from 1st of June to 28th of August, 2016
6th of September, 2016
Beginning of the 1st semester – 26th of September, 2016
Results of the 1st semester – 28th of February, 2016
Beginning of the 2nd semester – 13th of February, 2017
Results of the 2nd semester – 29th of July, 2017
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.
TeachersAntónio 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.
TeachersAssis 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.
TeachersDomennico 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.
TeachersCecília Azevedo (UM), Isabel Pereira (UA), Margarida Brito (UP), M. Emilia Athayde (UM)
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:
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.
TeachersCorália Vicente (UP), Delfim Torres (UA), Fernando Lobo Pereira (UP), Lisa Santos (UM), Sílvio Gama (UP), Sofia Castro (UP)
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.
TeachersCláudia Mendes Araújo (UM), Lisa Santos (UM)
Specialised seminars will be organised during the academic year.
“An overview of the Deep Learning Method” , Stéphane Clain, October 11, 14h30 – 15h15, UM
“Mathematical models for infectious diseases and optimal control”, Cristiana Silva, October 18, 14h30 – 15h15, UA
“Longitudinal data analysis in biostatistics”, Inês Sousa, October 25, 14h30 – 15h15, UM
“Models for time series of counts”,Eduarda Silva, November 8, 14h30 – 15h15, UP
“Research challenges in stochastic frontier analysis with maximum entropy estimation“, Pedro Macedo, November 9 14h30 – 15h15, UA
“Applicable generalized inverses of matrices“, Pedro Patrício, November 15, 14h00 – 14h45, UP
“Learning from data streams“, João Gama, November 15, 15h00 – 15h45, UP
“Abstract regular polytopes“, Elisa Fernandes, November 22, 14h30 – 15h15, UA
“An ODE model of immune response by T cells“, Bruno Oliveira, November 29, 14h30 – 15h15, UP
“Dynamic neural fields: theory and applications“, Wolfram Erlhagen, November 30, 14h30 – 15h15, UM
“Mathematical problems of general relativity“, Filipe Mena, December 6, 14h30 – 15h15, UM
“A graph of all numerical semigroups“, Manuel Delgado, December 9, 14h30 – 15h15, UP
“Linear statistical models: an overview“, A. Manuela Gonçalves and Susana Faria, December 13, 14h00 – 14h45, UM
“Representations of fundamental groups of surfaces“, Peter Gothen, December 13, 15h00 – 15h45, UP
“Multi-vehicle identification and tracking of oceanic lagrangian coherent structures“, João Tasso, December 20 14h00 – 14h45, UP
“An optimal control framework for sustainable resources management in agriculture“, Fernando Lobo Pereira, December 20, 14h00 – 14h45, UP
“Research themes in Logic and Computation“, José C. Espírito Santo, January 10, 14h00 – 14h45, UM
“On co-algebraic dualities“, Dirk Hofmann, January 10, 15h00 – 15h45, UA