STUDENTS WORKSHOP, 29 & 30 JUNE 2017
UNIVERSIDADE DO MINHO
CAMPUS DE GUALTAR, BRAGA
REGISTRATION UNTIL JUNE 22ND
J.C. Espirito Santo [UM] – 14:30 | 15:20
- A logical toolkit
- Abstract: This talk is about applications of a certain correspondence between proof theory,type theory, and category theory. On the one hand, the correspondence allows the transfer of methods from one theory to another – one example will be given, viz., the solution of a certain coherence problem in category theory by proof-theoretical methods. On the other hand, the correspondence equips the working computer scientist with a versatile toolkit – applications to Facebook engineering or quantum computing taken from the literature will be reported. The point is to illustrate the “unusual effectiveness of logic in computer science”.
Rita Guerra [UA] – 15:25 | 15:45
- Properties of an oscillatory integral operator with different trigonometric phases
Abstract: The main aim of this work is to obtain Heisenberg uncertainty principles for a specific oscillatory integral operator with different trigonometric phases. Additionally, invertibility theorems, Parseval type identities, Plancherel type theorems and convolutions associated with that operator are also obtained.
Manuel Baptista [CST AG] – 15:50 | 16:10
- Mathematics in EM Simulations
Over the years, simulations have become more and more important in the industrial context. Improvements in geometric representation, in the algorithms simulating physical processes, in results post-processing, analysis and visualisation, together with the huge development and wide availability of computational power, make simulations a valuable technique for the design and testing of products and services. One of the most cited examples is the simulation of car crashes which save huge amounts of money that physical prototypes would cost. In Electromagnetics there are numerous examples across many industries: electrical machines, particle accelerators, medical devices, home electronics, to mention a few. Antenna placement, optimisation of electronic layouts, electromagnetic interference/compatibility are among the problems to solve during the conception and development of new products containing electronics. In my talk I will try and describe the general aspects required to set up and run an EM (Electromagnetic) Simulation and what branches of Mathematics come into play, trying to raise awareness for the differences, similarities and possible synergies between academy and industry.
coffee break – 16:15 | 16:45
Weronica Wojtak [UM] – 16:45 | 17:10
- A novel dynamic field model supporting a continuum of bump amplitudes
Abstract: Dynamic Neural Fields (DNFs) formalized by nonlinear integro-differential equations have been introduced as models of the large-scale dynamics of spatially structured networks of neurons, in which the associated integral kernel represents the spatial distribution of synaptic connections between neurons. Neural fields have been used in the past to explain a wide range of neurobiological phenomena, including persistent population activity in higher brain areas associated with cognitive functions. For instance, the memory of sensory events is commonly believed to be stored in persistent activity patterns (“bumps”) that are initially triggered by brief inputs, and subsequently become self-sustained by recurrent interactions within the neural population. In classical neural field models, the fixed amplitude of the bump attractor is solely defined by the strength of the recurrent interactions. In this talk, I will present a novel field model consisting of two reciprocally coupled equations of Amari-type which supports a continuum of bump amplitudes. The model has been applied to explain persistent population activity that increases monotonically with the time integral of the input (parametric working memory). It robustly memorizes the strength and/or duration of inputs,and allows for changing the memory strength at any time by new relevant information. I will show how numerical simulations and analytical tools can be applied to better understand the pattern formation process.
Pedro Soares [UP] – 17:10 | 17:30
- Coupled cell networks and systems
Abstract:A coupled cell network is a directed graph where we label cells (vertices) and inputs (edges) with types. To each cell, we associate a dynamical system depending on the cell’s type.Moreover, the dynamical systems are coupled according the connections between the cells in the graph and their types. The resulting dynamical system respects the network structure and it is called a coupled cell system associated to the network. We explore some examples and results about networks and coupled cell systems. One striking example is the relation between balanced colorings in the network and the existence of flow invariant subspaces, given by the equality of some cell coordinates, for any coupled cell system. We also see how coupled cell systems can support non-generic phenomena, like heteroclinic networks and bifurcations of high degeneracy.
Delfim Torres [UA] – 17:35 | 18:25
- Optimal control of a delayed HIV model: A personal experience from PDMA Research Lab
Abstract: We propose a model for the human immunodeficiency virus type 1 (HIV-1) infection with intracellular delay and prove the local asymptotical stability of the equilibrium points. Then we introduce a control function, representing the efficiency of reverse transcriptase inhibitors, and consider the pharmacological delay associated with the control. Finally, we propose and analyze an optimal control problem with state and control delays. Through numerical simulations, extremal solutions are proposed for minimization of the virus concentration and treatment costs.
