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Mención Internacional en el título de doctor SUMMARY, REMARKS AND FUTURE WORK. Culture is one of the classical examples that is normally used to define a complex system. The inherent complexity of humans interacting directly provides enough diversity of phenomena, but if we also consider sociological problems that live in wider timescales, then it becomes boundless. The subject of the interaction is no more the person alone, but the different social constructs that coexist within a society, as, for example, different kind of social norms and expressions. The growing complexity of social systems is also confusing because it has not been traditionally understood in this way, which is logical because we have studied society from the different perspectives that have appeared historically. Humanities have studied the different holistic products that appear from complexity, like literature, philosophy or the different types of artistic studies. As we stated in the introduction, a different perspective appeared when the scientific method started comparative studies of society, bringing new tools, mainly modelling of general effects and statistics. Herbert Spencer popularized the analogy of considering the society as a human being, this is, as a cooperative organization of the different organs and tissues in order to provide a collective structure. And this is the main spirit of complexity sciences, which started to have more importance in the end of the XX century, giving rise to a new wave on how to understand quantitatively social systems. At this moment, we recall the title of this thesis: Behavior and context. The problem in the description of the context is a methodological one. Context is the conjunction of relevant circumstances correlated with the social description of an individual. Traditionally, the social description of a variable has been related with socioeconomic indicators, where the presence of the social structure is only understood with scalar variables. In this work, we show the importance of the network structure in the resemblance of this social profile, and how geometry may act as a proxy of this social variables containing, potentially, more information than just scalar variables. The behavior is correlated with the geometry of this social profile. The importance of the network version of the social profile can be checked by its correlation with the cultural variables of the ego. This is important because it reveals an idiosyncratic manner of making social relationships, and, therefore, the role of social institutions. But the network version is not only a proxy for cultural variables, it is a true identifier of people. A proof of this is the connection between personal profiles and probability of connection, showing that the similarity in the close structural network may be a proxy for social homophily, connected, therefore, to personal and cultural values. Therefore, we have identified a unit component (in terms of measure) of the individual, social and cultural influence on individuals, providing a personal signature around which social homophily appears, even in a complex context where culture and social values play a key role. At the end of the day, this thesis could be summarized by saying that the geometric structure of the close social environment is a signature of the individual and of its social and cultural values. Similarity in this signature is related to social homophily. The other main message that can be extracted from this thesis is related to the importance of ethnic groups in social modelling. Social modelling can not be replicated easily with a markovian model with binary interactions. This is due to the inherent complexity that was pointed out at the beginning of the chapter. Social phenomena exist in several timescales, and because of this they have a lot of implicit information. An ethnic group may exist because of this implicit information, due to, for example, a pre-existing social symbolism; but, in order to arise spontaneously from the social outcome of a binary process (in the way game theory works), some optimization is needed. This is, the existence of a marker does not provide, necessarily, a social separation depending on that variable, if this variable does not provide some kind of improvement in the fitness. This advantage may exist due to an asymmetrical social configuration, for example, in the case of a hierarchy. Previous works have shown the spontaneous appearance of this kind of structures [102], remarking again the importance of the different timescales in this kind of phenomena. An asymmetric social configuration allows the function to be optimized with ethnic markers, and the arena where cultures may compete, reinforcing the roles of their individuals, and building therefore, the different social profiles. With respect to the experimental part, the results that we obtain suggest that ethnic markers experiment should be replicated in contexts with different social configurations, studying, again, as the title suggests, the dependence on the context. It would be interesting the replication of simple ethnic separation altogether with a hierarchy or some reputation system. Once more, following the spirit of all the thesis, connects the importance of the ethnic, personal identifications with the social organization, that regulates its importance in the social equilibrium. In view of these conclusions, future works in this field should go deeper