Basque Center for Applied Mathematics – BCAM has welcomed to the center a **new group of PhD students** during this 2022.

They are going to work mostly in **mathematics** and **informatics**, but in the different research groups at BCAM. Below you will find more information about the doctoral theses they will be working on:

*I received a degree in Mathematics and a master’s degree in Advanced Mathematics from the University of Málaga. Currently, I am a member of the Harmonic Analysis group at BCAM since October 2021, as Research Technician at first and as PhD Student since September 2022.*

*I find the research project very interesting and I hope to learn a lot during these years.*

**Fractional Poincaré-Sobolev inequalities**

Poincaré and Poincaré-Sobolev inequalities are key tools in the study of the regularity of elliptic PDEs. Real Analysis and Harmonic Analysis provide new methods and more precise tools that improve these results avoiding the potential theory. The main goal is to understand degenerate fractional Poincaré and Poincaré-Sobolev inequalities, especially those with an extra gain obtained by Bourgain-Brezis-Mironescu and Maz’ya-Shaposhnikova.

*I obtained my Bachelor’s Degree in Mathematics at the University of the Basque Country in 2019 with an extraordinary award. After that I studied a Master’s Degree in Mathematics at the University of Bonn until 2021. Since then I worked at BCAM as a research technician in preparation for my PhD which I am about to start.*

*During my years at BCAM I hope I will get to learn a lot of Mathematics and get the opportunity to contribute to new and exciting theories.*

**Singularities in Agebraic Geometry and its interactions with Topology, Metric Geometry and Siplectic Geometry**

Currently there is an intense research activity in the metric and Lipschitz study of singularities and degenerations, in particular with the introduction of a metric homology theory a few years ago, named Moderately Discontinuous Homology. The main goal of the thesis is to further study this homology theory and the Algebraic Topology around it, in order to formulate a theory in the non-arquimedean setting and study its properties and consequences. This will involve a substantial learning in non-arquimedean metric geometry and a deep knowledge of Moderately Discontinous Homology. In te way we will likely find interesting intermediate problems which we will feel free to focus on.

*PhD Student at BCAM. I obtained a Bachelor’s degree in Physics in 2019 at the University of Murcia (UMU) with an exchange year at the University of La Laguna (ULL). Subsequently, I received a Master’s degree in Data Science at the Open University of Catalonia (UOC) in 2021.*

*Throughout the doctorate I hope to nourish myself with the knowledge of the health professionals and supervisors who support me along the thesis. Furthermore, I hope to help with my publications to improve the society in which we live.*

**Machine Learning applied to Health**

Throughout the PhD program, machine learning techniques will be applied to health data from the Basque Country. The purpose is to provide information of interest to health professionals that can help them in making their decisions. Our last contribution is a battery of COVID-19 mortality prediction models. Physicians and hospital managers can choose the model that best fits for a specific epidemiological situation. Currently, we are still working with COVID-19 data but we are focusing on deterioration. We are trying to create a consistent model that takes into consideration the ordinality of the deterioration and features that are only available during the training process and not during testing (Privileged Information).

*I have been a Research Technician at BCAM within the KTU group. Previously, I obtained my MSc on Particle Physics at the Autonomous University of Barcelona (UAB) and the BSc on Physics at the University of the Basque Country (UPV/EHU).*

*I am highly motivated to research, study, discuss and also question the most fundamental issues of the physics, to go deep on the building blocks of the quantum theory and be able to share my results with other researchers.*

**Quantum Measurements**

Quantum engineering requires accurate methods for measuring various properties of quantum systems. Quantum measurements are much more diverse and complex than their classical counterparts. There are many different kinds: accurate (strong), inaccurate (weak), impulsive consecutive, finite-time, continuous, etc. The uncertainty principle implies that a non-negligible back action is produced whenever a measuring device destroys coherence between otherwise interfering alternatives. One objective of the project would, therefore, involve systematic studies of varios aspects of quantum measurement theory. Equally important are the practical realisations of the quantum meters (detectors), which often involve a large number of degrees of freedom. Among such “hybrid devices” are the electronic point-contact and its bosonic counterpart, the bosonic junction. The second aim of the project is the modelling of such devices in varios regimes.

*I am an Italian PhD student in Computational Neuroscience at the Basque Center for Applied Mathematics (BCAM) enrolled in the ASTROTECH project. My academic studies began at the University of Milan-Bicocca where I obtained the bachelor’s degree in Physics. Subsequently, I moved to the University of Turin where I enrolled in the master’s degree class in Physics of Complex systems. *

*I am very happy and honored to participate in this great project. I want to study a lot of mathematics and try to build models that have a strong impact in research in the field of Neuroscience, trying to help in the understanding of some diseases, such as glioma , ischemia, epilepsy and depression.*

**Computational Glioscience approach to volume signalling transmission in the Neuron-Glial-Vascular Unit**

The past four decades demonstrated that non-neuronal cells, called astrocytes, are emerging as crucial players for brain function and dysfunction. A major obstacle of previous and current initiatives on neurotechnologies is a lack of focus on astrocytes and most of the tools used to probe and sense astrocytes are derived from those developed to study neurons. To go further, the ASTROTECH Consortium — which combines 11 funding entities and 14 partners in the academia, public research centers and industrial labs, from 9 European and non European countries — aims at pioneering the field of “Glial Engineering”, to develop a consistent range of tools to record, study, and manipulate astrocytes in the healthy and diseased brain. Pursuing this vision, the objective of my PhD project is to develop a mathematical model of neuron-glia-vascular unit (NGVU), based on biophysical principles of volume transmission in the brain parenchyma. Furthermore, during a training period provided by ASTROTECH within a group of 15 early stage researchers, another important part of the project is related to a tight collaboration with industry representatives and the implementation of the model in an open-source simulation platform adopted in the clinical practice. Moreover, in conjunction with biologists and medical doctors, another key point of the project is to test the NGVU model by brain-wide simulations of epileptic seizures with emphasis on new venues for their diagnosis and treatment.

*Graduated in Mathematics at the University of Salamanca and Master in Mathematical Research at the Polytechnic University of Valencia, my interest in mathematics was born when I discovered the concept of “demonstration”. At the same time, an interest in programming grew in me which, later on, would lead my mathematical interests to take a new direction towards numerical analysis and computational geometry and, more recently, towards machine learning.*

*From my time at BCAM I hope to grow both intellectually and personally; I believe that BCAM can provide me with the ideal environment to enrich myself with knowledge of all kinds both in terms of the people and the experiences it can provide me with.*

**Machine Learning applications to CNC machining: machine rute planning and tool shape optimization**

The naval and aeronautical industry makes extensive use, for obvious reasons, of objects with curved geometries, such as turbines or propellers. In order for these components to have the desired properties, they have to be manufactured with very high precision, usually in the order of micrometres. Nowadays, the most widely used technique for the manufacture of these parts involves the use of numerical control machines (CNC), which usually employ flat tools (cylindrical or conical), which are not properly adapted to the construction of curved geometries, having to use a large number of machining paths to obtain the target geometry, making the manufacturing process long and costly. In this sense, the main objective of the project is to use and design machine learning algorithms to calculate optimal manufacturing tools (which, typically, will not be flat) for these parts with curved geometry, as well as to automate the process of selecting the optimal machining paths for these tools and the target geometry, improving not only the construction accuracy but also the time taken to build the part in question.