# 1st Year

## Advanced Topics in Algebra, Logic and Computacion (ALC)

Three of the following topics are taught each year: Computer algebra: introduction to some computer algebra system; development of topics in computational number theory or in computational group theory. Automata, languages, and semigroups: regular languages; recognizability by finite state automata and by semigroups; (option 1) varieties of semigroups and languages, Eilenberg's theorem; (option 2) Chomsky's hierarchy, decidability problems. Algebraic logic: elements of universal algebra; algebraization of classical, intuitionistic and modal logics; abstract algebraic logic. Category theory: universal properties; constructions in categories; natural transformations and adjunctions; monads. Proof theory: lambda-calculus; intuitionistic logic and Curry-Howard correspondence; proof systems. Matrix theory: elementary divisors and invariant factors, minimal polynomial; canonical forms of a matrix; nonnegative matrix, irreducibility and primitivity.

### ECTS

6### Hours

56### Teachers

António Machiavelo (UP), Luis Pinto (UM), Manuel Delgado (UP), Yulin Zhang (UM)## Advanced Topics in Analysis and Optimization (AO)

Vector spaces: normed linear spaces; Banach spaces; separability; Lp-spaces. Hahn-Banach Theorem: Open Mapping Theorem; dual spaces; reflexivity; weak and weak-* topologies. Hilbert spaces: the Projection Theorem; Stampacchia e Lax-Milgram Theorems; Riesz Representation Theorem. Application of the Hahn-Banach theorem to minimum norm problems. Optimization of functionals: Gateaux and Frechet derivatives; Euler-Lagrange equations; problems with constraints; convex-concave functionals; conjugate functionals; dual optimization problems. Global constrained optimization: Lagrange multipliers; sufficiency; sensitivity; duality. Local constrained optimization: Inverse function equality and inequality constraints. Application to optimal control: Pontryagin maximum principle.

### ECTS

6### Hours

56### Teachers

Assis Azevedo (UM), Fernando Lobo Pereira (UP)## Advanced Topics in Dynamics and Geometry (DG)

Elementary geometry of submanifolds of R^n: Parametrisations (or charts), tangent bundle, differentiable functions, submanifolds, transversality. Differential forms, de Rham cohomology. Basic concepts of dynamics in R^n (or in submanifolds of R^n): Differential equations, stability of equilibria and of periodic solutions, hyperbolicity, stable and unstable manifolds, Poincaré map. Structural stability and bifurcations. The same concepts for the dynamics of recurrence relations.

### ECTS

6### Hours

56### Teachers

Domennico Catalano (UA), Isabel Labouriau (UP)## Advanced Topics in Probability and Statistics (PE)

Measurable spaces. Sequences of events. Measurable functions. Measures. Random variables, probability measures, fundamental properties. Probability spaces, types of probability laws. Integration in probability spaces and expectation. Inequalities. Some probability distributions. Independence and conditioning. Characteristic functions. Modes of convergence of sequences of random variables. Laws of large numbers. Central limit theorems. Multivariate distributions, conditional laws. Conditional expectation. Statistical models. Decision theory: risk functions, decision rules, criteria. Exponential families. Sufficiency. Point estimation, comparison of estimators, asymptotic properties, methods of estimation with emphasis on likelihood based inference. Hypothesis tests and confidence sets.

### ECTS

6### Hours

56### Teachers

Cecília Azevedo (UM), Isabel Pereira (UA), Margarida Brito (UP), M. Emilia Athayde (UM)## Optional Courses

Information about the syllabi can be found here here.

During the academic year 2016/2017, with the authorization of the scientific committee and in accordance with the preferences of students, the optional courses that will be taught will be chosen, making the most effective use of resources, from the following list of courses:

### Teacher

## Research Project in Mathematics

It is intended that each student, with the help of his/her supervisor, study a recent topic/field of research in mathematics and its applications. This study leads to (i) the written Report (thesis proposal) and (ii) an oral presentation. The Report lays out the plan of the research, describing the state-of-the-art, the scientific foundations, the methodology to be used and the objectives that are expected to achieve.

### ECTS

9### Hours

90### Teachers

Corália Vicente (UP), Delfim Torres (UA), Fernando Lobo Pereira (UP), Lisa Santos (UM), Sílvio Gama (UP), Sofia Castro (UP)## Seminar

The students are expected to attend the regularly organised seminars given by the program's teaching staff and write a summary with discussion for part of these seminars. Each student is also expected to choose one of the research themes proposed on these seminars and prepare a talk on that subject, under the advice and guidance of a supervisor of his/her choice.