into the importance of the geometric social profile of the ego networks as an individual identifier. In this case, more data needs to be collected, in order to research other sociological variables like different aspects of culture, religion, political views or socioeconomic measures. Note that in the case of the present work, we have focused on the nationality, as the dataset provided us with a diverse sample of nationalities and the control group. In order to study other sociological variables, like wealth, race, gender or a particular social norm, new data is needed that fits those necessities. Moreover, we have focused on networks centered in the ego, where we have a perspective on the close social structure. It would be also interesting to study how much information is needed in order to study different kinds of networks. This is interesting from a practical point of view, as it could help to use different datasets to provide further proofs of the existence of the social signature, but also to check more properties of this measure. It would be interesting, for example, to study if family networks provide us enough information to talk about the different social values and/or material conditions of different families. If the information about the signatures still can be extracted, new datasets could be reused to make a stronger point about the cultural signature, saving the time of collecting new ego networks. Another possible path to follow in the future could be the study of signed networks together with the cultural profile. In our works, we have studied the importance of the social signature in order to determine the cultural profile, and its importance in the outcome of a relationship. It would be interesting to show the connection between these two elements, providing a relationship between cultural or socioeconomic variables and the friends or enemies an individual may have. The final suggestion for future work is related to the measure of properties of this social signature.We have stressed in the introduction (and in several chapters of the thesis) that the cultural values are a holistic product, which may be assumed and created by collectivities. In this work we have focused on a model that is based in the relationship between individual and its environment (in a mean-field like approach), but this is not needed. Considering again the example of the family, perhaps the family could be understood as a whole, and study the problem of its close social environment from a perspective of networks. Given that the problem of culture exists in several scales, it would be interesting to check if new relationships arise from this point of view. At the end of the day, all of these suggestions are basically related to checking the potential of the measure. This poses also an ethical question: if the potential of the social signature is shown in more contexts, and it correlates with personal information, it is important that these studies (public or private) are carried with the utmost concert about their possible implications and misuses. Programa de Doctorado en Ingeniería Matemática por la Universidad Carlos III de Madrid Presidente: Jesús Gómez Gardeñes.- Secretario: Alberto Antonioni.- Vocal: Andreia Sofía Monteiro Teixeira

Mención Internacional en el título de doctor This doctoral thesis was supported through the FPI contract BES-2017-079755 associated to the grant FIS2016-78313-P from Ministerio de Ciencia e Innovación (MCIN)/Agencia Estatal de Investigación (AEI)/ 10.13039/501100011033/ and FEDER Una manera de hacer Europa. Additionally the author also acknowledge finantial support from MCIN/AEI/10.13039/501100011033 through grant BADS, no. PID2019-109320GB-100 Programa de Doctorado en Ingeniería Matemática por la Universidad Carlos III de Madrid Presidente: Luis Guillermo Morelli.- Secretaria: María Pilar Guerrero Contreras.- Vocal: Vicente Mariscal Romero

Programa de Doctorado en Ingeniería Matemática por la Universidad Carlos III de Madrid Presidente: José Navarro Salas.- Secretario: Salvador José Robles Pérez.- Vocal: Iñaki Garay Elizondo

Mención Internacional en el título de doctor Online web services have grown dramatically in size and diversity in the last years, becoming essential components of our daily life and allowing us to conduct elementary tasks like working, getting informed, or keeping in contact with relatives and friends. However, all the changes and evolution experimented on by the online web services had not have been possible without implementing a profitable economic model that sustains it. Despite a suitable percentage of these services being fee-based, they represent a lucrative business that generates billions of dollars, allowing the creation of some of the biggest companies in the world in terms of market capitalization, like Alphabet Inc. or Meta Inc. (Previously known as Facebook Inc.). Being costless and lucrative is possible due to an advertising-based monetization model, which consists of delivering ads to the users in exchange for their services (e.g., Facebook or YouTube). Although online advertising dates back to the middle of the 90s, its popularity has experienced an increase among brands and advertising agencies in the last decade, mainly due to its capacity to reach precise audiences at a low cost. Converting online web services into advertising walls is a double-edged sword for the users. The capacity offered by online advertising to segment their audiences requires a massive collection of personal data