Access profile

The doctoral programme in Applied Mathematics is designed for students with abstract reasoning abilities and interest in solving problems, in addition to having good work habits, being dedicated to studying and having a passion for mathematics. Applicants must have a solid background in mathematics. Along this line, the main undergraduate and master’s degrees that will provide access to this doctoral programme are the following:

Undergraduate degrees:
• Bachelor’s degree in Mathematics
• Bachelor’s degree in Physics
Engineering:
• Bachelor’s degree in Computer Science
• Bachelor’s degree in Statistics

Output profile

Doctoral candidates who complete a doctoral degree will have acquired the following competencies, which are needed to carry out quality research (Royal Decree 99/2011, of 28 January, which regulates official doctoral studies):

a) A systematic understanding of the field of study and a mastery of the research skills and methods related to the field.
b) An ability to conceive, design or create, put into practice and adopt a substantial process of research or creation.
c) An ability to contribute to pushing back the frontiers of knowledge through original research.
d) A capacity for critical analysis and an ability to assess and summarise new and complex ideas.
e) An ability to communicate with the academic and scientific community and with society in general as regards their fields of knowledge in the manner and languages that are typical of the international scientific community to which they belong.
f) An ability to foster scientific, technological, social, artistic and cultural progress in academic and professional contexts within a knowledge-based society.

The award of a doctoral degree must equip the graduate for work in a variety of settings, especially those requiring creativity and innovation. Doctoral graduates must have at least acquired the personal skills needed to:

a) Develop in contexts in which there is little specific information.
b) Find the key questions that must be answered to solve a complex problem.
c) Design, create, develop and undertake original, innovative projects in their field.
d) Work as part of a team and independently in an international or multidisciplinary context.
e) Integrate knowledge, deal with complexity and make judgements with limited information.
f) Offer criticism on and intellectually defend solutions.

Finally, with respect to competencies, doctoral students must:
a) have acquired advanced knowledge at the frontier of their discipline and demonstrated, in the context of internationally recognised scientific research, a deep, detailed and well-grounded understanding of theoretical and practical issues and scientific methodology in one or more research fields;
b) have made an original and significant contribution to scientific research in their field of expertise that has been recognised as such by the international scientific community;
c) have demonstrated that they are capable of designing a research project that serves as a framework for carrying out a critical analysis and assessment of imprecise situations, in which they are able to apply their contributions, expertise and working method to synthesise new and complex ideas that yield a deeper knowledge of the research context in which they work;
d) have developed sufficient autonomy to set up, manage and lead innovative research teams and projects and scientific collaborations (both national and international) within their subject area, in multidisciplinary contexts and, where appropriate, with a substantial element of knowledge transfer;
e) have demonstrated that they are able to carry out their research activity in a socially responsible manner and with scientific integrity;
f) have demonstrated, within their specific scientific context, that they are able to make cultural, social or technological advances and promote innovation in all areas within a knowledge-based society;
g) have demonstrated that they are able to participate in scientific discussions at the international level in their field of expertise and disseminate the results of their research activity to audiences of all kinds.

Number of places

20

Duration of studies and dedication regime

Duration
The maximum period of study for full-time doctoral studies is three years, counted from the date of admission to the programme to the date of submission of the doctoral thesis. The academic committee of the doctoral programme may authorise a doctoral candidate to pursue doctoral studies on a part-time basis. In this case, the maximum period of study is five years, counting from the date of admission to the programme to the date of submission of the doctoral thesis. For calculating these periods, the date of admission is considered to be the date of the first enrolment for tutorials, and the date of submission the moment in which the Doctoral School officially deposits the doctoral thesis.

For full-time doctoral candidates, the minimum period of study is two years, counted from the date of an applicant's admission to the programme until the date on which the doctoral thesis is deposited; for part-time doctoral candidates it is four years. When there are justified grounds for doing so, and the thesis supervisor and academic tutor have given their authorisation, doctoral candidates may request that the academic committee of their doctoral programme exempt them from the minimum period of study requirement.

The calculation of periods of study will not include periods of absence due to illness, pregnancy or any other reason provided for in the regulations in force. Students who find themselves in any of these circumstances must notify the academic committee of the doctoral programme, which, where appropriate, must inform the Doctoral School. Doctoral candidates may also temporarily withdraw from the programme for up to one year, and this period may be extended for an additional year. Doctoral candidates who wish to interrupt their studies must submit a justified request to the academic committee of the doctoral programme, which will decide whether or not to approve the request. Each programme will establish conditions for readmission to doctoral studies.

Extension
If full-time doctoral candidates have not applied to deposit their thesis by the end of the three-year period of study, the academic committee of the programme may authorise an extension of up to one year. In exceptional circumstances, a further one-year extension may be granted, subject to the conditions established by the corresponding doctoral programme. In the case of part-time doctoral candidates, an extension of two years may be authorised. In both cases, in exceptional circumstances a further one-year extension may be granted by the Doctoral School's Standing Committee, upon the submission of a reasoned application by the academic committee of the doctoral programme.

Dismissal from the doctoral programme
A doctoral candidate may be dismissed from a doctoral programme for the following reasons:

• The doctoral candidate submitting a justified application to withdraw from the programme.
• The maximum period of study and of extensions thereof ending.
• The doctoral candidate not having enrolled every academic year (unless he or she has been authorised to temporarily withdraw).
• The doctoral candidate failing two consecutive assessments.
• The doctoral candidate having disciplinary proceedings filed against him or her that rule that he or she must be dismissed from the UPC.

Dismissal from the programme implies that doctoral candidates cannot continue studying at the UPC and the closing of their academic record. This notwithstanding, they may apply to the academic committee of the programme for readmission and the committee must reevaluate them in accordance with the criteria established in the regulations.

Enrollment aid

Organization

• Rue Perna, Juan Jose

• Burillo Puig, Pep
• Franch Bullich, Jaume
• Guardia Munarriz, Marcel
• Huguet Casades, Gemma
• Muñoz Romero, Jose Javier
• Rue Perna, Juan Jose
• Serra Albo, Oriol

• School of Mathematics and Statistics (PROMOTORA)
• Barcelona Supercomputing Center-Centro Nacional de Supercomputación
• Department of Civil and Environmental Engineering
• Department of Mathematics
• Institute of Robotics and Industrial Informatics

Agreements with other institutions

Access profile

The doctoral programme in Applied Mathematics is designed for students with abstract reasoning abilities and interest in solving problems, in addition to having good work habits, being dedicated to studying and having a passion for mathematics. Applicants must have a solid background in mathematics. Along this line, the main undergraduate and master’s degrees that will provide access to this doctoral programme are the following:

Undergraduate degrees:
• Bachelor’s degree in Mathematics
• Bachelor’s degree in Physics
Engineering:
• Bachelor’s degree in Computer Science
• Bachelor’s degree in Statistics

Access requirements

Applicants must hold a Spanish bachelor’s degree or equivalent and a Spanish master’s degree or equivalent, provided they have completed a minimum of 300 ECTS credits on the two degrees (Royal Decree 43/2015, of 2 February)

In addition, the following may apply:

• Holders of an official degree awarded by a university in Spain or any other country in the European Higher Education Area, pursuant to the provisions of Article 16 of Royal Decree 1393/2007, of 29 October, which establishes official university course regulations, who have completed a minimum of 300 ECTS credits on official university degrees, of which at least 60 must be at the master's degree level.
• Holders of an official Spanish bachelor’s degree comprising at least 300 credits, as provided for by EU regulations. Holder of degrees of this kind must complete bridging courses unless the curriculum of the bachelor’s degree in question included research training credits equivalent in value to those which would be earned on a master's degree.
• Holders of an official university qualification who, having passed the entrance examination for specialised medical training, have completed at least two years of a training course leading to an official degree in a health-sciences specialisation.
• Holders of a degree issued under a foreign education system. In these cases, homologation is not required, but the UPC must verify that the degree certifies a level of training equivalent to an official Spanish master's degree and qualifies the holder for admission to doctoral studies in the country where it was issued. Admission on this basis does not imply homologation of the foreign degree or its recognition for any purpose other than admission to doctoral studies.
• Holders of a Spanish doctoral qualification issued under previous university regulations.

Note 1: Doctoral studies entrance regulations for holders of an undergraduate degree awarded before the introduction of the EHEA (CG 47/02 2014)

Note 2: Governing Council Decision 64/2014, which approves the procedure and criteria for assessing the fulfilment of academic admission requirements for doctoral studies by holders of non-homologated foreign degrees (CG 25/03 2014)

Admission criteria and merits assessment

During the admission process, the following elements will be considered (with their associated weights):

• Academic transcript from the applicant’s undergraduate and master’s programmes (70%).

• CV (10%).

• Motivation letter and/or personal interview (10%).

• Two letters of recommendation from professors/researchers (10%).

These criteria will be published on the programme’s web page. The final score will be used as the admission criteria in the case that demand exceeds the number of available seats.

There is no provision for entrance exams or any specific entrance tests. If candidates’ backgrounds in mathematics or the subject of the candidates’ doctoral theses so require, they will be required to take bridging courses in the master’s programme in Advanced Mathematics and Mathematical Engineering (MAMME), the official master’s programme offered by the FME that is associated with this doctoral programme.

