Flow Problem - Simplex algorithm; particular cases (transportation and assignment problems); shortest path and maximal flow problems as a particular problem.

Nonlinear Programming
I - Unconstrained nonlinear optimization (line searches, trust regions, nonlinear conjugate gradients, practical Newton methods, quasi-Newton methods, derivative calculations). II - Theory of constrained nonlinear optimization (optimality conditions, constraint qualifications, duality). III - Constrained nonlinear optimization (penalty, barrier and augmented Lagrangian methods, sequential quadratic programming).

Number Theory

Numerical Analysis I
1-Introduction to Numerical Computation. Computer Representation of real numbers; Roundoff errors; Computing with Floating Point Numbers; Truncation Errors ; Propagation Errors; Direct and Inverse Problems. Condicioning and Stability. 2-Systems of Linear Simultaneous Equations. Direct Methods: Gauss Elimination; L-U Factorization; Tridiagonal matrices. Iterative Methods: Jacobi, Gauss-Seidel and Relaxation and convergence conditions. 3- Nonlinear Equations: Localization and first approximation of the roots. Iterative Methods: Bissection; Secant; Newton-Raphson; Fixed - Point. Convergence analysis. Roots of Polynomials. Systems of Nonlinear Equations. 4-Interpolation: Introduction; Polynomial interpolation: Lagrange and Newton formulas. Analysis of the error. Hermite Interpolation. Interpolation for fuctions of two independent variables. 5-Least Square Curve Fitting :Polynomial least square curve fitting and applications. 6-Numerical Evaluation of Derivatives and Integrals: Newton-Cotes formulas and Errors. 7-Numerical Solution of Ordinary Differencial Equations (Initial Value Problems): one step and Multistep methods. References: Conte, S. D. and De Boor, C. -Elementary Numerical Analysis (1980). Valenca, M. R. - Metodos Numericos (1990). A. Wood- Introduction to Numerical Analysis. Addison-Wesley (1999).

Numerical Analysis II
I Approximation theory 1.1 Introduction 1.2 Polynomial approximation 1.3 Trigonometric approximation II Ordinary differential equations: initial value problems 2.1 Introduction 2.2 One-step methods 2.3 Multistep methods 2.4 Systems of ordinary differential equations III Ordinary differential equations: boundary value problems 3.1 Introduction 3.2 Some classical methods 3.3 Symmetric and nn-symmetric weak formulation 3.4 Variational formulation 3.5 The finite-element method 3.6 The finite-difference method

Numerical Linear Algebra
Direct Methods for Systems of Linear Equations with Square Matrices: The Decompositions and Stability. Band, Block and Sparse Matrices: Storage, Operations and Implementation of Direct Methods. Iterative Methods for Systems of Linear Equations: Convergence Analysis and Implementation. Direct and Iterative Methods for Systems of Linear Equations with Rectangular Matrices: Minimum Norm and Linear Least-Squares.

Numerical Methods for P.D.E.
1- Introduction: classification of p.d.es.as elliptic, parabolic or hyperbolic and different ways of obtainning analitic solutions. Some difficulties. 2- Computational methods. Finite Difference methods and some theoretical considerations: convergence, consistency and stability. The Lax Theorem. Initial and Initial-Boundary Value Problems. 3- Parabolic Equations: explicit, implicit and alternating direction implicit schemes. 4- Hyperbolic Equations. Initial and boundary conditions. Explicit and Implicit schemes. The Courant-Friedrichs-Lewy conditions. Dispersion and Dissipation. Elliptic p. d. es. Introduction to Finite Element Method.Error estimates. 5- Weighted Residual Methods- general formulation.
References: J.W. Thomas,Numerical Partial Differential Equations: Finite Difference Methods. Springer Verlag-1995; C. A. J. Fletcher, Computational Techniques for Fluid Dynamics, Springer Verlag, 1991; Claes Johnson, Numerical Solution of p. d. es. by the finite element method. Cambridge Univ. Press, 1990

Object-Oriented Programming
Introduction: the concepts of abstraction and modularity when programming-in-the-large. Classes and Objects has an implementation of ADTs. Classes hierarchies; inheritance, aggregation and association relations; polymorphism. Generic programming and error handling. The C++ programming language.