from the users, including their web browsing histories or even more invasive data such as age, gender, or location to infer the online profile of the users. This data collection is possible due to implementing a complex tracking ecosystem by online advertising companies from which multiple stakeholders collect, process, and exchange information. The many privacy cases of abuse inflicted by this industry motivated the implementation of new regulatory efforts to protect consumers’ privacy in the last years. Some notable examples are the General Data Protection Regulation (GDPR) [1] in the European Union or the California Consumer Privacy Act Regulations (CCPA) [2] in California, USA. Further, these privacy regulations typically contain specific provisions and strict requirements for websites that provide sensitive material to end users, including sexual, religious, and health services. Implementing new regulatory frameworks, alongside the growth of online web services, forces an endless evolution of current techniques to study and audit online web services. Furthermore, there is a need to emphasize the online advertising ecosystem, as it represents the primary economic support of a high percentage of web services. Also, the activities and abuses conducted by this ecosystem drove the implementation of current privacy regulations to control the use and collection of personal data. This dissertation falls within the topics of Internet measurements, tackling the need for new measurement techniques and methodological approaches to audit and study online web services. These efforts want to increase the limited knowledge about web subsystems offering sensitive material, including their regulatory compliance regarding current privacy regulations. Also, this dissertation tackles the need to study and measure how big ad tech companies create and use the online profiles of their users to distribute tailored ads. Furthermore, the work presented in this dissertation raises the need for a more in-depth understanding of fundamental tools for conducting Internet measurement works, including their limitations and suitability for academic research. Specifically, this dissertation presents three main contributions: The first one corresponds with implementing a novel methodology to audit sensitive web services’ privacy, transparency, and regulatory compliance. We validate our method by looking at pornographic websites concerning the GDPR in the European Union. We focus our analysis on such types of websites for two main reasons: (i) the GDPR establishes specific provisions and strict requirements on sensitive websites, including pornographic ones. (ii) big ad tech companies set strict constraints for porn-related publishers. As a result, it opened new market opportunities for other actors who have specialized in advertising and tracking technologies for adult sites, creating a semi-decoupled ecosystem from the rest of the web. We perform a holistic analysis of over 6,843 pornographic websites, finding a prevalent absence of regulatory compliance and very extended use of tracking techniques, including advanced ones such as fingerprinting. These results stress the importance of studying the World Wide Web subsets that have not been scrutinized by regulators, policymakers, and the research community in depth. Second, we empirically and comprehensively analyze 13 domain classification services to study their labeling strategy and performance. These services have multiple applications, from business applications such as online advertising to academic research works to conduct category-dependent measurements or to identify the purpose of a website or online service. We study each domain classification service’s methodologies, scalability limitations, label constellations, and suitability for academic research studies. In some cases, their findings depend on the results provided by the domain classification services. We find that the limitations and shortcomings of each domain classification service heavily affect their suitability and applicability, both for practical solutions and academic studies. In the third and last contribution, we implement a novel methodology with real users to study the performance and quality of the profiling and ad targeting algorithms from the two most important stakeholders in the online advertising business, Google and Meta (previously Facebook). We find that half of the categories associated with the profiles are incorrectly assigned. We also observe the presence of sensitive categories in Facebook users, posing a privacy risk and potential regulatory noncompliance. In summary, this dissertation brings new methodologies and results to increase our limited knowledge about the web. This work has been supported by IMDEA Networks Institute. Programa de Doctorado en Ingeniería Telemática por la Universidad Carlos III de Madrid Presidente: Nikolaos Laoutaris.- Secretario: Hamed Haddadi.- Vocal: Leyla Bilge

