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

• BELDA FERRÍN, GUILLEM: Conformal n-dimensional bisection for local refinement of unstructured simplicial meshes
  
  Author: BELDA FERRÍN, GUILLEM
  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: (DECA)
  Mode: Normal
  Deposit date: 14/09/2022
  Reading date: 28/10/2022
  Reading time: 11:00
  Reading place: FME_ Sala d'Actes de l'FME, Edifici U, Campus Sud
  Thesis director: ROCA NAVARRO, FRANCISCO JAVIER | RUIZ GIRONÉS, ELOI
  Committee:
       PRESIDENT: LEDOUX, FRANK
       SECRETARI: SARRATE RAMOS, JOSE
       VOCAL: CUETO FELGUEROSO, LUIS
  Thesis abstract: In n-dimensional adaptive applications, conformal simplicial meshes must be locally modified. One systematic local modification is to bisect the prescribed simplices while surrounding simplices are bisected to ensure conformity. Although there are many conformal bisection strategies, practitioners prefer the method known as the newest vertex bisection. This method guarantees key advantages for adaptivity whenever the mesh has a structure called reflectivity. Unfortunately, it is not known (i) how to extract a reflection structure from any unstructured conformal mesh for three or more dimensions. Fortunately, a conformal bisection method is suitable for adaptivity if it almost fulfills the newest vertex bisection advantages. These advantages are almost met by an existent multi-stage strategy in three dimensions. However, it is not known (ii) how to perform multi-stage bisection for more than three dimensions. This thesis aims to demonstrate that n-dimensional conformal bisection is possible for local refinement of unstructured conformal meshes. To this end, it proposes the following contributions. First, it proposes the first 4-dimensional two-stage method, showing that multi-stage bisection is possible beyond three dimensions. Second, following this possibility, the thesis proposes the first n-dimensional multi-stage method, and thus, it answers question (ii). Third, it guarantees the first 3-dimensional method that features the newest vertex bisection advantages, showing that these advantages are possible beyond two dimensions. Fourth, extending this possibility, the thesis guarantees the first n-dimensional marking method that extracts a reflection structure from any unstructured conformal mesh, and thus, it answers question (i). This answer proves that local refinement with the newest vertex bisection is possible in any dimension. Fifth, this thesis shows that the proposed multi-stage method almost fulfills the advantages of the newest vertex bisection. Finally, to visualize four-dimensional meshes, it proposes a simple tool to slice pentatopic meshes. In conclusion, this thesis demonstrates that conformal bisection is possible for local refinement in two or more dimensions. To this end, it proposes two novel methods for unstructured conformal meshes, methods that will enable adaptive applications on n-dimensional complex geometry.
  
• MONSÓ BURGUÉS, ENRIQUE PEDRO JAIME: Discrete operators and distances on subdivision networks
  