Training complements

The academic committee for the programme may require that doctoral students pass specific bridging courses. In such cases, the committee will keep track of the bridging courses completed and establish appropriate criteria to limit their duration. The courses could involve training in research or transversal education, but in no case will candidates be required to enrol in 60 or more ECTS credits.

Depending on the doctoral student activity report, the programme’s Academic Committee may propose measures, in addition to those established by current regulations, that would disassociate candidates who do not meet the established requisites.

Additionally, students who require it based on their previous education and the subject of their thesis, will have to take courses from the master’s programme in Advanced Mathematics and Mathematical Engineering (MAMME) offered by the School of Mathematics and Statistics. A detailed list of the subjects offered is listed hereafter:

Enrolment period for new doctoral students

All up-to-date information can be found on the FME’s web page: https://fme.upc.edu/ca/curs-actual/matricula/estudiants-de-doctorat

More information at the registration section for new doctoral students

Enrolment period

All up-to-date information can be found on the FME’s web page: https://fme.upc.edu/ca/curs-actual/matricula/estudiants-de-doctorat

More information at the general registration section

Procedure for the preparation and defense of the research plan

Doctoral candidates must submit a research plan, which will be included in their doctoral student activity report, before the end of the first year. The plan may be improved over the course of the doctoral degree. It must be endorsed by the tutor and the supervisor, and it must include the method that is to be followed and the aims of the research.

At least one of these annual assessments will include a public presentation and defence of the research plan and work done before a committee composed of three doctoral degree holders, which will be conducted in the manner determined by each academic committee. The examination committee awards a Pass or Fail mark. A Pass mark is a prerequisite for continuing on the doctoral programme. Doctoral candidates awarded a Fail mark must submit a new research plan for assessment by the academic committee of the doctoral programme within six months.

The committee assesses the research plan every year, in addition to all of the other activities in the doctoral student activity report. Doctoral candidates who are awarded two consecutive Fail marks for the research plan will be obliged to definitely withdraw from the programme.

If they change the subject of their thesis, they must submit a new research plan.

Formation activities

Procedure for assignment of tutor and thesis director

The academic committee of the doctoral programme assigns a thesis supervisor to each doctoral candidate when they are admitted or enrol for the first time, taking account of the thesis supervision commitment referred to in the admission decision.

The thesis supervisor will ensure that training activities carried out by the doctoral candidate are coherent and suitable, and that the topic of the candidate’s doctoral thesis will have an impact and make a novel contribution to knowledge in the relevant field. The thesis supervisor will also guide the doctoral candidate in planning the thesis and, if necessary, tailoring it to any other projects or activities undertaken. The thesis supervisor will generally be a UPC professor or researcher who holds a doctoral degree and has documented research experience. This includes PhD-holding staff at associated schools (as determined by the Governing Council) and UPC-affiliated research institutes (in accordance with corresponding collaboration and affiliation agreements). When thesis supervisors are UPC staff members, they also act as the doctoral candidate’s tutor.

PhD holders who do not meet these criteria (as a result of their contractual relationship or the nature of the institution to which they are attached) must be approved by the UPC Doctoral School's Standing Committee in order to participate in a doctoral programme as researchers with documented research experience.

The academic committee of the doctoral programme may approve the appointment of a PhD-holding expert who is not a UPC staff member as a candidate’s thesis supervisor. In such cases, the prior authorisation of the UPC Doctoral School's Standing Committee is required. A UPC staff member who holds a doctoral degree and has documented research experience must also be proposed to act as a co-supervisor, or as the doctoral candidate’s tutor if one has not been assigned.

A thesis supervisor may step down from this role if there are justified reasons (recognised as valid by the committee) for doing so. If this occurs, the academic committee of the doctoral programme will assign the doctoral candidate a new thesis supervisor.

Provided there are justified reasons for doing so, and after hearing any relevant input from the doctoral candidate, the academic committee of the doctoral programme may assign a new thesis supervisor at any time during the period of doctoral study.

If there are academic reasons for doing so (an interdisciplinary topic, joint or international programmes, etc.) and the academic committee of the programme gives its approval, an additional thesis supervisor may be assigned. Supervisors and co-supervisors have the same responsibilities and academic recognition.

The maximum number of supervisors of a doctoral thesis is two: a supervisor and a co-supervisor.

For theses carried out under a cotutelle agreement or as part of an Industrial Doctorate, if necessary and if the agreement foresees it this maximum number of supervisors may not apply. This notwithstanding, the maximum number of supervisors belonging to the UPC is two.

More information at the PhD theses section

Permanence

The academic committee of the programme may authorise an extension of up to one year for full-time doctoral candidates who have not applied to deposit their thesis by the end of the three-year period of study, in the terms outlined in the Academic Regulations for Doctoral Studies of the Universitat Politècnica de Catalunya. In the case of part-time candidates, an extension of two years may be authorised. In both cases, in exceptional circumstances a further one-year extension may be granted by the Doctoral School's Standing Committee, upon the submission of a reasoned application by the academic committee of the doctoral programme.

A doctoral candidate may be dismissed from a doctoral programme for the following reasons:

• The doctoral candidate submitting a justified application to withdraw from the programme.
• The maximum period of study and of extensions thereof ending.
• The doctoral candidate not having enrolled every academic year (unless he or she has been authorised to temporarily withdraw).
• The doctoral candidate failing two consecutive assessments.
• The doctoral candidate having disciplinary proceedings filed against him or her that rule that he or she must be dismissed from the UPC.

Dismissal from the programme implies that doctoral candidates cannot continue studying at the UPC and the closing of their academic record. This notwithstanding, they may apply to the academic committee of the programme for readmission and the committee must reevaluate them in accordance with the criteria established in the regulations.

International Mention

The doctoral degree certificate may include International Doctorate mention. In this case, the doctoral candidate must meet the following requirements:

a) During the period of study leading to the award of the doctoral degree, the doctoral candidate must have spent at least three months at a respected higher education institution or research centre outside Spain to complete courses or do research work. The stays and activities carried out must be endorsed by the thesis supervisor and authorised by the academic committee of the programme. The candidate must provide a certifying document issued by the person responsible for the research group of the body or bodies where the stay or activity was completed. This information will be added to the doctoral student’s activity report.
b) Part of the thesis (at least the summary and conclusions) must be written and presented in one of the languages commonly used for science communication in the relevant field of knowledge, which must not be an official language of Spain. This rule does not apply to stays and reports in Spanish or to experts from Spanish-speaking countries.
c) At least two PhD-holding experts belonging to a higher education institution or research centre outside Spain must have issued officially certified reports on the thesis.
d) The thesis examination committee must have included at least one PhD-holding expert from a higher education or research institution outside Spain who was not responsible for the candidate’s stay abroad (point a) above).
e) The thesis defence must have taken place on UPC premises or, in the case of joint programmes, at the location specified in the collaboration agreement.

List of authorized thesis for defense

• VRIONI, BRIKENA: A census for Curves and Surfaces with Diophantine Stability over Finite Fields
  
  Author: VRIONI, BRIKENA
  Thesis file: (contact the Doctoral School to confirm you have a valid doctoral degree and to get the link to the thesis)
  Programme: DOCTORAL DEGREE IN APPLIED MATHEMATICS
  Department: Department of Mathematics (MAT)
  Mode: Normal
  Deposit date: 21/07/2021
  Reading date: 29/10/2021
  Reading time: 16:30
  Reading place: FME: Defensa per videoconferència per COVID-19: https://meet.google.com/heg-sxds-ygg
  Thesis director: LARIO LOYO, JOAN CARLES
  Committee:
       PRESIDENT: BRAS AMOROS, MARIA
       SECRETARI: RIO DOVAL, ANA
       VOCAL: PUJOLAS BOIX, JORDI
  Thesis abstract: An algebraic variety defined over a field is said to have Diophantine stability for an extension of this field if the variety does not acquire new points in the extension.Diophantine stability has a growing interest due to recent conjectures of Mazur and Rubin linked to the well-known Lang conjectures, generalizing the celebrated Faltings theorem on rational points on curves of genus grater or equal than 2. Their framework is characteristic zero, and we shall focus on the analogous and related questions in positive characteristic.More precisely, the aim of the thesis is to initiate the study of Diophantine stability for curves and surfaces defined over finite fields. First we prove the finiteness of the finite field extensions where an algebraic variety can exhibit Diophantine stability (DS) in terms of its Betti numbers (the genus in the case of curves, the Hodge diamond in the case of surfaces, etc.) Then, we analyze the existence of curves with Diophantine stability. More precisely, for curves of genus g<=3 we give the complete list of (isomorphism classes of) DS-curves, and we also provide data on the candidate Weil polynomials for DS-curves of genus g=4 and 5. For curves of large genus, we exhibit certain families of DS-curves: Deligne-Lusztig curves, Carlitz curves, ....Finally, we also aim to make a contribution on surfaces defined over finite fields with Diophantine stability. From the classification of surfaces of Enriques-Munford-Bombieri we derive partial results and a census of DS-surfaces.
  

Last update: 16/10/2021 04:47:07.