Mención Internacional en el título de doctor Recent technological advances have led to an exponential growth in the volume of data generated. The quest to make sense of these data, some of which are usually complex, has led to recent interest in development of statistical methods for analysing data with complex structures. One such field of interest is functional data analysis (FDA), which deals with the analysis of data that can be considered as functions, curves, or surfaces observed over a domain set. Outlier detection is a challenging but important part of the exploratory analysis process in FDA because functional observations can exhibit outlyingness in various ways compared to the bulk of the data. This thesis addresses the problem of detecting and classifying outliers in functional data with three main contributions. First, the fdaoutlier R package is presented in Chapter 2. The package contains implementations of some of the state-of-the-art functional outlier detection methods in the literature. Some of the methods implemented include directional outlyingness, magnitude-shape plot, sequential transformations, total variation depth, and modified shape similarity index. Detailed illustrations of the functions of the package are provided, using various simulated and real functional datasets curated from the functional outlier detection literature. Overviews of the functional outlier detection methods implemented in the package are also presented in Chapter 2. This chapter therefore, serves as a review of some of the current literature in outlier detection for functional data. Next, two new methods, named ‘Semifast- MUOD’ and ‘Fast-MUOD’, are presented in Chapter 3. These methods work by computing for each curve three indices (magnitude, amplitude and shape index) that measure the outlyingness of that curve in terms of its magnitude, amplitude and shape. ‘Semifast- MUOD’ computes these indices with respect to (w.r.t.) a random sample of the dataset, while ‘Fast-MUOD’ computes these indices w.r.t. to the point-wise or L1 median. The classical boxplot is then used as a cutoff on the three indices to identify curves that are outliers of different types. A byproduct of the methods is an unsupervised classification of the outliers into different types, without the need for visualisation. Performance evaluation of the methods, using various real and simulated datasets, shows that Fast-MUOD is the better of the two new proposed methods for outlier detection, in addition to being very scalable. Comparisons with latest functional outlier detection methods in the literature also show superior or comparable outlier detection performance. In Chapter 4, some theoretical properties of the Fast-MUOD indices are presented. These include some definitions of the indices, as well as convergence proofs of the sample approximations. Some properties of the indices under simple transformations are also presented in this chapter. Finally, three techniques are presented in Chapter 5 for extending the Fast-MUOD indices to outlier detection in multivariate functional data observed on the same domain. These techniques include the use of random projections and identifying outliers on the marginal components of the multivariate functional data. The use of random projections showed the best result in performance evaluations with various real and simulated datasets. Chapter 6 contains some concluding remarks and possible future research work. This work has been supported by IMDEA Networks Institute Programa de Doctorado en Ingeniería Matemática por la Universidad Carlos III de Madrid Presidente: Francisco Javier Prieto Fernández.- Secretario: Alba María Franco Pereira.- Vocal: Fabian Scheipl

Wireless communication is a transformative technology that has changed the way we work, communicate and enjoy our free time. The number of connected devices is expected to increase to 29.4 billion by 2030. As a result, there is a demand for higher data rates, lower delays and constant connectivity of wireless devices. These demands have driven the networking community to seek new technologies as these requirements are beyond the capabilities of current networks. Communication at higher frequencies, beyond 10 GHz where most of current communications systems operate, could be the game changing technology for the next generation of networks. At millimeter-wave frequencies, 30-300 GHz, the available spectrum is larger than all spectrum currently allocated to cellular and wireless area networks (WLAN). The unlicensed spectrum at 60 GHz alone can offer 10 to 100 times more spectrum than it is available in current unlicensed WLANs. The larger bandwidth allocations allow for increased datarates. These multi-gigabit per second rates and milisecond latencies are now achievable thanks to new directional high gain antennas and cost-effective CMOS technology that can operate at mm-wave frequencies. However, with the use of this new technology new challenges arise. For example, since the communication is directional in mm-wave, the best direction to communicate needs to be decided, and updated in mobility scenarios. Aside from the technical deployment challenges at this frequency, another challenge is how to provide increased capabilities for the network such as localization and passive localization. The specific characteristic of propagation at mm-wave frequencies such as the quasi optical patterns, the dense deployments and the multi path diversity warrant designs of localization schemes aimed at exploiting these characteristic. Finally, stepping away from conventional network design, the future generation of networks can be enhanced by Machine Learning. Machine Learning has shown to be a powerful tool in many domains and has increased the autonomy of chatbots and self-driving cars. As we go further in frequency, ML can help design the different layers of the future communication networks to embrace the hardware imperfections, that are more relevant at higher frequencies, and to serve applications in the optimum way in different environments. The thesis