  Author: MONSÓ BURGUÉS, ENRIQUE PEDRO JAIME
  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: 27/07/2022
  Reading date: 20/10/2022
  Reading time: 11:00
  Reading place: FME Sala d'actes. Edifici U Campus SUD
  Thesis director: CARMONA MEJIAS, ANGELES | MITJANA RIERA, MARGARIDA
  Committee:
       PRESIDENT: MORO CARREÑO, JULIO
       SECRETARI: JIMÉNEZ JIMÉNEZ, M. JOSÉ
       VOCAL: ABIAD MONGE, AIDA
  Thesis abstract: The research we have carried out lies within the domain of diffusion problems for elliptic operators that are defined on discrete structures called networks, that are graphs that are capable of discriminating between edges and between vertices per se as well.Typically elliptic problems are posed in the continuum setting, defined on surfaces or manifolds and are a corner stone of applied mathematics. Partial differential equations and the geometry of the domain where they are to be satisfied, as well as boundary conditions that a solution must fulfill are the ingredients of this critical tool in the development of mathematics for more than the last two hundred years. We are concerned with elliptical problems that refer to the discrete case. Diffusion in a discrete network is understood as the distribution of a physical entity between all different adjacent vertices of a given vertex. Although diffusion is conceived as a local phenomena that becomes globally widespread, our work is situated in a model that permits adjacencies with vertices that are not only in the spatial vicinity but can be situated far away, at the other end of the structure. Therefore allows an idea of diffusion that overcomes the classical concept. To carry out our research we have applied techniques of the so called Discrete Potential Theory. So the first framework where our work fits in is that of discrete boundary value problems. Specifically we have dealt with solving Poisson problems where the elliptical operator is the combinatorial Laplacian, the normalized Laplacian and Schrödinger type operators as well. The elementary graph operation of subdivision of an edge and, by extension, subdivision of all edges is very useful for deciding planarity of a given graph. But at the same time it can be intimately and undeniably related with the series connection of two elements in an electrical circuit. In our work we have developed a new concept of electrical subdivision (two versions) and we have analyzed the behavior of different diffusion problems in these contexts. Specifically, our main interest has been focused on relating the solution of a DBVP posed on an electrically subdivided network to the solution of another DBVP, related to the previous one, and posed on the pre-existing network, before the subdivision process has been applied. So, the main results obtained determine the relationship between the Green operators associated with an elliptical discrete difference operator that corresponds to a subdivided network and the Green operator associated with the same operator but corresponding to the initially given network. A second target that has been set ourselves, and on which we have been able to find nice results is the analysis of the effective resistances and which is the relationship that is stablished between effective resistances associated with a subdivision network and effective resistances corresponding to the network without being electrically subdivided. Finally a third main objective has been the study of the well known Kirchhoff index as a topologycal network invariant. We have managed to obtain the existing relation between Kirchhoff indices of an initial given network and an electrically compatible subdivision network obtained from it. Since both, the considered elliptic discrete operators that we have taken into account but their associated Green operators as well can be represented by matrices, a possible lecture of our results can be set in the field of computing generalized inverses matrices. Specifically we have found out entrywise expressions of Moore-Penrose inverse matrices of high-dimensional matrices in terms of the contents of Moore-Penrose matrices of smaller matrices. In this sense our results may be related to the well-known Sherman-Morrison-Woodbury expressions.
  

Last update: 06/10/2022 04:45:23.

List of lodged theses

• GIRALT MIRON, MAR: Homoclinic and chaotic phenomena to L3 in the Restricted 3-Body Problem
  
  Author: GIRALT MIRON, MAR
  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: 04/10/2022
  Reading date: pending
  Reading time: pending
  Reading place: pending
  Thesis director: BALDOMA BARRACA, INMACULADA CONCEPCION | GUARDIA MUNARRIZ, MARCEL | BALDOMA BARRACA, INMACULADA CONCEPCION | GUARDIA MUNARRIZ, MARCEL
  Committee:
       PRESIDENT: FONTICH JULIA, ERNEST
       SECRETARI: MARTINEZ-SEARA ALONSO, MARIA TERESA
       VOCAL: NIEDERMAN, LAURENT JOSUÉ
  Thesis abstract: The Restricted 3-Body Problem models the motion of a body of negligible mass under the gravitational influence of two massive bodies, called the primaries. If the primaries perform circular motions and the massless body is coplanar with them, one has the Restricted Planar Circular 3-Body Problem (RPC3BP). In synodic coordinates, it is a two degrees of freedom autonomous Hamiltonian system with five critical points, L1,¿...,L5, called the Lagrange points. The Lagrange point L3 is a saddle-center critical point which is collinear with the primaries and is located beyond the largest one. This thesis focuses on the study of the one dimensional unstable and stable manifolds associated to L3 and the analysis of different homoclinic and chaotic phenomena surrounding them. We assume that the ratio between the masses of the primaries is small. First, we obtain an asymptotic formula for the distance between the unstable and stable manifolds of L3. When the ratio between the masses of the primaries is small the eigenvalues associated with L3 have different scales, with the modulus of the hyperbolic eigenvalues smaller than the elliptic ones. Due to this rapidly rotating dynamics, the invariant manifolds of L3 are exponentially close to each other with respect to the mass ratio and, therefore, the classical perturbative techniques (i.e. the Poincaré-Melnikov method) cannot be applied. In fact, the formula for the distance between the unstable and stable manifolds of L3 relies on a Stokes constant which is given by the inner equation. This constant can not be computed analytically but numerical evidences show that is different from zero. Then, one infers that there do not exist 1-round homoclinic orbits, i.e. homoclinic connections that approach the critical point only once. The second result of the thesis concerns the existence of 2-round homoclinic orbits to L3, i.e. connections that approach the critical point twice. More concretely, we prove that there exist 2-round connections for a specific sequence of values of the mass ratio parameters. We also obtain an asymptotic expression for this sequence. In addition, we prove that there exists a set of Lyapunov periodic orbits whose two dimensional unstable and stable manifolds intersect transversally. The family of Lyapunov periodic orbits of L3 has Hamiltonian energy level exponentially close to that of the critical point L3. Then, by the Smale-Birkhoff homoclinic theorem, this implies the existence of chaotic motions (Smale horseshoe) in a neighborhood exponentially close to L3 and its invariant manifolds. In addition, we also prove the existence of a generic unfolding of a quadratic homoclinic tangency between the unstable and stable manifolds of a specific Lyapunov periodic orbit, also with Hamiltonian energy level exponentially close to that of L3.
  

