## Research Themes

### Algebraic generalized invertibility

With this project, we aim to study algebraically the notion of generalized invertibility, and explore the similarities and differences in various algebraic settings: matrices over fields, rings, semigroups, operator spaces or categories. The class of generalized inverses is vast, and includes the "classic" such as Moore-Penrose, Drazin, von Neumann, and the "new" generation such as core and (b,c) inverses. These can also be characterized by the inverse along an element, or the Mary inverse, for a particular choice of element. Generalized inverses appear in numerous areas of applied mathematics, namely linear estimation, differential and difference equations, Markov chains, graphics, cryptography, coding theory, incomplete data recovery and robotics, some of which may be addressed.

Supervisors and contacts: Pedro Patricio (UM|CMAT), Xavier Mary (Laboratoire MODAL’X|Université Paris Nanterre, France)

### Algebraic topology and concurrency

Methods from algebraic topology have been used successfully in concurrency theory. The purpose of this project is to develop new topological techniques for the analysis of higher-dimensional automata and/or other topological models of concurrent systems.

Supervisors and contacts: Thomas Kahl (UM|CMAT)

### Coinductive proof search

Proof search is a paradigmatic computational process in logic with applications in automated reasoning and logic programming. CMAT researchers and collaborators are developing an original and fruitful approach to the area based on typed lambda-calculi.

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

### Contributions on multivariate data analysis and forecasting

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|UAveiro)

### 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|UAveiro)

### Contributions to species distribution modelling of spatial big data** **

The ongoing climate changes and the need to improve knowledge about their impact on biodiversity call for the analysis of spatially extensive data on species communities to understand and forecast distributional changes of the underlying stochastic processes. The research program aims to investigate the problem of dealing with numerous species efficiently, while explicitly accounting for spatial structure in the data. The current applications are generally limited to relatively small spatial datasets due to computational burden problems that arise as the number of spatial locations increases. It becomes urgent to consider statistical methods able to overcome the usual scability constraints associated to joint species modelling over large regions. The theoretical methods to be investigated will be strongly motivated by fishery and biological applications.

Supervisors and contacts: Raquel Menezes (UM|CMAT) and Isabel Natário (UNL|CMA-FCT)

### 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-calculus does correspond to the sequent calculus?

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

### Differential Equations: theory, modelling and applications

Ordinary and Partial Differential Equations (ODEs and PDEs) are of great relevance when one pretends to describe natural or physical phenomena using a universal and accurate language such as mathematics.

Within this range of mathematical theory, modelling and applications, several problems can be addressed in a Ph.D. project, according to the student’s profile and interests.

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

eurica@utad.pt; ajsoares@math.uminho.pt

### Dynamics of Holder maps:

Smooth dynamical systems are nowadays a quite well establish theory whereof its stability and genericity are very well studied. We intend to pursued towards weaker topologies namely what kind of results still prevails under Holder regularity?

Supervisors and contacts: Davide Azevedo (UM|CMAT) and Mário Bessa (UP|CMUP)

### 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

### Focused proof systems for polarized logics

Focused proof systems originated with linear logic but are now known to be applicable to intuitionistic and classical logics or their polarized variants. Focused proof systems have mostly been developed in the perspective of proof search, while the perspective of proofs-as-programs is comparatively under-developed.

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

jes@math.uminho.pt

### Kinetic Models and Applications

Kinetic models can be used for the description of several interdisciplinary real-world problems arising, for example, in Physics, Biology, Engineering and Economy. Many challenging mathematical problems can be studied for these models, including for example existence, uniqueness and properties of solutions, hydrodynamic limits and long-time asymptotics, as well as application-oriented problems. Under this project, different topics can be proposed for a Ph.D. 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

### Machine Learning Methods for Longitudinal Data Prediction

In longitudinal studies it is of great interest to make predictions for individual longitudinal trajectories. I this project we intend to compare the traditional likelihood methods and machine learning methods when using them for longitudinal data prediction. It is our hypothesis to observe better performance of machine learning methods with the increase of size in the data.

Supervisors and contacts: Inês Sousa (UM|CMAT)

### 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

### Nonlinear Partial Differential Equations

Nonlinear partial differential equations (pdes) model several physical and biological phenomena, being for that of great interest and relevance, and give rise to quite interesting and defiant questions. Many is done but there are still several questions to be answered (namely concerning anisotropic regimes and doubly nonlinear pdes).

This proposal of a PhD thesis comprehends a review of the literature (known results) and aims to point out open problems and to give a contribution to solve them; possibly in co-supervision with national or foreign collaborators.

Supervisors and contacts: Eurica Henriques (UTAD|CMAT)

### 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

### Pseudo-Riemannian geometry on homogenous manifolds

Lie groups, symmetric spaces or more generally homogenous manifolds are good model spaces for many problems arising in pseudo-Riemannian geometry. Under this topic, different projects can be proposed for a PhD thesis, according to the interest of the student, possibly in co-superivsion with a foreign collaborator.

Supervisors and contacts: Ana Cristina Ferreira (UM|CMAT)

### Qualitative Analysis of Bio-inspired Models

Dynamical systems theory is widely used to describe phenomena in nature that change over time. To seek a better understanding of these dynamic, bio-inspired models, the project will explore new techniques and mechanisms to find tipping points, strange attractors, chaos stabilization, and bifurcations.

Supervisors and contacts: Flora Ferreira (UM|CMAT), Wolfram Erlhagen (UM|CMAT)

wolfram.erlhagen@math.uminho.pt

### Quantitative type systems

Type system with non-idempotent intersection types are able to give logical characterizations of dynamic aspects of the normalization process in proof systems. The recent extension of this methodology to natural deduction with general elimination rules is a sign of its wide applicability.

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

jes@math.uminho.pt

### Quaternionic polynomials: algorithms and applications

The aim of this project is to study polynomials defined over the 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

### Recent advances in the statistical analysis of failure time data

This project aims to develop new approaches for the analysis of complex event history data. The methods will be used to model real data from medical studies that may involve multivariate failure time data, censure and/or truncation.

Supervisors and contacts: Luís Meira-Machado (UM|CMAT) and Carla Moreira (UM|CMAT)

### Risk based modelling and applications

The aim of this project is to develop risk measures, to analyse their mathematical properties and to establish corresponding risk models and risk based classification methods. Applications to the modelling of choice problems and to portfolio optimization problems will be considered.

Supervisors and contacts: Irene Brito (UM|CMAT)

### Statistical learning for spatial and temporal streaming data

Streams of data are all around us, and the need to process that data instantly —or as close to real time as possible— has become evident. 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|UAveiro)

### Topological complexity and related invariants

The project aims to study the notion of topological complexity and some of its approximations for certain classes of topological spaces using techniques from algebraic topology.

Supervisors and contacts: Lucile Vandembroucq (UM|CMAT)

### 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)