is divided into two parts, first we study the challenges in the optimization and features of conventional mm-wave networks, and second, we consider a new way of designing mm-wave communications using Machine Learning. In the first part, we study the challenge of scaling mm-wave networks to dense deployments, and secondly we increase the capabilities of mm-wave networks by proposing a passive localization mechanism that is standard compliant. Currently the beam training mechanisms deployed to establish a directional link result in a large overhead in communications and this prohibits dense deployments. This is specially due to station beam training. While the APs can beam train with multiple stations simultaneously, the stations have to train one at a time. This overhead is even larger in mobile deployments where the devices require constant re-training to maintain communication. To overcome this overhead and allow dense deployments we propose a low-overhead beam training mechanism, SPIDER, that eliminates the station beam training overhead entirely. To this end, stations carry out phase-coherent measurements by switching through multiple receive beam patterns on a time-scale of tens of nanoseconds when receiving a packet preamble to calculate the AoA to the AP and select the best beam pattern for communication. Our results show that our algorithm achieves highly accurate angle estimation used to drive the beam steering decisions and that it reduces the overhead by an order of magnitude compared to IEEE 802.11ad beam training. The second work presented in the first part of this thesis is regarding the use of the network as a sensor in mm-wave frequencies. Joint communication and sensing approaches would allow for an extremely distributed cost-effective sensor network. At higher frequencies, due to the larger bandwidth, the theoretical precision of localization information is higher than at lower frequencies. However, localization algorithms for mm-wave frequencies can not directly build from the lower frequency ones. At lower frequencies, these algorithms leverage information from multiple antennas with separated RF chains, while in mm-wave phased antenna arrays are usually connected to a single RF chain, so the phase information per antenna can not be obtained. However, several properties of mm-wave communications have implications in the design of localization systems. The high path loss attenuation due to the high frequency implies that the line of sight paths (LOS) can be distinguished from the NLOS paths. Additionally, reflections on surfaces have limited scattering. These properties generate channels with sparse multi paths, for which geometrical propagation assumptions become appropriate. Additionally, the need for directional communication due to the high path loss results in the need for mechanisms that enable beam selection, which are the focus of the earlier works. Such beam training mechanisms become relevant for localization in mm-wave frequencies. Since during the beam training the environment is scanned using beam patterns in different directions, these information can be used to detect and track moving reflections. We present POLAR, the first passive mm-wave localization algorithm for mm-wave commercial devices with zero overhead. We use commercial APs whereas the station design is based on a full-bandwidth 802.11ad compatible FPGA-based platform with a phased antenna array. The stations exploit the preamble of the beam training packets of the APs to obtain channel state information measurements for all antenna patterns. With this, we determine distance and angle information for the different multipath components in the environment to passively localize a mobile object. We evaluate our system with multiple APs and a moving robot with a metallic surface. Our results show that is is possible to obtain 6.5cm mean error accuracy and sub-meter accuracy in 100% of the cases performing joint localization and sensing in a mm-wave testbed. In the second part of this thesis, we study a new way to design communications with the concept of AI-native air-interface framework in mm-wave. In this area of research, some works have proposed to learn to communicate using an end-to-end communication approach using deep learning (DL). However, a channel model is needed in order to train these models, and analytical channels were shown to produce a reduced performance in experimental scenarios. In order to extend this to real channels, model-free approaches have been presented in which the model can be learned implicitly from data. Building on this works to obtain an explicit channel model would also allow for transfer learning to speed up the training on time-variant channels. In this thesis, we present a neural network architecture based on Mixture Density Networks to explicitly learn the channel model for end-to-end communications for SISO and MIMO systems. Our results show that it is possible to learn complex stochastic channel models from data with differentiable methods that are compatible with the training of the autoencoder structure. These end-toend learning methods will become increasingly important in future generations of wireless networks, where the system complexity will increase due to a higher number of antennas, higher frequencies, wider bandwidths, and more complex hardware, making ML-based physical layer design highly appealing. Second, we study the scaling properties of end-to-end ML models for large-scale MIMO systems. We conclude that