Sofia Castro [UP] – 10:00 | 10:50
- Problems in economics and mathematical models
Abstract: Several problems in economics are understood by constructing models that high-light all (or, some of) the relevant questions addressed. Often these models are mathematical in nature.
This presentation will report on some models used in different branches of economics and will describe some of the mathematics that proved useful in understanding them. Some of the mathematics is new and only came into being because of the economics problem. Some mathematics is standard but proved useful when applied to the particular problem.
The economics problems include topics such as industry location, oligopoly theory, growth and price setting in general equilibrium. The mathematics ranges from finding zeros of polynomials to bifurcation and singularity theory in dynamical systems, as well as game theory.
Filipe Martins [UP] – 10:55 | 11:15
The fundamental bifurcation for evolutionary matrix models with multiple traits
Abstract: One fundamental question in biology is population extinction and persistence, i.e., stability/instability of the extinction equilibrium and of non-extinction equilibria. In the case of nonlinear matrix models for structured populations, a bifurcation theorem answers this question when the projectionmatrix is primitive by showing the existence of a continuum of positive equilibria that bifurcates from the extinction equilibrium as the inherent population growth rate passes through 1. This theorem also characterizes the stability properties of the bifurcating equilibria by relating them to the direction of bifurcation, which is forward (backward) if, near the bifurcation point, the positive equilibria exist for inherent growth rates greater (less) than 1. In this paper we consider an evolutionary game theoretic version of a general nonlinear matrix model that includes the dynamics of a vector of mean phenotypic traits subject to natural selection.We extend the fundamental bifurcation theorem to this evolutionary model. We apply the results to an evolutionary version of a Ricker model with an added Allee component. This application illustrates the theoretical results and, in addition, several other interesting dynamic phenomena, such as backward bifurcation induced strong Allee effects.
coffee break – 11:20 | 11:50
Benjamim Anwasia [UM] – 11:50 | 12:10
- Reaction Diffusion Asymptotics of the Simple Reacting Sphere Kinetic Model
Abstract: We consider a system of Boltzmann-type equations that describe a multi-species gaseous mixture participating in a bimolecular reversible chemical reaction under isothermal condition and perform an asymptotic analysis of a physical situation in which the dominant role in the evolution of the species is played by elastic collisions, while chemical reactions are slow, to obtain a coupled system of equation referred to as the reaction diffusion system of Maxwell-Stefan type, as the hydrodynamic limit of the simple reacting sphere kinetic model.
Liliana Silva [UP] – 12:15 | 12:30
- Stability of quasi-simple heteroclinic cycles
Abstract: The stability of heteroclinic cycles may be obtained from the value of the local stability index along each connection of the cycle. We establish a way of calculating the local stability index for quasi-simple cycles : cycles whose connections are one dimensional and contained in ow-invariant spaces of equal dimension. These heteroclinic cycles exist both in symmetric and non-symmetric contexts. We make one assumption on the dynamics along the connections to ensure that the transition matrices have a convenient form. Our method applies to all simple heteroclinic cycles of type Z and to various heteroclinic cycles arising in population dynamics, namely nonsimple heteroclinic cycles, as well as to cycles that are part of a heteroclinic network. We illustrate our results with a quasi-simple (non-simple) cycle present in a heteroclinic network of the Rock-Scissors-Paper game.
lunch – 12:40 | 14:30
António Machiavelo [UP] – 14:30 | 15:20
- Secret Codes and Mathematics
Abstract: Modern “secret codesî, or more precisely cryptosystems, have an increasingly mathematical nature. We will explain why this is so, while attempting to give an overview of some relevant parts of modern cryptology (= cryptography + cryptanalysis). All the mathematical concepts needed to understand this talk will be explained.
Pedro Delgado [BOSCH] – 15:250 | 15:45
Nuno Paiva [NOS] – 15:50 | 16:10
coffee break – 16:15 | 16:45
Round Table – 16:35 | 18:15
- Fernando Lobo Pereira [UP]
- Luis Tiago Paiva [SYSTEC/FEUP]
- Manuel Baptista [CST AG]
- Nuno Paiva [NOS]
- Paulo Pinto [Critical Software]
- Pedro Delgado [BOSCH]
- Tiago Santos [SMARTWATT]
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