Last update: 06/10/2022 04:30:28.

List of defended theses by year

• CÓRDOBA PAÑELLA, PAULA: Local Preconditioning for Parallel Iterative Solvers
  
  Author: CÓRDOBA PAÑELLA, PAULA
  Thesis link: http://hdl.handle.net/10803/674737
  Programme: DOCTORAL DEGREE IN APPLIED MATHEMATICS
  Department: (DECA)
  Mode: Normal
  Reading date: 22/04/2022
  Thesis director: HOUZEAUX, GUILLAUME PASCAL
  

  Committee:
       PRESIDENT: MOULINEC, CHARLES
       SECRETARI: FERNANDEZ MENDEZ, SONIA
       VOCAL: MAGOULÈS, FRÉDÉRIC
  Thesis abstract: This thesis aims at improving the convergence of iterative solvers, used for algebraic systems coming from the discretization of partial differential equations (PDE), in the context of large scale simulations and high performance computing (HPC). The methodology followed consists in adapting some existing preconditiong techniques to the physics and numerics of convection-dominated transport and boundary layer problems in flows.For convection-dominated flows, a physics-based permutation algorithm is presented, which consists in renumbering the mesh in the direction of convection. This renumbering is then used together with a Gauss-Seidel preconditioner to propagate the result of the matrix-vector products along the convection. The robutsness and effectiveness of this preconditioner is proved in several test cases solving the heat equation as well as the Navier-Stokes equations in both sequential and in parallel using the Message Passing Interface library MPI.Additionally, the composition of preconditioners is proposed to solve cases where different local physical behaviors co-exist in the same flow. In particular, we focus on such problems where of a highly convective flow encounters an obstacle. Such problems involve a zone with high convection far from the obstacle and the development of a boundary layer in the vicinity of the obstacle. In numerical terms, these local behaviors translate into specific matrix structures that we will take advantage of to adapt the preconditioner locally. On the one hand, the linelet preconditioner is a well-known efficient preconditioner for boundary layers where the mesh is highly anisotropic, in particular to solve the Poisson equation. On the other hand, the streamline linelet that we propose in this thesis (Gauss-Seidel together with a mesh renumbering in the convection direction) is well adapted for locally hyperbolic flows. Both preconditioners will be composed (combined) in different ways to investigate their robustness in terms of convergence as well as their costs to solve the proposed transport problems. We will study as well their performances in terms of parallelization.
  

• HERNÀNDEZ I BARRIOS, VÍCTOR: p-adic L-functions, p-adic Gross-Zagier formulas and plectic points
  
  Author: HERNÀNDEZ I BARRIOS, VÍCTOR
  Thesis link: http://hdl.handle.net/10803/674193
  Programme: DOCTORAL DEGREE IN APPLIED MATHEMATICS
  Department: Department of Mathematics (MAT)
  Mode: Normal
  Reading date: 19/04/2022
  Thesis director: ROTGER CERDÀ, VICTOR | MOLINA BLANCO, SANTIAGO
  