the bottleneck that prevents the scaling of such networks, in the state-of-the-art architectures, is due to the input and output representations. Large MIMO systems constitute highly demanding scenarios in which training would require unmanageable memory resources. We study typical approaches available in DL to reduce the memory allocation and show that a bit-wise architecture for MIMO allows to reduce the memory allocation by several orders of magnitude. Our results show that with the bit-wise architecture it is possible to extend the end-to-end approaches to large systems of antennas. This study is an important first step towards developing practical end-to-end learning algorithms for large scale MIMO systems. So far, end-to-end systems have only been applied in a per symbol manner as training these systems for blocks of symbols is complex as it introduces implicit channel coding. Optimizing the receiver only would allow to reduce the complexity and train in blocks of symbols while helping to deal with imperfections and other non-linearities without having to explicitly correct them. We explore the gains that can be achieved by using an ML based design on the receiver of single carrier mm-wave systems for frequency selective channels, where due to the multi-path and the resulting inter symbol interference block wise treatment is required. Our results show that it is possible to outperform baseline typical receivers in static scenarios. In summary, mm-wave technologies will open the door to the next generation of applications, however, new challenges need to be tackled. We provide new solutions to enable scalable deployments and show that the beam training station overhead can be completely eliminated. We introduce the feature of passive localization and sensing for mm-wave frequencies, and show the high accuracy achievable in an experimental FPGA based testbed. Finally, explore new ways of designing the physical layer to reduce the effect of hardware imperfections in mm-wave and show gains compared to state-of-the-art receivers. This work has been supported by IMDEA Networks Institute Programa de Doctorado en Ingeniería Telemática por la Universidad Carlos III de Madrid Presidente: Tobias Mirco Koch.- Secretario: Andrés García Saavedra.- Vocal: Dani Korpi

This thesis investigates the metric and tetrad formulations of three gravitational field theories in manifolds with timelike boundaries within the covariant phase space program. With the recently developed relative bicomplex framework, we explore the space of solutions and presymplectic structures associated with each action principle and analyse their equivalence. The first action we consider is the Einstein-Hilbert (EH) action with the Gibbons- Hawking-York boundary term. By including the appropriate boundary terms in the variational principles, we show that the metric and tetrad formulations derived from them are equivalent. Furthermore, we show that their solution spaces are the same and that their presymplectic structures and associated charges coincide. The second action we consider is the Palatini action with the Obukhov boundary term, assuming torsion and non-metricity, and we prove the equivalence between its metric and tetrad formulations. Furthermore, we show that the metric and tetradsector of the first-order Palatini formulation are equivalent to the metric and tetrad formulations of the EH action. Lastly, we introduce the Hojman-Mukku-Sayed (HMS) action, a generalisation of the Palatini action plus the Holst term in the presence of boundaries with non-metricity and torsion. We prove that the space of solutions of the HMS and Palatini actions coincided and conclude that HMS’s metric and tetrad sectors are identical to their corresponding versions of the EH action. Additionally, we prove that the Palatini and HMS Lagrangians are not cohomologically equal despite defining the same space of solutions. Consequently, a careful analysis is required for the presymplectic structures and the charges because they may differ. However, we show that the covariant phase spaces of both theories were equivalent. This sheds light on some open problems regarding the equivalence of their associated charges in different formulations. Programa de Doctorado en Ingeniería Matemática por la Universidad Carlos III de Madrid Presidente: José Navarro Salas.- Secretario: Bogar Díaz Jiménez.- Vocal: Mercedes Martín Benito

Mención Internacional en el título de doctor This research has been funded in part by MCIN/AEI/10.13039/501100011033 and by “ERDF A way of making Europe” through grant PGC2018-098186-B-I00 (BASIC). Programa de Doctorado en Ingeniería Matemática por la Universidad Carlos III de Madrid Presidente: Sandro Meloni.- Secretaria: Francesca Lipari.- Vocal: Giulia Andrighetto

Mención Internacional en el título de doctor Programa de Doctorado en Ingeniería Matemática por la Universidad Carlos III de Madrid Presidente: Francisco José Marcellán Español.- Secretario: Bernardo de la Calle Ysern.- Vocal: Arnoldus Bernardus Jacobus Kuijlaa

The concept of Gromov hyperbolicity grasps the essence of negatively curved spaces like the classical hyperbolic space and Riemannian manifolds of negative sectional curvature. It is remarkable that a simple concept leads to such a rich general theory. The study of hyperbolic graphs is an interesting topic since the hyperbolicity of any geodesic metric space is equivalent to the hyperbolicity of a graph related to it. In this Ph. D. Thesis we characterize the hyperbolicity constant of interval graphs and circular-arc graphs. Likewise, we provide relationships between dominant sets and the hyperbolicity constant. Finally, we study the invariance of the hyperbolicity constant when the graphs are transformed by several operators. Programa de Doctorado en Ingeniería Matemática por la Universidad Carlos III de Madrid Presidente: Domingo de Guzmán Pestana Galván.- Secretaria: Ana Portilla Ferreira.- Vocal: Eva Tourís Lojo