List of lodged theses

Last update: 16/10/2021 04:30:36.

List of defended theses by year

• CARDONA AGUILAR, ROBERT: The geometry and topology of steady Euler flows, integrability and singular geometric structures
  
  Author: CARDONA AGUILAR, ROBERT
  Thesis link: http://hdl.handle.net/10803/672150
  Programme: DOCTORAL DEGREE IN APPLIED MATHEMATICS
  Department: Department of Mathematics (MAT)
  Mode: Normal
  Reading date: 26/05/2021
  Thesis director: MIRANDA GALCERÁN, EVA
  

  Committee:
       PRESIDENT: PERALTA SALAS, DANIEL
       SECRETARI: DELSHAMS I VALDES, AMADEU
       VOCAL: MUNDET RIERA, IGNASI
       VOCAL: GHRIST, ROBERT
       VOCAL: CANNAS DA SILVA, ANA
  Thesis abstract: n this thesis, we make a deep investigation of the geometry and dynamics of several objects (singular or not) appearing in nature. The main goal is to study rigidity versus flexibility dynamical behavior of the objects considered. In particular, we inspect normal forms, h-principles, classifications, and existence theorems. These concern a series of objects which are either close or far away from what we call ¿integrable situations" in the sense of Frobenius theorem and the existence of first integrals. Such dynamical systems arise in the context of symplectic and contact geometry (and their singular counterparts), as well as in the Euler equations on Riemannian manifolds.As integral objects, we consider integrable systems appearing in symplectic manifolds but also on singular symplectic manifolds. Singularities show up naturally on these phase spaces by considering spaces with cylindrical ends and studying b-symplectic forms as initiated by Guillemin-Miranda-Pires. Other types of singularities are folded structures originally considered by Martinet and then by Cannas da Silva, Guillemin, Woodward for geometrical purposes. We give classification results of steady Euler flows which admit a Morse-Bott first integral using techniques coming from the symplectic world, and study obstructions arising from the ambient topology. Our analysis includes the existence of action-angle coordinates on folded symplectic manifolds, and a correspondence between the recently introduced b-contact forms and Beltrami fields on b-manifolds. As examples of systems that are ¿far from integral" we consider the case of contact manifolds and their close allies in the study of Euler flows (Beltrami vector fields). This gives us the possibility to extend the h-principles from the contact realm to that of Beltrami vector fields. This last observation enables us to consider universality properties, as introduced by Tao, of steady Euler flows by analyzing those of high-dimensional Reeb flows in contact geometry. In the same spirit, we address the construction of steady Euler flows in dimension 3 which simulate a universal Turing machine, using tools coming from symbolic dynamics. In particular, these solutions have undecidable trajectories. In all these discussions, a key role is played by different classes of vector fields such as geodesible, Beltrami, and Eulerisable fields. We set up the study of the relations between such classes in higher odd-dimensions, showing that new phenomena arise as soon as one leaves the realm of three-dimensional manifolds. For these high dimensional Euler flows (or more generally, flows admitting a strongly adapted one-form), we show that they satisfy the periodic orbit conjecture, which was known to be satisfied with the weaker assumption of geodesibility.
  

• CODONY GISBERT, DAVID: Mathematical and computational modeling of flexoelectricity at mesoscopic and atomistic scales
  
  Author: CODONY GISBERT, DAVID
  Thesis link: http://hdl.handle.net/10803/671925
  Programme: DOCTORAL DEGREE IN APPLIED MATHEMATICS
  Department: School of Mathematics and Statistics (FME)
  Mode: Normal
  Reading date: 01/03/2021
  Thesis director: ARIAS VICENTE, IRENE
  

  Committee:
       PRESIDENT: SHARMA, PRADEEP
       SECRETARI: RODRIGUEZ FERRAN, ANTONIO
       VOCAL: CIRAK, FEHMI
  Thesis abstract: This PhD thesis focuses on the development of mathematical and computational models for flexoelectricity, a relatively newelectromechanical coupling that is present in any dielectric at the micron and sub-micron scales. The work is framed in thecontext of both continuum and quantum mechanics, and explores the gap between these two disciplines.On the one hand, the focus is put on the mathematical modeling of the flexoelectric effect by means of continuum (electro-)mechanics, and the development of computational techniques required to numerically solve the associated boundary valueproblems. The novel computational infrastructure developed in this work is able to predict the performance of engineereddevices for electromechanical transduction at sub-micron scales, where flexoelectricity is always present, without anyparticular restrictions in geometry, material choice, boundary conditions or nonlinearity. The numerical examples within thisdocument show that flexoelectricity can be harnessed in multiple different ways towards the development of breakthroughapplications in nanotechnology.On the other hand, the flexoelectric effect is also studied at an atomistic level by means of quantum mechanics. This workproposes a novel methodology to quantify the flexoelectric properties of dielectric materials, by means of connecting ab-initioatomistic simulations with the proposed models at a coarser, continuum scales. The developed approach sheds some lighton a controversial topic within the density functional theory community, where large disagreements among differenttheoretical derivations are typically found. The ab-initio computations serve not only to assess the material parameters withinthe continuum models, but also to validate their inherent assumptions regarding the relevant physics at the nanoscale.
  

• COSTA MIRADA, NURIA: Long-term privacy in electronic voting systems
  
  Author: COSTA MIRADA, NURIA
  Thesis link: http://hdl.handle.net/10803/672120
  Programme: DOCTORAL DEGREE IN APPLIED MATHEMATICS
  Department: Department of Mathematics (MAT)
  Mode: Normal
  Reading date: 14/04/2021
  Thesis director: MORILLO BOSCH, MARIA PAZ
  

  Committee:
       PRESIDENT: RÀFOLS SALVADOR, CARLA
       SECRETARI: HERRANZ SOTOCA, JAVIER
       VOCAL: GUASCH CASTELLÓ, SANDRA
  Thesis abstract: This PhD thesis focuses on lattice-based cryptography and how to apply it to build post-quantum online voting systems. It is the result of the research done by the author at Scytl in close collaboration with Dr. Paz Morillo, from the Department of Applied Mathematics at UPC and Ramiro Martínez, PhD student. As part of her work at the electronic voting company Scytl, the author has participated in the design of several electronic voting systems as well as in their implementation, by providing support to the development team. Nevertheless, all these systems use standard and well-known cryptographic primitives, i.e., not lattice-based primitives, to ensure that the security requirements are fulfilled. Due to this, one of the main challenges of this PhD has been to start researching on a field which was not familiar to the author and contribute to its state of the art. This has allowed the company to enter the post-quantum world by participating in a project which aims to implement a lattice-based online voting system. The thesis has the following contents: an introduction to the lattice theory by describing some of its basic concepts and the computational problems in which the security of lattice-based cryptosystems relies. In this first part it is also described in detail those cryptosystems that are used as building blocks of three new protocols proposed in the thesis: a lattice-based coercion-resistant cast-as-intended protocol, a post-quantum mix-net and a fully post-quantum proof of a shuffle. The former is the lattice version of an existing protocol and allows the voter to check that the vote cast contains the selected voting options. The second and third protocols are the result of the research on lattice-based mix-nets. Two constructions are proposed: the first one allows to demonstrate that a mix-node has permuted and re-encrypted a list of RLWE ciphertexts without modifying them, but it cannot be considered fully post-quantum since the binding property of the commitment scheme relies on classical computational problems. The second one is fully post-quantum since all the cryptographic schemes used for building it, i.e., commitment scheme and zero-knowledge proofs, are based on lattices. Last but not least, for this second proposal a security definition and a proof of security are also provided. Finally, the last part of the thesis consists of building a post-quantum online voting system using as building blocks the protocols already presented and existing lattice-based constructions. This system is considered secure under quantum attacks and provides long-term privacy. It also guarantees vote anonymity, vote authenticity, vote integrity, individual verifiability and receipt-freeness. The algorithms involved in each phase are described in detail as well as the interaction among the participants. An implementation of this system is not given as part of this thesis although a lattice-based online voting system based on that is already being implemented at the company.
  

