Works presented

Summary
The main goal of this work is to explore and analyze several methods to solve the diophantine equations that appear in mathematical olympiads. It tries to understand how these equations, that require integer solutions, are used in competition problems and the way in which the theoretical concepts translate into technics useful to solve them. In this way, the work is divided in three chapters. The first one, about the history of these problems. The second one, about different types of diophantine equations and their resolution. And finaly, a sellection of problems that can be found in local, national and international mathematical olympiads.

Direction
GAGO COUSO, FELIPE (Tutorships)
RIVERO SALGADO, OSCAR (Co-tutorships)

Court
RODRIGUEZ CASAL, ALBERTO (Chairman)
ALONSO TARRIO, LEOVIGILDO (Secretary)
SALGADO SECO, MODESTO RAMON (Member)

Summary
In this project we will consider some of the fundamental aspects of the research papers that conform the beginning of the research on elliptic partial differential equations with overdetermined boundary conditions, that is where both Dirichlet and Neumann boundary conditions are present. In the first place we will analyze Serrin's Theorem, along with its proof, which is based on finding symmetries using the moving plane method along with the maximum principles. We will also study Weinberger's alternative and more concise proof in which more classical analytical methods are applied. Lastly we will also propose a series of examples in the field of physics in which elliptic partial differential equations where overdetermined boundary conditions appear with the goal of to show the usefulness and importance of the research done in this area.

Direction
DOMINGUEZ VAZQUEZ, MIGUEL (Tutorships)

Court
RODRIGUEZ CASAL, ALBERTO (Chairman)
ALONSO TARRIO, LEOVIGILDO (Secretary)
SALGADO SECO, MODESTO RAMON (Member)

Summary
Game theory is a mathematical discipline that studies decision problems that involve various agents. We differenciate between cooperative and non cooperative games, which are distinguished by the existence or not of mechanisms for establish binding agreements. Two basic concepts are the Shapley value and the subgame perfect equilibrium, taken from cooperative games with transferable utility and games in extensive form. In this work we will use the tools mentioned above to understand and explain a recent investigation about fish aggregating devices. This leads to a possible increase of fishing firms profits, besides a beneficial contribution for the enviroment in terms of fuel reduction and CO2 emissions. Along with theoretical considerations, the aim is also to show an empirical analysis of this problem.

Direction
CASAS MENDEZ, BALBINA VIRGINIA (Tutorships)

Court
RODRIGUEZ CASAL, ALBERTO (Chairman)
ALONSO TARRIO, LEOVIGILDO (Secretary)
SALGADO SECO, MODESTO RAMON (Member)

Summary
In the last decades, several topological tools for data analysis in different areas have been developed. The present work aims to explain simplicial homology and persistent homology, and their application in biology as a method to study and predict the viral evolution, not very well known nor controlled. Specifically, we will focus on the flu virus (Influenza A) and the Human Immunodeficiency Virus (HIV), both for their prevalence and mortality rate in humans, as well as the disposition of its data and the suitability of the explained topological methods for its study.

Direction
Gómez Tato, Antonio M. (Tutorships)

Court
RODRIGUEZ CASAL, ALBERTO (Chairman)
ALONSO TARRIO, LEOVIGILDO (Secretary)
SALGADO SECO, MODESTO RAMON (Member)

Summary
During the history of computing, routing problems have attracted great interest due to their multiple applications in different fields, such as planning and logistics. This study focuses on the traveling salesman problem or TSP. Specifically, on the techniques to solve it in an approximate way in polynomial time, the metaheuristics. The main objective of this study is to provide a guide to understand four of the most important ones, both theoretically and computationally. For this purpose, a literature review was performed, finding relevant information of them and synthesizing it. They are: tabu search, simulated annealing, genetic algorithm and ant colony optimization. For the computational part, R implementations of all metaheuristics were performed and evaluated with different instances of the TSPLIB library. As a result, it was obtained that there is no metaheuristic better than the rest in all aspects. Tabu search and ant colony optimization obtain very promising results in terms of distance to optimal cost, however, they are temporarily more expensive than the other two. Simulated annealing obtains somewhat worse results than the previous ones, but in a very fast way. Finally, the genetic algorithm obtains very bad results in a relatively acceptable time. In conclusion, this work serves as a guide to people who want to understand these concepts.

Direction
CASAS MENDEZ, BALBINA VIRGINIA (Tutorships)

Court
RODRIGUEZ CASAL, ALBERTO (Chairman)
ALONSO TARRIO, LEOVIGILDO (Secretary)
SALGADO SECO, MODESTO RAMON (Member)

Summary
The objective of this work is to present, in a clear and simple way, the evolution of mathematics with a focus on logic from its beginnings in Ancient Greece to the 19th and 20th centuries. The necessary basic concepts will be defined and the development of Set Theory, which encompasses fundamental concepts such as the Axiom of Choice, the Continuum Hypothesis and the Zermelo-Fraenkel Axiomatization, will be studied. In addition, the relationship between these and their impact on modern mathematics will be analyzed.