Self-organization processes, in which some form of collective behavior arises from local interactions in a physical system, are promising mechanisms in the context of manufacturing at ultra small scales, where processing techniques are technically challenging. In many cases, the approach is to induce self-organization or self-assembly phenomena on the system surface |which indeed acquires an increased importance at small scales because of the enhanced surface-to-volume ratio| as a result of which a desired surface morphology is achieved, with different properties depending on its intended application. In this thesis, we consider two important examples of self-organization processes which take place at the surfaces of many small non-equilibrium systems. One of them, kinetic roughening, reflects the dominance of fluctuations in the surface morphology, with strong correlations which are quite similar to those of an equilibrium system at a continuous phase transition. The second, opposite self-organization process is the formation of ordered patterns. In the thesis, we will deal with the control of the level and type of various surface properties, like roughness and other, and the conditions for the emergence of a varying degree of spatial order in patterns via spontaneous physical processes. We work in the context of ultrathin fluid films on solid substrates, focusing on the role of the physical effects that become relevant at these very small scales -but not so much at larger scales-, while other loose their relevance. One of these aspects is the thermal noise. The other one is the interaction between the uid surface and the substrate, that play a key role in the two reference physical systems we are going to study: an ultrathin fluid film falling down an inclined plane |where the dynamics of the fluid surface follows the celebrated Kuramoto-Sivashinsky (KS) equation -and a ferrofluid ultrathin film under a magnetic field. Both kinetic roughening and pattern formation usually exhibit some kinds of universal behavior. The universality classes in the kinetic roughening processes that occur in several approximations of the KS equation are widely studied in this thesis. These universality classes are characterized by both how the fluctuations scale with space and time and how these fluctuations are statistically distributed. We will deal with the emergence of symmetries in the fluctuation distributions that are unexpected considering the bare microscopic interactions; the non-trivial relation between the universality class of closely related models; with a novel physical mechanism that induces the transition between different universality (sub)classes as the system temperature and hence the dominant nature of the fluctuations (chaotic or stochastic) changes, and finally with some anomalous kinetic roughening processes in the limit of vanishing viscosity and surface tension. Finally, on the other hand, the formation of highly ordered patterns is assessed in the context of ultrathin ferrofluid films under a magnetic field, due to the spontaneous physical break-up of the film into drops. The conditions under which higher levels of order are achieved will be described. This is intended as a proof-of-concept, previous step that could encourage experiments being performed for this type of systems. This Doctoral Thesis has been supported by Ministerio de Educación, Cultura y Deporte (Spain) through Formación del Profesorado Universitario scolarship No. FPU16/06304. Additionally, research by the author has been partially supported by Ministerio de Economía y Competitividad, Agencia Estatal de Investigación, and Fondo Europeo de Desarrollo Regional (Spain and European Union) through grant Nos. FIS2015-66020-C2-1-P and PGC2018-094763-B-I00. Programa de Doctorado en Ingeniería Matemática por la Universidad Carlos III de Madrid Presidente: Miguel Ángel Rodríguez Díaz.- Secretario: Juan Jesús Ruiz Lorenzo.- Vocal: Silvia Noemi Santalla Arribas

Mención Internacional en el título de doctor El principal objetivo de esta tesis es presentar y probar la viabilidad de un método no estándar para controlar el estado de un sistema cuántico, cuya dinámica está gobernada por la ecuación de Schrödinger, modificando sus condiciones de frontera en lugar de interaccionar con el sistema a través de campos externos para dirigir su estado. El control del estado de sistemas cuánticos está volviéndose más y más significativo dado el asombroso avance experimental que está teniendo lugar motivado por la carrera para alcanzar tecnologías cuánticas efectivas. Un requisito natural para cualquier procesador de información cuántico es el control de los estados de un cúbit. Por ejemplo, los cúbits basados en espín pueden controlarse a través de puertas universales basadas en la rotación del espín un cierto ángulo sobre un eje dado [60]. Una demostración de la viabilidad del control y de la lectura de alta fidelidad de un cúbit basado en espín puede encontrarse en [55, 59]. El marco matemático necesario para trabajar con el control de sistemas cuánticos basados