• DEMPSEY BRADDELL, ROISIN MARY: New Geometrical and Dynamical Techniques for Problems in Celestial Mechanics
  
  Author: DEMPSEY BRADDELL, ROISIN MARY
  Thesis link: http://hdl.handle.net/10803/671704
  Programme: DOCTORAL DEGREE IN APPLIED MATHEMATICS
  Department: Department of Mathematics (MAT)
  Mode: Normal
  Reading date: 17/02/2021
  Thesis director: DELSHAMS I VALDES, AMADEU | MIRANDA GALCERÁN, EVA
  

  Committee:
       PRESIDENT: HARO PROVINCIALE, ÀLEX
       SECRETARI: MARTIN DE LA TORRE, PAU
       VOCAL: SCOTT, GEOFFREY
  Thesis abstract: In this thesis, we study the application of symplectic geometry, regular and singular, to symplectic dynamical systems.We start with a motivating case: the relation between symplectic foliations and global transverse Poincaré sections, showing that meaningful dynamical information can be gleaned by simple observations on the geometry of the phase space - in this case, the existence of a symplectic foliation on a hypersurface of the phase space.We then go on study dynamical systems of particular importance in geometry - those given by a group action on a manifold. In particular, we consider a singular symplectic manifold (specifically, a manifold equipped with a symplectic form which blows up in a controlled manner on a hypersurface of that manifold, namely, a b-symplectic form) with a group action preserving the geometry and give a b-symplectic slice theorem which provides an equivariant normal form of the b-symplectic form in the neighbourhood of an orbit. Particular examples of b-symplectic group symmetries are then explored: those given by the cotangent lift of group translation on so-called b-Lie groups.The second part of this thesis focuses on symplectic and b-symplectic dynamical systems coming from celestial mechanics. In particular, the separatrix map of the stable and unstable manifolds of the fixed point at infinity of the planar circular restricted three-body problem is examined and an estimate of the width of the stochastic layer is given - that is the existence of a K.A.M. torus which acts as a boundary to bounded motions is proved. Due to the delicate nature of the problem - namely issues coming from the parabolic nature of the fixed point and exponentially small nature of the splitting, careful control of the errors of the separatrix map is paramount. This is achieved by employing geometric methods, namely, by taking full advantage of generating functions which exist by virtue of the symplectic nature of the system.Finally, motivated by the important role of symplectic geometry in the systems of celestial mechanics in mind, we give examples of degenerate and singular symplectic structures occurring in systems of celestial mechanics which cannot be equipped with a symplectic form.
  

• DEUTSCHMANN, INA MARIA: Disentangling ecological networks in marine microbes
  
  Author: DEUTSCHMANN, INA MARIA
  Thesis file: (contact the Doctoral School to confirm you have a valid doctoral degree and to get the link to the thesis)
  Programme: DOCTORAL DEGREE IN APPLIED MATHEMATICS
  Department: Department of Mathematics (MAT)
  Mode: Normal
  Reading date: 29/09/2021
  Thesis director: LOGARES MAURIE, RAMIRO
  

  Committee:
       PRESIDENT: BITTNER, LUCIE
       SECRETARI: CASANELLAS RIUS, MARTA
       VOCAL: LOPEZ, PHILIPPE
  Thesis abstract: There is a myriad of microorganisms on Earth contributing to global biogeochemical cycles, and their interactions are considered pivotal for ecosystem function. Previous studies have already determined relationships between a limited number of microorganisms. Yet, we still need to understand a large number of interactions to increase our knowledge of complex microbiomes. This is challenging because of the vast number of possible interactions. Thus, microbial interactions still remain barely known to date. Networks are a great tool to handle the vast number of microorganisms and their connections, explore potential microbial interactions, and elucidate patterns of microbial ecosystems. This thesis locates at the intersection of network inference and network analysis. The presented methodology aims to support and advance marine microbial investigations by reducing noise and elucidating patterns in inferred association networks for subsequent biological down-stream analyses. This thesis¿s main contribution to marine microbial interactions studies is the development of the program EnDED (Environmentally-Driven Edge Detection), a computational framework to identify environmentally-driven associations inside microbial association networks, inferred from omics datasets. We applied the methodology to a model marine microbial ecosystem at the Blanes Bay Microbial Observatory (BBMO) in the North-Western Mediterranean Sea (ten years of monthly sampling). We also applied the methodology to a dataset compilation covering six global-ocean regions from the surface (3 m) to the deep ocean (down to 4539 m). Thus, our methodology provided a step towards studying the marine microbial distribution in space via the horizontal (ocean regions) and vertical (water column) axes.
  

• FELIPE NAVARRO, JUAN CARLOS: Qualitative properties of solutions to integro-differential elliptic problems
  
  Author: FELIPE NAVARRO, JUAN CARLOS
  Thesis link: http://hdl.handle.net/10803/672313
  Programme: DOCTORAL DEGREE IN APPLIED MATHEMATICS
  Department: Department of Mathematics (MAT)
  Mode: Normal
  Reading date: 14/07/2021
  Thesis director: CABRE VILAGUT, XAVIER
  

  Committee:
       PRESIDENT: SOLÀ-MORALES RUBIÓ, JUAN DE LA CRUZ DE
       SECRETARI: MAS BLESA, ALBERT
       VOCAL: ROS OTON, XAVIER
       VOCAL: SERRA MONTOLÍ, JOAQUIM
       VOCAL: ORTEGA CERDA, JOAQUIM
  Thesis abstract: The thesis is devoted to the analysis of elliptic PDEs and related problems. It is mainly focused on the study of qualitative and regularity properties of solutions to integro-differential equations. The study of such equations has attracted much attention recently since they arise naturally in different areas when dealing with processes where long range interaction phenomena appear. The canonical example of integro-differential operators is the fractional Laplacian, which is translation, rotation, and scale invariant. The thesis is divided into three parts. Part I concerns the study of uniqueness and regularity properties of solutions to integro-differential linear problems. First, we prove, by following a nonlocal Liouville-type method, the uniqueness of solutions in the one-dimensional case, in the presence of a positive solution or of an odd solution vanishing only at zero. As an application, we deduce the nondegeneracy of layer solutions (bounded and monotone solutions) to semilinear problems of Allen-Cahn type. Next, we establish the first boundary regularity result for the Neumann problem associated to the fractional Laplacian. We prove that weak solutions are Hölder continuous up to the boundary by developing a delicate Moser iteration with logarithmic corrections on the boundary. We also establish a Neumann Liouville-type theorem in a half-space, which is used together with a blow-up argument to show higher regularity of solutions. Part II of the thesis is focused on the study of the saddle-shaped solution to the integro-differential Allen-Cahn equation. These solutions, whose zero level set is the Simons cone, are expected to be the simplest minimizer which is not one-dimensional to the local and nonlocal Allen-Cahn equation in high enough dimensions. It plays, thus, the same role as the Simons cone in the theory of minimal surfaces. First, we study the saddle-shaped solution for the fractional problem by using the extension problem. We establish its uniqueness and, in dimensions greater or equal than 14, its stability. As a byproduct, we give the first analytical proof of a stability result for the Simons cone in the nonlocal setting for such dimensions. The key ingredient to prove these results is a maximum principle for the linearized operator. Next, we study saddle-shaped solutions for any rotation invariant and uniformly elliptic integro-differential operator. In this scenario, we need to develop some new nonlocal techniques since the extension approach is not available. In this respect, our main contribution is a characterization of the kernels for which one can develop a theory of existence and uniqueness of saddle-shaped solutions. Under these assumptions, we establish an energy estimate for doubly radial odd minimizers and some properties of the saddle-shaped solution, namely: existence, uniqueness, asymptotic behavior, and a maximum principle for the linearized operator. Finally, in Part III we develop a nonlocal Weirstrass extremal field theory. In analogy to the local theory, we construct a calibration for the nonlocal functional in the presence of a foliation made of solutions when the nonlocal Lagrangian satisfies an ellipticity condition. The model case in our setting corresponds to the energy functional for the fractional Laplacian, for which such a calibration was still unknown. The existence of such a calibration allows us to prove that any leaf of the foliation is automatically a minimizer for its own exterior datum, with no need to have an existence result of minimizers, neither to know their regularity.
  

• GARROTE LOPEZ, MARINA: Algebraic and semi-algebraic phylogenetic reconstruction
  
  Author: GARROTE LOPEZ, MARINA
  Thesis link: http://hdl.handle.net/10803/672316
  Programme: DOCTORAL DEGREE IN APPLIED MATHEMATICS
  Department: Department of Mathematics (MAT)
  Mode: Normal
  Reading date: 22/07/2021
  Thesis director: CASANELLAS RIUS, MARTA | FERNANDEZ SANCHEZ, JESUS
  