Direction
ALONSO TARRIO, LEOVIGILDO (Tutorships)

Court
ALONSO TARRIO, LEOVIGILDO (Student’s tutor)

Summary
Ordinary differential equations (ODEs) are a fundamental tool for modeling dynamic processes in various disciplines. This work focuses on the application of ODEs to biological models, particularly those related to the spread of infectious diseases. The study provides a detailed analysis of the SIR model and its extensions, such as the SEIR and SIRS models, aiming to understand epidemiological dynamics and the stability of equilibrium states. Furthermore, a specific case study on the spread of HIV in Cuba is presented using a nonlinear extension of the SIR model. The analysis includes both analytical and numerical solutions, as well as an evaluation of the impact of different control and eradication strategies. The results highlight the importance of the basic reproduction number R0 and interventions such as vaccination in mitigating disease transmission.

Direction
Rodríguez López, Jorge (Tutorships)

Court
Rodríguez López, Jorge (Student’s tutor)

Summary
It is well known that there is no general rule for solving first-order ordinary differential equations(ODEs), but rather a variety of methods, many of which can be expressed in the language of integrating factors. Unfortunately, there is no technique that allows for the explicit determination of integrating factors for an arbitrary differential equation. However, the Norwegian mathematician Sophus Lie (1842-1899) developed, based on the symmetries of differential equations, a unified procedure for their determination. The aim of this work is to study symmetries and integrating factors as a method of solving first-order ordinary differential equations.

Direction
BUEDO FERNANDEZ, SEBASTIAN (Tutorships)
SANMARTIN LOPEZ, VICTOR (Co-tutorships)

Court
BUEDO FERNANDEZ, SEBASTIAN (Student’s tutor)
SANMARTIN LOPEZ, VICTOR (Student’s tutor)

Summary
Within the field of game theory, weighted majority games play a fundamental role in the analysis of voting processes in parliaments and committees. This work introduces this class of games, focusing on the study of power indices, a solution concept that assigns a measure of influence or power to the players involved in the voting process. Among the power indices available in the literature, we will consider five: Shapley-Shubik, Banzhaf, Johnston, Colomer-Martínez, and Johnston-Colomer-Martínez. Their mathematical properties will be examined, practical applications will be provided, and their computational cost will be assessed. To facilitate the computation of these five indices, we will develop methods based on generating functions, which are combinatorial tools that allow us to derive, through polynomials, the necessary components for their calculation. Furthermore, we will model a new scenario in which players can form alliances, leading to what are known as games with coalition structure. For these games, we will introduce two additional power indices: Owen and Banzhaf-Owen, along with computation methods based on generating functions. Finally, these concepts will be applied to a practical case: the analysis of the Spanish Parliament. We will examine changes in the distribution of power among political parties between the general elections held in November 2019 and July 2023, as well as the consequences of members of parliament switching between parliamentary groups during the XV Legislature. The \textit{powerindexR} library within the statistical software R will be used to compute the power indices in these scenarios.

Direction
SAAVEDRA NIEVES, ALEJANDRO (Tutorships)
DAVILA PENA, LAURA (Co-tutorships)

Court
CABADA FERNANDEZ, ALBERTO (Chairman)
BORRAJO GARCIA, MARIA ISABEL (Secretary)
MUÑOZ SOLA, RAFAEL (Member)

Summary
The study of the qualitative behaviour of differential equations seeks to obtain properties of the solutions without the need to know them explicitly. This approach acquires special relevance when parameters are incorporated into the equation, since small variations in them can lead to very significant changes, having effects on the number of singular points, their stability or the appearance of oscillatory solutions. This is the idea behind the theory of bifurcations, which will be explored in depth by means of the most typical examples in one and two dimensions: the tangent, transcritical, pitchfork and Hopf bifurcations. For each of them, the qualitative behaviour of a type equation will be explored, followed by a generic study in which the conditions that characterise it will be obtained.

Direction
BUEDO FERNANDEZ, SEBASTIAN (Tutorships)
LOIS PRADOS, CRISTINA (Co-tutorships)

Court
CABADA FERNANDEZ, ALBERTO (Chairman)
BORRAJO GARCIA, MARIA ISABEL (Secretary)
MUÑOZ SOLA, RAFAEL (Member)

Summary
This work provides an introduction to geostatistics, focusing particularly on the concept of the variogram, a structure that quantifies spatial dependence, and the Kriging spatial interpolation method. To this end, the theoretical foundations of spatial dependence are presented as the basis for the development of the variogram, including both its experimental and theoretical conception, as well as the different existing models and the reasons why it may fail to properly model spatial dependence. Next, the theory behind the Kriging interpolation method is introduced, along with its different variants: ordinary, universal, and multivariate Kriging. Finally, a practical case is presented to illustrate the usefulness of these concepts, aiming to model the interpolation of the pollutants SO2, PM10 and NOx in the Galician territory using the R libraries gstat and sm

Direction
FEBRERO BANDE, MANUEL (Tutorships)

Court
CABADA FERNANDEZ, ALBERTO (Chairman)
BORRAJO GARCIA, MARIA ISABEL (Secretary)
MUÑOZ SOLA, RAFAEL (Member)

Summary
Over the course of this dissertation, we present and explore the use of Kriging models in fitting a solution to the geostatistical problem of estimating the total size of the Greenland ice sheet, both in volume and extention. In addition, we evaluate the prediction performance of these and other models, comparing their precision in relation to their respective complexity.