Operating Systems

Partial Differential Equations
I Introduction 1. Basic concepts for PDE's 2. Where PDE's come from: flows, vibrations and diffusions 3. Initial and boundary conditions 4. Well posed problems, Stability theory 5. Types of second order equations
II The method of characteritics 1. Linear case 2. Quasi -linear case 3. Non linear case
III Wave equation, Diffusion equation, Poisson equation 1. Wave equation 2. Diffusion equation 3. Poisson equation
IV Fourier series and PDE's 1. Separation of variables 2. The Sturm-Liouville problem and Fourier series 3. Series solutions of initial and boundary value problems 4. Inhomoheneous equations
V Integral transforms and PDE's 1. One dimensional Fourier transforms 2. Fourier sine and cosine transform 3. Integral solutions of initial and boundary value problems

Probability
I - An introduction to probability theory II - Real random variables and probability laws III - Real random vectors. Independence of real random variables IV - Characteristic function of a real random variable V - Stochastic convergences
References: E. Gonçalves, N. Mendes Lopes(1999) Probabilidades-Princípios teóricos (book in press); Rohatgi, V. K.(1976) An introduction to probability theory and mathematical statistics, Wiley.

Probability Theory

Programming Methods

Specification and Verification
Introduction: formal programming, the support for abstraction and modularity in the many different programming paradigms. Functional Programming: The concepts of Set Theory needed for the understanding of a functional language. The programming language Haskell. Algebraic Specification: Signatures; Equational Specification and Spec-algebras; Initial Semantics and Quotient Term Algebras. The programming language CafeOBJ.

Stage at a Secondary School

Statistics
I - Descriptive and mathematical statistics - an introduction. II - Descriptive statistics: univariated and bivariated frequency distributions. III - Introduction to statistical inference. Sample statistics and sampling distributions. Law of large numbers. Central limit theorem. IV - Parametric estimation. Estimators, maximum likelihood method, confidence intervals. V - Tests of hypotheses. Neyman-Pearson tests. Chi-square tests. Fundamental references: A.M. Mood, Graybill, Franklin and Boes, Duane C., An introduction to the theory of Statistics, 3rd edition, Mc Graw-Hill, 1974. Rohatgi, V. K., An introduction to probability theory and mathematical statistics, Wiley, 1976.

Stochastic Forecasting
I. Additional topics on random vectors I.1 Conditional distributions I.2 Regression of the first and second kind II. Linear regression models II.1 Introduction II.2 Estimation of the regression parameters II.3 Statistical inference about the parameters: confidence intervals and tests II.4 Prediction of future responses II.5 Assessing the model: index of fit and analysis of residuals III. Time series models III.1 Introduction III.2 Traditional methods for modelling trend and seasonal components III.3 Second order stationary processes III.3.1. The autocovariance and partial autocorrelation functions: properties and sample estimates III.3.2 The spectral density function and the peridiogram III.3.3 Shift operators III.4 Autoregressive and Moving Average models (ARMA) III.4.1 Stationarity and invertibility III.4.2 Summary of time relationships III.4.3 Preliminary estimation III.4.4 Model-building and forecasting: the Box-Jenkins approach

Stochastic Processes and Queues
I - General stochastic models. II - Stochastic processes: definition; Kolmogorov's theorem. Classical types of stochastic processes: Gaussian process; second order processes; stationary processes; processes with stationary and independent increments; Brownian motion process. III - Discrete time Markov chains: introduction and main definitions. Classification of states of a Markov chain: recurrence, transience and periodicity. Random walks. Branching processes. IV - Continuous time Markov chains. Poisson and pure birth processes. Birth and death processes. V -An introduction to queueing theory. M/M/1 model. Queues with parallel channels. Non-Poissonian models - an introduction. M/D/1 model. Fundamental references: Karlin, S. and H. Taylor, A first course in stochastic processes, Academic Press, 1975. Gross, D. and C. Harris, Fundamentals of queueing theory, Wiley, 1985.

Topology
The course is a study of the basic properties of three types of spaces: metric, topological and uniform spaces. We follow "General Topology" by Stephen Willard, Addison - Wesley Publishing Company, 1970 and "Topology and uniformities" by I. M. James, Springer, 1999.