  Committee:
       PRESIDENT: PETROVIC, SONJA
       SECRETARI: ALVAREZ MONTANER, JOSEP
       VOCAL: ZWIERNIK, PÌOTR
  Thesis abstract: Phylogenetics is the study of the evolutionary history and relationships among groups of biological entities (called taxa). The modeling of those evolutionary processes is done by phylogenetic trees whose nodes represent different taxa and whose branches correspond to the evolutionary processes between them. The leaves usually represent contemporary taxa and the root is their common ancestor. Nowadays, phylogenetic reconstruction aims to estimate the phylogenetic tree that best explains the evolutionary relationships of current taxa using solely information from their genome arranged in an alignment. We focus on the reconstruction of the topology of phylogenetic trees, which means reconstructing the shape of the tree considering labels at the leaves.To this end, one usually assumes that DNA sequences evolve according to a Markov process ruled by a prescribed model of nucleotide substitutions. These substitution models specify some transition matrices at the edges of the tree and a distribution of nucleotides at the root. Given a tree T and a substitution model, one can compute the distribution of nucleotide patterns at the leaves of T in terms of the model parameters. This joint distribution is represented by a vector whose entries can be expressed as polynomials on the model parameters and satisfy certain algebraic relationships. The study of these relationships and the geometry of the algebraic varieties defined by them (called phylogenetic varieties) have provided successful insight into the problem of phylogenetic reconstruction. However, from a biological perspective we are not interested in the whole variety, but only in the region of points that arise from stochastic parameters (the so-called phylogenetic stochastic region). The description of these regions leads to semi-algebraic constraints which play an important role since they characterize distributions with biological and probabilistic meaning. One of the main motivations for this thesis follows from the following question. Could the use of semi-algebraic tools improve the already existent algebraic tools for phylogenetic reconstruction?To answer this question, we compute the Euclidean distance of data points arising from an alignment of nucleotide to the phylogenetic varieties and their stochastic regions in a some scenarios of special interest in phylogenetics, such as trees with short external branches and/or subject to the long branch attraction phenomenon. In some cases, we compute these distances analytically and we can decide which tree has stochastic region closer to the data point. As a consequence, we can prove that, even if the data point was close to the phylogenetic variety of a given tree, it might be closer to the stochastic region of another tree. In particular, considering the stochastic phylogenetic region seems to be fundamental to cope with the phylogenetic reconstruction problem when dealing with the long branch attraction phenomenon.However, incorporating semi-algebraic tools into phylogenetic reconstruction methods can be extremely difficult and the procedure to do it is not evident at all. In this thesis, we present two phylogenetic reconstruction methods that combine algebraic and semi-algebraic conditions for the general Markov model. The first method we present is SAQ, which stands for Semi-Algebraic Quartet reconstruction method. Next, we introduce a more versatile method, ASAQ (for Algebraic and Semi-Algebraic Quartet reconstruction method}), which combines SAQ with the method Erik+2 (based on certain algebraic constraints). Both are phylogenetic reconstruction methods for DNA alignments on four taxa which have been proven to be statistically consistent.We test the suggested methods on simulated and real data to check their actual performance in several scenarios. Our simulation studies show that both methods SAQ and ASAQ are highly successful, even when applied to short alignments or data that violates their assumptions.
  

• GATTI, FRANCESCA: Special values of the triple product p-adic L-function
  
  Author: GATTI, FRANCESCA
  Thesis link: http://hdl.handle.net/10803/671936
  Programme: DOCTORAL DEGREE IN APPLIED MATHEMATICS
  Department: Department of Mathematics (MAT)
  Mode: Normal
  Reading date: 19/03/2021
  Thesis director: ROTGER CERDÀ, VICTOR
  

  Committee:
       PRESIDENT: XARLES, XAVIER
       SECRETARI: QUER BOSOR, JORDI
       VOCAL: LONGO, MATTEO
  Thesis abstract: Let E be an elliptic curve defined over the field of rational numbers and let f be the corresponding newform of weight 2. Let g,h be two weight one newforms whose nebentype characters are mutually inverse. Let V be the tensor product of the Artin representations attached to g and h, let L be the number field generated by the Fourier coefficients of g and h and let H be the number field cut by V. The points of E having coefficients in H form a finitely generated abelian group E(H), and one can attach to the pair (E,V) the finite-dimensional L-vector space MW(E,V) consisting of the homomorphisms from V to E(H) which are equivariant with respect to the action of the absolute Galois group of the rational numbers. We call algebraic rank of (E,V) the L-dimension of MW(E,V). On the other hand, a complex L-function L(E,V;s), which satisfies a functional equation whose center of symmetry is s=1, can be attached to (E,V). The Galois-equivariant version of the Birch and Swinnerton¿Dyer conjecture predicts that the algebraic rank of (E,V) should equal the order of vanishing of L(E,V;s) at s=1, that we call analytic rank of (E,V). An interesting approach to these sort of problems (which are in general still unsolved) is via p-adic analysis. In our setting, we will consider the triple product p-adic L-function Lp(F,G,H)(k,l,m) attached by Darmon and Rotger to a triple of Hida families F,G,H passing through f,g,h respectively at weights 2,1,1. The function Lp(F,G,H)(k,l,m) interpolates p-adically the central values of the complex L-functions attached to the specialisations of the families F,G and H at classical weights k,l,m for which the weight l is greater or equal to the sum of the other two (in this case we say that l is ¿dominant¿). In particular, the triple (2,1,1), which corresponds to the pair (E,V) lies outside the region of classical interpolation of the triple product L-function. In this thesis we study the value Lp(F,G,H)(2,1,1), and more in general the values of Lp(F,G,H) at weights (k,l,m) with k dominant, in different settings. In the case in which the ranks of (E,V) are 2, with the elliptic Stark conjecture Darmon, Lauder and Rotger predict (and prove, in some special cases) that Lp(F,G,H)(2,1,1) can be expressed as a p-adic regulator involving the p-adic logarithm of points on E(H), divided by the logarithm of a so- called Stark unit. We generalise this conjecture to general weights (k,l,m) with k dominant and we prove some cases of this conjecture. Here p is an odd prime which does not divide the conductors of g and h and divides at most once the conductor of E. If p does not divide any of the conductors and if the analytic and the alge- braic ranks of (E,V) are 0 we prove, under the additional hypothesis that the Selmer group of (E,V) is trivial, a formula for the value Lp(F,G,H)(2,1,1) in terms of the Bloch¿Kato logarithm of a canonical non-cristalline class which belongs to the p-relaxed Selmer group of (E,V), along a certain cristalline direction. As a corollary we prove that, in the special case in which g and h are theta series of Hecke characters of the same imaginary quadratic field in which p splits, the triple product L-function vanishes at (2,1,1). Finally we study the case in which E has multiplicative reduction at p, g and h are theta series of the same imaginary quadratic field K in which p is inert, p does not divide the conductors of g and h and the analytic rank of (E,V) is 0. In this setting, Bertolini and Darmon constructed a so¿called Kolyvagin class with interesting arithmetic properties, which lies in the relaxed Selmer group of E over the field H cut out by V. We prove a formula which relates the value Lp(F,G,H)(2,1,1) to the p-adic logarithm of the projection along a cristalline direction of the Kolyvagin class.
  

• GIL MUÑOZ, DANIEL: An effective method to study the Hopf-Galois module structure of certain extensions of fields
  
  Author: GIL MUÑOZ, DANIEL
  Thesis link: http://hdl.handle.net/10803/672317
  Programme: DOCTORAL DEGREE IN APPLIED MATHEMATICS
  Department: Department of Mathematics (MAT)
  Mode: Normal
  Reading date: 02/07/2021
  Thesis director: RIO DOVAL, ANA | CRESPO VICENTE, MARIA TERESA
  

  Committee:
       PRESIDENT: GÓMEZ MOLLEDA, MARIA DE LOS ANGELES
       SECRETARI: VELA DEL OLMO, MARIA MONTSERRAT
       VOCAL: TRUMAN, PAUL JAMES
  Thesis abstract: We develop a method to compute a basis of the associated order in a Hopf Galois structure H of the ring of integers O_L of an extension of number or p-adic fields L/K. We state and prove a necessary and sufficient condition for a given element in O_L to be a free generator of O_L as module over its associated order in H. Whenever it exists, one can use such a free generator and a basis of this associated order to build a basis which can be seen as an analog of the normal integral basis in the Galois case. We use this method to determine the associated order and the existence of those normal integral bases generators for different classes of extensions of fields, such as Galois extensions of degrees 2, 3 and 4, and separable degree p extensions of Q_p with normal closure having Galois group isomorphic to the dihedral group D_p of 2p elements. We shall use the theory of induced Hopf Galois structures to study the same problem for the normal closure itself, i.e. a dihedral degree 2p extension of Q_p. We give complete answers for the cases p=3 and p=5.
  

• LATORRE IBARS, ERNEST: ACTIVE SUPERELASTICITY IN THREE-DIMENSIONAL EPITHELIA OF CONTROLLED SHAPE
  
  Author: LATORRE IBARS, ERNEST
  Thesis link: http://hdl.handle.net/10803/671767
  Programme: DOCTORAL DEGREE IN APPLIED MATHEMATICS
  Department: (DECA)
  Mode: Normal
  Reading date: 19/02/2021
  Thesis director: ARROYO BALAGUER, MARINO | TREPAT GUIXER, XAVIER
  

  Committee:
       PRESIDENT: SCHWARTZ, ULRICH
       SECRETARI: MUÑOZ ROMERO, JOSE JAVIER
       VOCAL: RUPRECHT, VERENA
  Thesis abstract: Fundamental processes in development and physiology are determined by the three-dimensional architecture of epithelial sheets. How these sheets deform and fold into complex structures has remained unclear, however, because their mechanical properties in three-dimensions have not been accessed experimentally. By combining measurements of epithelial tension, shape, and luminal pressure with mathematical modeling, here we show that epithelial cell sheets are active superelastic materials. We develop a new approach to produce massive arrays of epithelial domes with controlled basal shape and size. By measuring 3D deformations of the substrate and curvature of the dome we obtain a direct measurement of luminal pressure and epithelial tension. Observations over time-scales of hours allow us to map the epithelial tension-strain response, revealing a tensional plateau over several-fold areal strain reaching 300%. We show that these extreme nominal strains are accommodated by a highly heterogeneous stretching of individual cells, with barely deformed cells coexisting with others reaching 1000% areal strain, in seeming contradiction with the measured tensional uniformity. This phenomenology is reminiscent of superelasticity, a mechanical response generally attributed to microscopic material instabilities in metal alloys. We provide evidence that this instability is triggered in epithelial cells by a stretch-induced dilution of the actin cortex and rescued by the intermediate filament network. Finally, we implement a mathematical model that captures both the tension/strain relationship and strain heterogeneity. Our study unveils a new type of mechanical behavior -active superelasticity- that enables epithelial sheets to sustain extreme stretching under constant tension.
  