Direction
FEBRERO BANDE, MANUEL (Tutorships)

Court
CABADA FERNANDEZ, ALBERTO (Chairman)
BORRAJO GARCIA, MARIA ISABEL (Secretary)
MUÑOZ SOLA, RAFAEL (Member)

Summary
This Bachelor's Thesis focuses on the analysis of certain ordinary differential equations applied to the study of economic models. Throughout this work, five key models are addressed: the Phillips curve, the Harrod-Domar model, the Solow-Swan model, the Goodwin model, and the dynamic Leontief model, which allow for the description of fundamental economic phenomena, from the relationship between unemployment and wages to the interaction among productive sectors. Each model has been contextualized, solved, and analyzed in detail, highlighting both its contributions and its limitations, with the aim of better understanding its usefulness and exploring possible improvements for its application in modern economies.

Direction
Rodríguez López, Rosana (Tutorships)

Court
Rodríguez López, Rosana (Student’s tutor)

Summary
Quantum computing is a field of computer science that uses principles of quantum physics to solve problems more efficiently than classical computing, especially in areas such as optimization. Bioinformatics, on the other hand, is a field that combines elements of biology and computer science to analyze large biological data sets. A prominent example of this discipline is genomics, which includes the generation of phylogenetic trees, key tools for understanding the biological evolution of species. The reconstruction of these trees represents a computational problem that is very difficult to solve due to its complexity. This work explores whether quantum computing can offer effective solutions to address this problem. In this context, the performance of quantum computation and quantum optimization algorithms has been studied, with emphasis on Quantum Annealing and the Quantum Approximate Optimization Algorithm (QAOA). Based on these approaches, a quantum algorithm capable of reconstructing phylogenies by cutting graphs has been developed. The proposed algorithm was implemented and tested on currently available quantum hardware, obtaining satisfactory results that demonstrate its potential to solve complex problems in the area of bioinformatics.

Direction
Fernández Pena, Anselmo Tomás (Tutorships)
PICHEL CAMPOS, JUAN CARLOS (Co-tutorships)

Court
ARIAS RODRIGUEZ, JUAN ENRIQUE (Chairman)
Querentes Hermida, Raquel Esther (Secretary)
PIÑEIRO POMAR, CESAR ALFREDO (Member)

Summary
Nowadays there are multiple tools to perform image classification processes, such as convolutional neural networks and transformers. However, the Zara brand continues to perform labeling manually, resulting in a set of inaccurate labels. For this reason, this study explores the implementation of automated methods to improve the results obtained manually. The purpose of this research is to evaluate and analyze the effectiveness of the AWS Rekognition Custom Labels service for labeling garments. The adopted strategy aims to identify the limits of the service for the referred task through a feasibility analysis of the source dataset. The project development starts with a preliminary analysis of the dataset to determine its suitability for model training. Subsequently, an examination of the service constraints is performed, considering five main variables: the total number of images, the interrelationship between labels, the type of label, the number of images available for each label, and the influence of each label on the others. To achieve this, several resources will be used such as the service itself, an initial dataset and a REST API developed for this project. The main findings include the low relevance of the total number of images, as well as the limitations associated with the type of tag and the importance of the tags not being overly related.

Direction
CARREIRA NOUCHE, MARIA JOSE (Tutorships)
Rodríguez Díez, Helio (Co-tutorships)

Court
ARIAS RODRIGUEZ, JUAN ENRIQUE (Chairman)
Querentes Hermida, Raquel Esther (Secretary)
PIÑEIRO POMAR, CESAR ALFREDO (Member)

Summary
In the area of remote sensing, there is great interest in collecting land cover information to identify and classify the different types of surfaces present on the ground, such as vegetated areas, water bodies, urban soils, grasslands, forests or agricultural areas, among others. On the other hand, semantic image segmentation allows assigning a label to each pixel of the image, classifying them into different categories or specific classes, which facilitates the interpretation and analysis of satellite or aerial images. The use of deep learning techniques has proven to be effective in the field of computer vision, specifically in semantic segmentation tasks. However, these models are very computationally expensive, and often require the use of specialised hardware and optimisation techniques to improve the efficiency and feasibility of training and inference. In this Bechelor Thesis, the aim is to test different models with encoder-decoder architecture, trying to improve the efficiency and feasibility of training even with large amounts of data. From the existing parallelism techniques for multiGPU training, data parallelism will be used, selecting a PyTorch module that implements it in an efficient way. In addition, using 16-bit mixed floating-point precision reduces memory usage and makes better use of the GPU hardware, performing training in half the time without affecting the quality of the segmentation.

Direction
Argüello Pedreira, Francisco Santiago (Tutorships)
Blanco Heras, Dora (Co-tutorships)

Court
ARIAS RODRIGUEZ, JUAN ENRIQUE (Chairman)
Querentes Hermida, Raquel Esther (Secretary)
PIÑEIRO POMAR, CESAR ALFREDO (Member)