  Committee:
       PRESIDENT: MASDEU SABATÉ, MARC
       SECRETARI: QUER BOSOR, JORDI
       VOCAL: FORNEA, MICHELE
  Thesis abstract: In this work we generalize the construction of p-adic anticyclotomic L-functions associated to an elliptic curve E/F and a quadratic extension K/F, by defining a measure µ_f^p attached to K/F and an automorphicform. In the case of parallel 2, the automorphic form is associated with an elliptic curve E/F.The first main result is a p-adic Gross-Zagier formula: if E has split multiplicative reduction at p and p does not split at K/F, we compute the first derivative of the p-adic L-function by relating it with the conjugate difference of a Darmon point twisted by a character ¿. The proof uses the reciprocity mapprovided by class field theory as a natural way to interpret conjugate differences of points in E(Kp) aselements in the augmentation ideal for the aluation at the character ¿. This generalizes a result ofBertolini and Darmon. With a similar argument, after discovering the work of Fornea and ehrmann onplectic points, we prove an exceptional zero formula which relates a higher order derivative of In this work we generalize the construction of p-adic anticyclotomic L-functions associated to an elliptic curve E/F and a quadratic extension K/F, by defining a measure µ_f^p attached to K/F and an automorphic form. In the case of parallel 2, the automorphic form is associated with an elliptic curve E/F. The first main result is a p-adic Gross-Zagier formula: if E has split multiplicative reduction at p and p does not split at K/F, we compute the first derivative of the p-adic L-function by relating it with the conjugate difference of a Darmon point twisted by a character ¿. The proof uses the reciprocity map provided by class field theory as a natural way to interpret conjugate differences of points in E(Kp) as elements in the augmentation ideal for the evaluation at the character ¿. This generalizes a result of Bertolini and Darmon. With a similar argument, after discovering the work of Fornea and Gehrmann on plectic points, we prove an exceptional zero formula which relates a higher order derivative of µ_f^S with plectic points. We find an interpolating measure µ_F^p for µ_f^p attached to an interpolating Hida family F for f. Here µ_F^p can be regarded as a two variable p-adic L-function, which now includes the weight as a variable. Then we define the Hida-Rankin p-adic L-function Lp(f^p, ¿, k) as the restriction of µ_F^p to the weight space. Finally, we prove a formula which relates the weight-leading term of Lp(f^p, ¿, k) with plectic points. In short, the leading term is an explicit constant times Euler factors times the logarithm of the trace of a plectic point. This formula is a generalization of a result of Longo, Kimball and Hu, which has been used to prove the rationality of a Darmon point under some hypotheses.
  

• VENTURINI, CHIARA: Multiscale clutch models for cell mechanics
  
  Author: VENTURINI, CHIARA
  Thesis link: http://hdl.handle.net/10803/675004
  Programme: DOCTORAL DEGREE IN APPLIED MATHEMATICS
  Department: (DECA)
  Mode: Normal
  Reading date: 22/07/2022
  Thesis director: SAEZ VIÑAS, PABLO
  

  Committee:
       PRESIDENT: GARCÍA AZNAR, JOSÉ MANUEL
       SECRETARI: MUÑOZ ROMERO, JOSE JAVIER
       VOCAL: SUNYER BORRELL, RAIMON
  Thesis abstract: Integrin-based cell adhesion is a key mechanism in fundamental biological processes such as cell migration and wound healing, and in diseases like osteoporosis and cancer. For example, during migration, cells form adhesion complexes as they move, exerting tractions on the extracellular matrix (ECM), which enable the formation of protrusions. So far experimental research has demonstrated the role of ligand spacing and substrate rigidity in adhesion dynamics. Moreover, some mathematical models have provided fundamental understanding in cell adhesion. Specifically, the Clutch Model has been very successful in explaining how cells can sense force and consequently, how they respond to substrate rigidity. To exploit its potential, we have used it to address single cell durotaxis, the migration of a cell due to a stiffness gradient in the substrate. We provide a mechanistic rationale of durotaxis by integrating continuum models of cell migration with the stochastic clutch model. We show that a gradient in the ECM stiffness activates an asymmetric intracellular retrograde flow during the initial cell spreading. The competition between this flow and the polymerization velocity at the cell membrane, creates a polarized state that establishes cell migration directionality. Our theoretical framework confirms previous experimental observations and rationalizes why some cell types follow positive stiffness gradients, while others move toward softer region.Unfortunately, current clutch models also present some limitations. Adhesion complexes present a number of different adaptor proteins, and are able to link the actomyosin cortex in the interior of the cell to the ECM. The study of their composition and function has shown that the integrin-talin-actin chain serves as a mechanosensor and mechanotransducer of the external and internal forces. Current models simplify the composition and organization of the adhesion complex and use a phenomenological approach to reproduce cell scale variables. However, they fail in modelling the sub-scale molecular behaviour. These simplifications have limited our mechanistic understanding of cell adhesion. Therefore, we propose a multiscale clutch model that, from the physical and biological laws of the cell components, reproduces the mechanics of an adhesion complex as well as the dynamics of the subcellular molecules. Because the ligands distance is fundamental for cell adhesion mechanics, we include a space-dependent ligand distribution and compute the displacement of the substrate point-wise thanks to Green¿s functions. We also implement a detailed description of the talin rod. Given the importance of the recruitment of adaptor proteins in an adhesion complex, we implement the presence of vinculin binding sites (VBSs) and actin binding sites (ABSs) in the talin rod. Our results reproduce previous cell scale variables, but also reproduce the behavior of single adhesion molecules within the adhesion complex. Moreover, our results suggest that cell traction is mostly dependent on the complex adhesion size rather than ligand density. Our computational framework also allows us to easily manipulate VBSs and ABSs and analyze the effect of their depletion in cell adhesion mechanics. Our computational results reproduce experimental data of mouse embryonic fibroblasts, both with full-length talin and with different combinations of talin domains depletion, showing how the traction force changes with respect to the talin configuration. We believe that our detailed multi-scale model represents an important step forward in the understanding of cell adhesion, which can be further extended to account for other adhesion molecules and, e.g. , for the study of adhesion maturation. The multi-scale durotaxis model can be also used in a wider biological context, to engineer better biomimetic tissue constructs or to propose strategies for arresting tumor invasion.
  