• PORRO MARTORELL, OLGA: A Hesitant Fuzzy Perceptual-Based Approach to Model Linguistic Assessments
  
  Author: PORRO MARTORELL, OLGA
  Thesis link: http://hdl.handle.net/10803/672127
  Programme: DOCTORAL DEGREE IN APPLIED MATHEMATICS
  Department: Department of Mathematics (MAT)
  Mode: Normal
  Reading date: 07/04/2021
  Thesis director: SANCHEZ SOLER, MONICA | AGELL JANÉ, NÚRIA
  

  Committee:
       PRESIDENT: TRAVÉ-MASSUYÈS, LOUISE
       SECRETARI: GIBERT OLIVERAS, CARINA
       VOCAL: VALLS MATEU, AÏDA
  Thesis abstract: Multiple-criteria or multiple-attribute group decision-making is a sub-field of operations research that seek to find a common and representative solution given the preferences elicited by a pre-defined group, over a set of alternatives and with respect to a set of coherent criteria (or attributes). Recently, the modelling of natural language in these processes has captured the attention of many researchers. Most of the evaluations in a group-decision making context are inherently imprecise, incomplete or vague, and therefore, experts feel more comfortable using their language rather than numerical values. The use of hesitant fuzzy linguistic term sets is one of the recent tools that enables the modelling of linguistic assessments in multiple-criteria decision-making. Nonetheless, advances in hesitant linguistic multi-attribute group decision making require the development of structures flexible enough to deal with unbalanced and multi-granular linguistic information. More tools are needed in order to really grasp the differences in the qualitative reasoning processes of each individual. This thesis, firstly, introduces a perceptual-based distance able to capture differences between unbalanced linguistic assessments, which is based on a lattice structure of hesitant fuzzy linguistic terms. Secondly, this distance is used to define a perceptual-based centroid or central opinion which, in turn, is used to define a consensus measure or degree of agreement within the group. Thirdly, with the aim to deal with multi-perceptual group decision-making contexts, where each decision maker has its own qualitative reasoning approach, a perceptual-based transformation function and a projected algebraic structure are defined. The developed tools can deal with different multi-granularity linguistic environments. Two applications are presented to demonstrate the utility, relevancy and feasibility of the methods. On the one hand, a specific perceptual-based classification and ranking method is introduced and applied to a real group decision making problem in an educational setting. This framework is used to classify and rank a set of secondary students according to their degree of entrepreneurial competency, which is based on real data provided by the Andorra Government. On the other hand, an extended fuzzy multi-perceptual linguistic TOPSIS is designed and applied to a real group decision making problem in the context of smart city governance. This perceptual extension is used to assess the criteria governing the strategic decision making process of energy multinational companies when deciding where to expand its sustainable services and products.
  

• RIVERO SALGADO, ÓSCAR: Arithmetic applications of the Euler systems of Beilinson-Flach elements and diagonal cycles
  
  Author: RIVERO SALGADO, ÓSCAR
  Thesis link: http://hdl.handle.net/10803/671865
  Programme: DOCTORAL DEGREE IN APPLIED MATHEMATICS
  Department: Department of Mathematics (MAT)
  Mode: Normal
  Reading date: 12/02/2021
  Thesis director: ROTGER CERDÀ, VICTOR
  

  Committee:
       PRESIDENT: DIEULEFAIT, LUIS VICTOR
       SECRETARI: MOLINA BLANCO, SANTIAGO
       VOCAL: LOEFFLER, DAVID
  Thesis abstract: The main objective of this dissertation is the exploration of certain arithmetic applications of the Euler systems of Beilinson--Flach elements and diagonal cycles. Euler systems have been proved to be a very powerful tool for the study of Iwasawa theory and Selmer groups. Roughly speaking, they are collections of Galois cohomology classes satisfying certain compatibility relations, and are typically constructed using the étale cohomology of algebraic varieties. The genesis of the concept comes from Kolyvagin, who used them to fully prove the Birch and Swinnerton--Dyer conjecture in analytic rank one, and also from Rubin, who proposed a systematic framework to understand this cohomological tool. In the last years, many new constructions and results around these Euler systems have been obtained, and the aim of this thesis is to look at some of their arithmetic applications towards exceptional zeros, special value formulas, and Eisenstein congruences.The first chapters deal with different kinds of exceptional zero phenomena. The main result we obtain is the proof of a conjecture of Darmon, Lauder and Rotger on special values of the Hida--Rankin p-adic L-function, which may be regarded both as the proof of a Gross--Stark type conjecture, or as the determination of the L-invariant corresponding to the adjoint representation of a weight one modular form. The proof recasts to Hida theory and to the ideas developed by Greenberg--Stevens, and makes use of the Galois deformation techniques introduced by Bellaïche and Dimitrov. We further discuss a similar exceptional zero phenomenon from the Euler system side, leading us to the construction of derived Beilinson--Flach classes. This allows us to give a more conceptual proof of the previous result, using the underlying properties of this Euler system. We also discuss other instances of this formalism, studying exceptional zeros at the level of cohomology classes both in the scenario of elliptic units and diagonal cycles.The last part of the thesis aims to start a systematic study of the Artin formalism for Euler systems. This relies on ideas regarding factorizations of p-adic $L$-functions, and also recasts to the theory of Perrin-Riou maps and the study of canonical periods attached to weight two modular forms. We hope that these results could be extended to different settings concerning the other Euler systems studied in this memoir.
  

• ROCA LACOSTENA, JORDI: The embedding problem for Markov matrices
  
  Author: ROCA LACOSTENA, JORDI
  Thesis link: http://hdl.handle.net/10803/672175
  Programme: DOCTORAL DEGREE IN APPLIED MATHEMATICS
  Department: School of Mathematics and Statistics (FME)
  Mode: Normal
  Reading date: 18/05/2021
  Thesis director: FERNANDEZ SANCHEZ, JESUS | CASANELLAS RIUS, MARTA
  

  Committee:
       PRESIDENT: SUMNER, JEREMY
       SECRETARI: SERRA ALBO, ORIOL
       VOCAL: KUBJAS, KAIE
  Thesis abstract: The goal of this thesis is to solve the embedding problem for Markov matrices, posed by Gustav Elfving in1937. A Markov matrix is non-negative square matrix whose rows sum to 1. We say that such a matrix M is embeddable if it can be written as the exponential of a rate matrix, that is a real square matrix with rows summing to zero and non-negative off-diagonal entries. In this case, we say that the rate matrix is a Markov generator for M. The embedding problem consists deciding whether a given Markov matrix is embeddable or not, thus solving the embedding problem results in giving a characterization of embeddablem matrices.This problem is motivated by Markov processes, which are used to model the change of state of a random variable over time. under the assumption that future is independent from past given the present, that is, the assumption that the substitution probabilities do not depend on previous changes of state. In this framework, the entries of Markov matrices represent the substitution probabilities along a fixed time interval. When these probabilities ar considered to be continuous (and differentiable) functions depending on time, the Markov process is ruled by the instantaneous rates of substitution between states. In order to keep continuous-time Markov processes tractable, the substitution rates are usually assumed to be constant over time. In this case, we say that the process is a homogeneous continuous-time process and one displays all the rates together in a matrix called rate matrix Q. Moreover, the Markov matrix encoding the substitution probabilities after time t can be written as M(t) = exp(Qt) for all t=0. Thus, M(t) is embeddable and Qt is a Markov generator for it. Therefore, in the context of substitution processes, the embedding problem consists on deciding whether a given transicion matrix can be obtained from a homogeneous Markov process in continuous-time.The embedding problem is solved for 2x2 and 3x3 Markov matrices and also for some particular cases such as matrices close to the identity or matrices with dferent real eigenvaleus. In this thesis we present a method to decide wheter a Markov matrix with different eigenvalues (real or not) is embeddable. This solves the embedding problem for a dense subset of nxn Markov matrices for any size n. We also study the rate identifiability problem, which is concerned about the unicity of Markov generators for a given embeddable matrix. Moreover, we give an explicit solution for the embedding problem for 4x4 Markov matrices. Not only this case was the smaller unsolved case, but it has practical consequences in phylogenetics as the embedding problem appears related to fundamental questions concerning the consistency of nucleotide substitution models. These models generally use a Markov process to describe the substitution of nucleotides over time in a given DNA sequence. We give the solution of the embedding problem restricted to some commonly used nucleotide substitution models such as Kimura models and the strand-symmetric model. For each of these models, one can define two different versions related by the embedding problem, a homogeneous continuous-time version and a more general version that avoids the time consideration. We shall see that the proportion of embeddale matrices (i.e. in the continuous-time version of the model) within the general version of the model decreases as the model hypothesis are relaxed. Indeed, we shall see that the proportion of embeddable matrix lies in the range between the 0.05% and the 75% depending on the complexity of the model. This shows that the time-continuous versions of these models are much more restrictive.
  

