## Research Themes

### Algorithms and applications of quaternionic polynomials

The aim of this project is to study polynomials defined over the algebra of quaternions and related algebras, from a theoretical and computational point of view, as well as their applications.

Supervisors and contacts: M. Irene Falcão (UM|CMAT) and Fernando Miranda (UM|CMAT)

mif@math.uminho.pt ; fmiranda@math.uminho.pt

### Coinductive proof search

Proof search is a paradigmatic computational process in logic with many applications. CMAT researchers and collaborators are developing an original and fruitful approach to the area.

Supervisors and contacts: José Espírito Santo (UM|CMAT) and Luís Pinto (UM|CMAT)

### Contributions on multivariate data analysis and modelling

This project proposes new contributions to regression and time series modelling for multidimensional response variables. For that, different processes for dimension reduction -clustering methods and principal component analysis (PCA) - it will be proposed to accurately predict and forecast models. These new proposed methodologies will be evaluated using both synthetic and real data (multidimensional data and high-dimensional data).

Supervisors and contacts: A. Manuela Gonçalves (UM|CMAT) and Marco Costa (CIDMA-Univ. Aveiro)

### Contributions on temporal and spatial disaggregation methods of time series models

This project intends to develop new methodologies for temporal and spatial disaggregation of time series which are specified at the high frequency although the observations are only available in aggregate form. The new methodologies must incorporate temporal and spatial disaggregation as developments, for instance, of Litterman and Fernández models or the Chow-Lin models. These new methodologies will be assessed by simulations studies and applied to economic and environmental data.

Supervisors and contacts: A. Manuela Gonçalves (UM|CMAT) and Marco Costa (CIDMA-Univ. Aveiro)

### Curry-Howard isomorphism for the sequent calculus

A question in the areas of proof theory and lambda-calculus is: given that the theory of combinators corresponds to Hilbert systems, and the lambda-calculus corresponds to natural deduction, what version of the lambda-calcus does correspond to the sequent calculus?

Supervisors and contacts: José Espírito Santo (UM|CMAT) and Luís Pinto (UM|CMAT)

Doubly nonlinear partial differential equations

The doubly nonlinear equations are a natural bridge between p-Laplace equations and porous medium equations. Many papers are devoted to this argument. Nevertheless and especially for the singular case, not all the aspects are clear. This proposal of PhD thesis comprehends a review of the literature (all the known results) and aims to point out the open problems and to give a contribution to solve them.

Supervisors and contacts: Eurica Henriques (UTAD|CMAT) and Vincenzo Vespri (Univ. Firenze)

### Dynamic field theory and machine learning

Dynamic neural fields (DNF) formalized by differential equations provide a mathematical language to explain and model cognitive behaviors in biological and artificial agents. The recent boom of deep learning methods has shifted the attention to the role of “big data” in implementing artificial intelligence. The project will explore the complementary roles of DNF-based and data-driven approaches through mathematical analysis and applications in Cognitive Neuroscience and Robotics.

Supervisors and contacts: Wolfram Erlhagen (UM|CMAT)

wolfram.erlhagen@math.uminho.pt

### Kinetic Models and Applications

Kinetic models can be used for the description of several problems arising, for example, in Physics, Biology, Engineering and Economy. The mathematical analysis of the associated problems can give rise to many interesting problems with relevance in applications. Under this project, different problems can be proposed for a PhD student, according to his or her preferences, possibly in co-supervision with foreign collaborators.

Supervisors and contacts: Ana Jacinta Soares (UM|CMAT)

### Machine-assisted theorem proving

Case studies in developing the meta-theory of proof systems with the assistance of Coq.

Supervisors and contacts: José Espírito Santo (UM|CMAT) and Luís Pinto (UM|CMAT)

jes@math.uminho.pt

### Mathematical modelling and analysis of biological systems

There exist many biological systems in which mathematical models and analysis can give a valuable contribution. The aim of this project is to study differential systems modelling biological processes, using analysis techniques and numerical simulations to describe and predict the dynamical behaviour.

Supervisors and contacts: Ana Jacinta Soares (UM|CMAT) and Maria Joana Torres (UM|CMAT)

ajsoares@math.uminho.pt ; jtorres@math.uminho.pt

### Mathematical models for the study of autoimmune disease

In this project we intend to develop mathematical models that describe the microscopic interactions between cells involved in the processes leading up to autoimmunity. Another objective of this project is to include, in the models obtained, the effect of immunotherapy on the control of the disease.

Supervisors and contacts: M. Piedade Ramos (UM|CMAT) and Carolina Ribeiro (UM|CMAT)

mpr@math.uminho.pt ; cribeiro@math.uminho.pt

Mathematical problems in General Relativity

In this project we intend to investigate problems related to geometry and analysis of Einstein´s equations in the context of the Theory of General Relativity. These problems can involve, for example, black holes, cosmological models, exact solutions and gravitational waves.

Supervisors and contacts: M. Piedade Ramos (UM|CMAT) and F. Mena (CMAT and IST)

mpr@math.uminho.pt ; fmena@math.uminho.pt

### Optimization for Deep Learning

Neural networks (NN) are trained using a gradient method for finding the optimal weights (parameters) of the NN model that minimizes the loss function (model error). Although NN are a powerful model, in practice, it is hard to train properly and can be computationally expensive. Among the main reasons why these models are so unwieldy are: i) the dataset and the number of weights can both be very large in practice; ii) gradient method converges very slowly in optimizing NN with many weights; iii) NN can suffer from vanishing and exploding gradient problems. In this work, we propose to investigate and develop optimization methods to overcome these issues, that take into account the error backpropagation when training NN.

Supervisors and contacts: Fernanda Costa (UM|CMAT) and Luís L. Ferrás (UM|CMAT)

mfc@math.uminho.pt ; llima@math.uminho.pt

Statistical learning for spatial and temporal streaming data

This project addresses the problem of predicting spatio-temporal processes with temporal patterns that vary across spatial regions, when data is available as a stream, meaning that the dataset is augmented sequentially. The methods to be proposed must be evaluated using both synthetic and real data, hopefully demonstrating their ability to accurately predict data missing in spatial regions over time.

Supervisors and contacts: Raquel Menezes (UM|CMAT) and M. Eduarda Silva (FEP-UP & CIDMA-Univ. Aveiro)

###

Topology of higher-dimensional automata

Higher-dimensional automata, i.e., labeled precubical sets, are a very expressive topological model for concurrent systems. The purpose of this project is to establish new structural results on the topology and especially the homology of HDAs. In particular, it will be investigated which graded submodules of exterior algebras may arise as homology languages of HDAs.

Supervisors and contacts: Thomas Kahl (UM|CMAT)

### Utility clustering theory and applications

The aim is to introduce the utility function to construct a clustering of data. The main idea is to gather the events in function of their utilities and not with the traditional metrics as the euclidean or Manhattan distance. We expect to develop a generalization of the clustering process by introducing additional information for classification. Applications to health and economic areas will be tackled.

Supervisors and contacts: Irene Brito (UM|CMAT) and Stéphane Clain (CF-UM-UP)

### Variational and quasi-varional inequalities with zero and/or first order constraints

This project aims to study the existence of scalar or vector solutions of variational and quasi-variational inequalities in convex sets defined by (convex) constraints on the functions or on their partial derivatives.

Supervisors and contacts: Lisa Santos (UM|CMAT)