• WÖTZEL, MAXIMILIAN: Probabilistic and Extremal studies in Additive Combinatorics
  
  Author: WÖTZEL, MAXIMILIAN
  Thesis link: http://hdl.handle.net/10803/673625
  Programme: DOCTORAL DEGREE IN APPLIED MATHEMATICS
  Department: Department of Mathematics (MAT)
  Mode: Normal
  Reading date: 26/01/2022
  Thesis director: RUE PERNA, JUAN JOSÉ | SERRA ALBO, ORIOL
  

  Committee:
       PRESIDENT: PERSON, YURY
       SECRETARI: PERARNAU LLOBET, GUILLEM
       VOCAL: CANDELA POKORNA, PABLO
  Thesis abstract: using probabilistic means. The topics in this thesis all lie in the intersection of these three areas, and concern the following problems. Integer solutions to systems of linear equations. Recent years have seen a lot of developments regarding threshold results for certain integer solutions to an arbitrary given system of linear equations, answering the question when one expects the binomial random subset of an initial segment of integers to contain solutions almost surely. The next logical question is the following. Suppose we are in the range that there will exist integer solutions in the binomial random set, how are these solutions distributed? In Chapter 1, we will take steps towards answering this question by providing sufficient conditions for when a wide variety of solutions follow a normal distribution. We will also discuss how in certain cases, these sufficient conditions are also necessary. Sets with bounded sumset. What can be said about the structure of two finite sets in an abelian group if their Minkowski sum is not much larger than the sets themselves? A classical result by Kneser says that this can happen if and only if the Minkowski sum is periodic with respect to a proper subgroup. In Chapter 3 we are going to establish two types of results. First, we are going to establish so-called robust versions of classical theorems of Kneser and Freiman. Robust here refers to the fact that instead of asking for structural information on the constituent sets with the knowledge that their sumset is small, we only require that this holds for a large subset. The second part of the chapter concerns the typical structure of pairs of sets with small Minkowski sum, that is, what if we only want to give a structural statement for almost all pairs of sets with a sumset of a given size? We give an approximate structure theorem that holds for almost the complete range of possible sumset sizes. Sidon set systems. Classical questions regarding Sidon sets are to determine their maximum size or to know when a random set is a Sidon set. In Chapter 4 we generalize the notion of Sidon sets to set systems and establish the corresponding bounds for these two questions. We also prove a so-called relative density result conditional on a conjecture on the specific structure of maximal Sidon systems. Independent sets in hypergraphs. The method of hypergraph containers is a general tool that can be used to obtain results on the number and structure of independent sets in hypergraphs. The connection with Additive Combinatorics appears because many additive problems can be encoded as studying independent sets in hypergraphs. In Chapter 2 we are going to establish an extension of a recent asymmetric version of the container method due to Morris, Samotij and Saxton to multipartite hypergraphs. This extension is one of the main ingredients in the proof of the typical structure theorem proved in Chapter 3.
  

Last update: 06/10/2022 05:01:04.

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
• MAPTHE-Matrix Analysis and Discrete Potential Theory
• 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