• RODRÍGUEZ DEL RÍO, ÓSCAR: Ejection-collision orbits in the Restricted Three-Body Problem
  
  Author: RODRÍGUEZ DEL RÍO, ÓSCAR
  Thesis link: http://hdl.handle.net/10803/672338
  Programme: DOCTORAL DEGREE IN APPLIED MATHEMATICS
  Department: Department of Mathematics (MAT)
  Mode: Normal
  Reading date: 06/07/2021
  Thesis director: OLLE TORNER, MERCEDES | SOLER VILLANUEVA, JAUME
  

  Committee:
       PRESIDENT: GOMEZ MUNTANÉ, GERARD
       SECRETARI: CORS IGLESIAS, JOSEP M.
       VOCAL: BARRABÉS VERA, ESTHER
  Thesis abstract: The main objective of this dissertation is the study of the ejection-collision (EC) orbits in the circular and planar Restricted Three Body Problem (RTBP from now on). In particular, we will focus on the analytical and numerical study of a very specific type of EC orbits, that we denote as n-EC orbits. An n-EC orbit is an orbit such that the particle ejects from one primary and reaches n times a relative maximum in the distance with respect to the primary from which it ejected before colliding with it. In this way, we will study numerically in depth this kind of orbits and we will show analytically that for a sufficiently large value of the Jacobi constant (for which we will give an expression in terms of the mass parameter and the value of n) there exist exactly four n-EC orbits with well-defined characteristics. These results generalize and improve the previous results for the particular case of n=1, and we will see that they can be easily extrapolated to the Hill problem. Besides, we will observe numerically that the evolution of these four original families of n-EC orbits present a very rich dynamics.It is well-known that the system that defines the motion of the particle is not well defined at the points where the primaries are located. For this reason, we have used two different techniques to regularize the collision, the McGehee regularization and the Levi-Civita regularization. Thus, in this dissertation we have analyzed the advantages and disadvantages of each regularization and the different methods that can be used to detect collisions. Since this dissertation will be mainly focused on values of the Jacobi constant greater than those associated to the equilibrium point L1, these two local regularizations will be enough. For less restrictive values of the Jacobi constant we will see that there exist other global regularizations or alternatively, we can simply work with local regularizations in a neighbourhood of each primary.On the other hand, from the numerical point of view we have analyzed the global behaviour of the ejection orbits in the RTBP. We have studied the relation between the family of Lyapunov periodic orbits around the equilibrium point L1 and the ejection orbits for values of the Jacobi constant such that the associated Hill regions only allow a bounded motion for these orbits. In particular, we have seen that a chaotic infinity of heteroclinic connections between one primary and the Lyapunov periodic orbits around the equilibrium point L1 are obtained. As a consequence a chaotic infinity of ejection-collision orbits is also derived. Besides, we will see that we can construct colour diagrams that allow to describe the global dynamics of the ejection orbits given a range of time. These colour diagrams provide a very precise understanding of the dynamics of these orbits.Finally, we have made a first exploration of the spatial case of the circular restricted three body problem (RTBP 3D). In this first approach we have not used the classical Kustaanheimo¿Stiefel regularization, instead we have decided to use a 3D version of the McGehee regularization. This presents some problems that we have analyzed and addressed,
  

• TOZZI, CATERINA: A theoretical and computational study of the interaction between biomembranes and curved proteins
  
  Author: TOZZI, CATERINA
  Thesis link: http://hdl.handle.net/10803/672132
  Programme: DOCTORAL DEGREE IN APPLIED MATHEMATICS
  Department: (DECA)
  Mode: Normal
  Reading date: 30/04/2021
  Thesis director: ARROYO BALAGUER, MARINO
  

  Committee:
       PRESIDENT: RANGAMANI, PADMINI
       SECRETARI: CAMPELO AUBARELL, FELIX
       VOCAL: PAGONABARRAGA MORA, IGNACIO
  Thesis abstract: Organelles are the smallest functional parts of eukaryotic cells. Among them, some are membrane-bound such as the nucleus, the endoplasmic reticulum, or the Golgi apparatus, each of them with essential biological functions. In order to accomplish cell functions, membranes enclosing these organelles continuously adapt their shapes through the out-of-equilibrium interaction with macro-molecules, notably proteins.During the life of cells, proteins are main actors in membrane bending dynamics since they have the ability to impinge their curvature onto the membrane, and generate transiently highly curved structures, such as tubes and spherical buds. How proteins can remodel the different organelles has been broadly studied in equilibrium, but a clear understanding of the complex chemo-mechanical problem that drives membrane reshaping out-of-equilibrium is still lacking.In the first Part of the thesis we develop a general theoretical and computational framework for the dynamics of curved proteins adhered to lipid membranes. The theory is based on a nonlinear Onsager's principle, a variational method for irreversible thermodynamics. The resulting governing equations and numerical simulations provide a foundation to understand the dynamics of curvature sensing, curvature generation, and more generally membrane curvature mechano-chemistry, as illustrated by a selection of test cases. We show that continuum modeling is a powerful instrument to describe the protein-membrane interaction. However, this model does not account for the orientational order of proteins and its derivation lacks a microscopic basis.To address these limitations, in the second Part of the thesis we develop a mean-field density functional theory to predict the orientational order and evaluate the free-energy of ensembles of elongated and curved objects, such as BAR proteins, on curved membranes. This kind of protein may adopt different states of orientational order, from isotropic to nematic. The theory is tightly coupled to the microscopic properties of the proteins and explains how a density-dependent isotropic-to-nematic transition is modified by the anisotropic underlying curvature of the membrane. This work lays the ground to understand the interplay between the molecular organization of proteins and the membrane shape dynamics. We explore the coexistence of isotropic and nematic phases on differently curved lipid membranes. We explain, both experimentally and through modelling, how a BAR protein binds on differently curved membrane templates and reshapes them based while modifying their microscopic organization. Our results broaden our understanding of the reshaping dynamics by BAR proteins on mechanically constrained membranes, and provide a framework to understand biological responses involving BAR proteins to membrane-mediated mechanical stimuli.
  

• VELONA, VASILIKI: A study on structure recovery and the broadcasting problem
  
  Author: VELONA, VASILIKI
  Thesis file: (contact the Doctoral School to confirm you have a valid doctoral degree and to get the link to the thesis)
  Programme: DOCTORAL DEGREE IN APPLIED MATHEMATICS
  Department: Department of Mathematics (MAT)
  Mode: Normal
  Reading date: 17/09/2021
  Thesis director: LUGOSI, GABOR | RUE PERNA, JUAN JOSÉ
  

  Committee:
       PRESIDENT: THILIKÓS TOULOUPAS, DIMITRIOS
       SECRETARI: SERNA IGLESIAS, MARIA JOSE
       VOCAL: BEN-HAMOU, ANNA
  Thesis abstract: The thesis contains a study of two problems of combinatorial statistics. The first one is structure learning for partial correlation graphs and the second one is the broadcasting problem on certain families of random recursive trees. In a precise language, the structure recovery problem that we study is the following: given access to individual entries of a covariance matrix S, learn the support of the inverse of S (let us denote this matrix by K) using only a small fraction of S. We call this problem `structure recovery¿ since the zero-entries of K define the adjacency matrix of a graph (called partial correlation graph). As an example of why this is an important graph to learn, consider that S corresponds to a Gaussian random vector (X1, ..., Xn). Then an entry sij is zero if and only if Xi and Xj are independent given the rest of the random variables. A series of algorithms is proposed to address the aforementioned question, assuming that our graphs satisfy certain sparsity conditions. The sparsity that is assumed is related to how much our graph resembles a tree; in particular we deal with trees, graphs of small 2 connected components, and graphs of small treewidth. The proposed algorithms can also be used to estimate K and not only learn the partial correlation graph. Moreover, they can be used to invert any symmetric positive definite matrix since the analysis can be detached from its statistical connection and impact.The motivation for the use of covariance entries is that S might be too large to even store, as it often happens in statistical settings. In fact, our goal is to learn the partial correlation graph using sub-quadratic number of queries, since quadratic time is needed just to write down and store the covariance matrix ¿ this is the starting point for a big part of the literature. The desired complexity bounds are achieved through our analysis.Moving to the second problem under study, we consider a broadcasting process on a graph to be the propagation of a message (let us say a bit value in {0, 1}) from one node to all the rest, possibly corrupted. Our goal is to guess the initial message. We consider that our graph is a tree and is created dynamically in times 0, 1..., n, in a way that at time i the i-th vertex enters the system and attaches with an edge to an existing vertex j (we then write i ~ j). We are interested in the case where i attaches uniformly at random to an existing vertex (uniform attachment) or where i attaches to a vertex with probability proportional to the outdegree of an existing vertex, plus some parameter ß > 0. The broadcasting process we consider is one where vertex 0 (the root) has a bit value that is propagated correctly to its neighbours with probability 1 - q and incorrectly with probability q. The broadcasting problem under study can be formulated in this way: given access to a random tree produced by either uniform attachment or preferential attachment and the bit values of the vertices, but without observing the time labels of the vertices, recover the bit of vertex zero. In a more difficult variant, we answer the same question given only the bits of vertices with outdegree zero (the leaves). In both variants of the problem in both models, we characterize the values of q for which the optimal reconstruction method has a probability of error bounded away from 1/2. We also show that the probability of error is bounded by a constant times q. Two simple reconstruction rules are analyzed in detail. One of them is the simple majority vote, the other is the bit value of the centroid of the tree (or the closest leaf to the centroid). We also analyze a third reconstruction rule which is more complex but works for all q where reconstruction is theoretically possible.
  

• VILA PÉREZ, JORDI: Low and high-order hybridised methods for compressible flows
  
  Author: VILA PÉREZ, JORDI
  Thesis link: http://hdl.handle.net/10803/671889
  Programme: DOCTORAL DEGREE IN APPLIED MATHEMATICS
  Department: School of Mathematics and Statistics (FME)
  Mode: Normal
  Reading date: 10/02/2021
  Thesis director: HUERTA CEREZUELA, ANTONIO | GIACOMINI, MATTEO
  

  Committee:
       PRESIDENT: PERAIRE GUITART, JAIME
       SECRETARI: FERNANDEZ MENDEZ, SONIA
       VOCAL: HASSAN, OUBAY
       VOCAL: SHERWIN, SPENCER
       VOCAL: REBAY, STEFANO
  Thesis abstract: The aerospace community is challenged as of today for being able to manage accurate overnight computational fluid dynamics (CFD) simulations of compressibleflow problems. Well-established CFD solvers based on second-order finite volume (FV) methods provide accurate approximations of steady-state turbulent flowsbut are incapable to produce reliable predictions of the full flight envelope. Alternatively, promising high-order discretisations, claimed to permit feasible highfidelitysimulations of unsteady turbulent flows, are still subject to strong limitations in robustness and efficiency, placing their level of maturity far away fromindustrial requirements. In consequence, the CFD paradigm is immersed at this point into the crossroads outlined by the inherent limitations of low-order methodsand the yet immature state of high-order discretisations. Accordingly, this thesis develops a twofold strategy for the high-fidelity simulation of compressible flowsintroducing two methodologies, at the low and high-order levels, respectively, based on hybridised formulations.First, a new finite volume paradigm, the face-centred finite volume (FCFV) method, is proposed for the formulation of steady-state compressible flows. Thepresent methodology describes a hybrid mixed FV formulation that, following a hybridisation process, defines the unknowns of the problem at the face barycentres.The problem variables, i.e. the conservative quantities and the stress tensor and heat flux in the viscous case, are retrieved with optimal first-order accuracyinside each cell by means of an inexpensive postprocessing step without need of reconstruction of the gradients. Hence, the FCFV solver preserves the accuracyof the approximation even in presence of highly stretched or distorted cells, providing a solver insensitive to mesh quality. In addition, the FCFV method is amonotonicity-preserving scheme, leading to non-oscillatory approximations of sharp gradients without resorting to shock capturing or limiting techniques. Finally,the method is robust in the incompressible limit and is capable of computing accurate solutions for flows at low Mach number without the need of introducingspecific pressure correction strategies.In parallel, the high-order hybridisable discontinuous Galerkin (HDG) method is reviewed in the context of compressible flows, presenting an original unifiedframework for the derivation of Riemann solvers in hybridised formulations. The framework includes, for the first time in an HDG context, the HLL and HLLEMRiemann solvers as well as the traditional Lax-Friedrichs and Roe solvers. The positivity preserving properties of HLL-type Riemann solvers are displayed,demonstrating their superiority with respect to Roe in supersonic cases. In addition, HLLEM specifically outstands in the approximation of boundary layers becauseof its shear preservation, which confers it an increased accuracy with respect to HLL and Lax-Friedrichs.An extensive set of numerical benchmarks of practical interest is introduced along this study in order to validate both the low and high-order approaches.Different examples of compressible flows in a great variety of regimes, from inviscid to viscous laminar flows, from subsonic to supersonic speeds, are presentedto verify the accuracy properties of each of the proposed methodologies and the performance of the introduced Riemann solvers.
  

Last update: 16/10/2021 05:11:51.

Research groups

• ACES-Advanced Control of Energy Systems
• ALBCOM-Algorithms, Computational Biology, Complexity and Formal Methods
• CoDAlab-Control, Dynamics and Applications
• COMBGRAPH-Combinatorics, Graph Theory and Applications
• DAMA-UPC-Data Management Group
• DCG-Discrete and Combinatorial Geometry
• gAGE-Research Group of Astronomy and Geomatics
• GEOMVAP-Geometry of Manifolds and Applications
• GIES-Geophysics and Earthquake Engineering
• GNOM-Mathematical Optimization Group
• GRAA-Research Group in Applied Analysis
• GREC-Knowledge Engineering Research Group
• GRHCT-Group of Research on History of Science and Technology
• GRTJ-Game Theory Research Group
• INSIDE-Innovation in Systems for Engineering Design and Training
• KEMLG-Knowledge Engineering and Machine Learning Group
• L'AIRE-Laboratory of Aeronautical and Industrial Research and Studies
• LACAN - UPC-Numerical Methods for Applied Sciences and Engineering
• LARCA-Laboratory of Relational Algorithmics, Complexity and Learnability
• MAK-Mathematics Applied to Cryptography
• SAC-Advanced Control Systems
• SARTI-MAR-Sistemes d'Adquisicio Remota de dades i Tractament de la Informacio en el Medi Mari
• TN-Number Theory Research Group
• VIS-Artificial Vision and Intelligent Systems

Teachers

• Alberich Carramiñana, Maria
• Alvarez Montaner, Josep
• Amoros Torrent, Jaume
• Arias Vicente, Irene
• Arroyo Balaguer, Marino
• Balbuena Martinez, M.Camino
• Baldoma Barraca, Inmaculada
• Barja Yañez, Miguel Angel
• Batlle Arnau, Carles
• Burillo Puig, Pep
• Cabre Vilagut, Xavier
• Carmona Mejias, Angeles
• Casanellas Rius, Marta
• Comellas Padro, Francesc de Paula
• Delshams i Valdes, Amadeu
• Diez Mejia, Pedro
• Encinas Bachiller, Andres Marcos
• Fabrega Canudas, Josep
• Farre Cirera, Rafel
• Fernandez Mendez, Sonia
• Fernandez Sanchez, Jesus
• Ferrer Biosca, Alberto
• Ferrer Llop, Josep
• Fiol Mora, Miguel Angel
• Franch Bullich, Jaume
• Freixas Bosch, Josep
• Garcia Planas, Maria Isabel
• Gargallo Peiró, Abel
• Giacomini, Matteo
• Gimenez Pradales, Jose Miguel
• Gracia Sabate, Xavier
• Guardia Munarriz, Marcel
• Guillamon Grabolosa, Antoni
• Herranz Sotoca, Javier
• Huerta Cerezuela, Antonio
• Huguet Casades, Gemma
• Lario Loyo, Joan C.
• Lazaro Ochoa, Jose Tomas
• Magret Planas, M. Dolors
• Martí Farré, Jaume
• Martin de la Torre, Pau
• Martinez-Seara Alonso, Maria Teresa
• Masdemont Soler, Josep Joaquim
• Miranda Galceran, Eva
• Mitjana Riera, Margarida
• Molina Blanco, Santiago
• Mora Gine, Mercè
• Morillo Bosch, Maria Paz
• Muñoz Lecanda, Miguel Carlos
• Muñoz Romero, Jose Javier
• Myers, Timothy
• Noy Serrano, Marc
• Olle Torner, Maria Mercedes
• Palacios Quiñonero, Francisco
• Pascual Gainza, Pere
• Pozo Montero, Francesc
• Puente del Campo, Maria Albina
• Quer Bosor, Jordi
• Ramirez Ros, Rafael
• Rio Doval, Anna
• Rodellar Benede, Jose
• Rodriguez Ferran, Antonio
• Roman Roy, Narciso
• Rotger Cerdà, Victor
• Rubio Massegu, Josep
• Rue Perna, Juan Jose
• Saez Viñas, Pablo
• Sainz Sapera, Luis
• Sanchez Soler, Mónica
• Sarrate Ramos, Jose
• Serra Albo, Oriol
• Silveira Isoba, Rodrigo Ignacio
• Sola-Morales Rubio, Joan
• Soler Villanueva, Jaume
• Susin Sanchez, Antonio
• Vazquez, Mariano
• Ventura Capell, Enric
• Vidal Segui, Yolanda
• Villar Santos, Jorge Luis

Trepat, Xavier

Dalfó, Cristina

Gomis, Joaquim

Valdinoci, Enrico

Chinesta, Francisco José

Kurz , Sascha

Larraia, Enrique

Gónzalez, Maria del Mar

Cajas, Juan Carlos

Streng, Marco

López, Susana Clara

Gómez, Gerard

Alvarez, Francesc

Owen, Herbert

Crespo, Maria Teresa

Monleon, Antonio

Fleixas, Gerard

Lugosi, Gabor

Agell, Nuria

Logares, Ramiro

Bose, Prosenjit

Aguado, Jazmín

Fernández, Ramsés

Guitart, Xavier

Ruiz, Eloi

Borrell, Richard

Lehmkuhl, Oriol

Badia, Santiago