en el espín es la Teoría de Control Geométrica (véase [19] para un repaso de este enfoque). También puede mencionarse [15, 56], dónde se realiza un estudio del control de sistemas cuánticos de espín utilizando dicha teoría. Una revisión del estado de la cuestión en control (óptimo) de sistemas cuánticos puede verse en [31]. Sin embargo, la Teoría de Control Geométrica encuentra problemas serios a la hora de tratar sistemas infinito dimensionales provenientes de las dificultades intrínsecas a la geometría en dimensión infinita. Éste constituye un problema estándar que se aborda también, por ejemplo, en Teoría de Campos y puede por lo tanto ser sorteado. Sin embargo, lo cierto es que tan sólo un pequeño número de resultados han aparecido en las fuentes bibliográficas, sobre todo referentes a la controlabilidad de sistemas bilineales (cf. sec. 3.2). Véanse por ejemplo los trabajos de Beauchard et al. [11, 12] y Chambrion et al. [18]. El enfoque habitual del control de sistemas cuánticos se basa en el uso de campos externos para manipular el estado del sistema. Desde un punto de vista tecnológico, aparecen dificultades al intentar controlar un sistema cuántico de este modo que tienen que ver con las complicaciones para manipular un sistema hecho de unas pocas partículas mientras se intenta mantener las correlaciones cuánticas. Como consecuencia, el sistema cuántico debe mantenerse a muy baja temperatura y las interacciones debe producirse muy rápidamente [5], lo cual es inconveniente para las aplicaciones. El paradigma de Control Cuántico en la Frontera aborda el control de sistemas cuánticos de una forma diametralmente opuesta al enfoque estándar. En lugar de buscar el control del sistema cuántico interactuando directamente con el sistema a través de un campo externo, el control se consigue manipulando las condiciones de contorno del sistema. El espectro de un sistema cuántico, por ejemplo un electrón moviéndose en un pozo de potencial, depende de las condiciones de contorno (típicamente Dirichlet o Neumann). Por lo tanto, modificando dichas condiciones de contorno puede modificarse el estado del sistema, lo cual permitiría en última instancia controlarlo [34]. Este tipo de interacción, que es más débil en cierto sentido, hace esperar que sea más sencillo mantener las correlaciones cuánticas. Llamamos sistema (cuántico) de control en la frontera a los sistemas definidos dentro de este paradigma de control. El marco establecido por el Control Cuántico en la Frontera ha sido usado para mostrar cómo generar estados entrelazados en sistemas compuestos modificando las condiciones de frontera del sistema [38]. La relación entre dicho paradigma y la topología del sistema ha sido explorada en [58] y recientemente usada para describir las propiedades físicas de sistemas con paredes móviles [25–28,30]. Sin embargo, a pesar del interés intrínseco que tiene, problemas básicos como la controlabilidad de sistemas sencillos dentro de este paradigma no ha sido aún estudiada. La mayoría de los sistemas cuánticos que aparecen en las aplicaciones para desarrollar ordenadores cuánticos, tales como las trampas de iones o los circuitos superconductores, son infinito dimensionales. Mientras tanto, los modelos usados para describirlos son aproximaciones finito dimensionales [49, 70], lo que introduce errores en dicha descripción. Es por tanto natural buscar mejores modelos matemáticos para estos sistemas. Buenos candidatos para estos modelos pueden construirse a partir de lo que llamamos Circuitos Cuánticos, una generalización de los grafos cuánticos (esto es, grafos métricos equipados con un operador diferencial en sus aristas junto con condiciones de contornos apropiadas que definan una extensión autoadjunta de dicho operador [46,47]). Por un lado, los Circuitos Cuánticos comparten con los dispositivos físicos la estructura tipo grafo y la dinámica en ellos está gobernada por operadores diferenciales (en concreto, consideraremos el operador de Laplace-Beltrami y Laplacianos magnéticos junto con algunos potenciales que representan campos magnéticos y eléctricos en los Circuitos Cuánticos). Por otro lado, los Circuitos Cuánticos son suficientemente sencillos como para encontrar soluciones de la ecuación de evolución numéricamente (o incluso analíticamente). Todo esto hace interesante considerar la dinámica y la controlabilidad de sistemas definidos en Circuitos Cuánticos. Algunos ejemplos concretos de Circuitos Cuánticos pueden encontrarse al final del Capítulo 6, véase la Figura 6.1. Más aún, los grafos cuánticos pueden verse como un caso concreto de Circuito Cuántico, por lo que los resultados presentados en esta tesis pueden aplicarse directamente a ellos. Y, además de el interés matemático de estudiar la dinámica de sistemas definidos en grafos cuánticos, creemos que éstos pueden ser utilizados para modelizar circuitos superconductores. Más concretamente, es sabido que las junturas de Josephson (que son un dispositivo fundamental en los circuitos superconductores, cf. [22, 70, 71]) pueden modelizarse usando condiciones de frontera [48,68]. Nótese que los circuitos superconductores son una de las prometedoras nuevas tecnologías en el desarrollo de ordenadores cuánticos. Para mostrar la viabilidad del método de Control Cuántico en la Frontera, definimos una familia de sistemas de control cuántico en la frontera sobre Circuitos Cuánticos. Para ello, utilizamos la caracterización de las extensiones autoadjuntas del operador de Laplace-Beltrami desarrolladas en [37], que parametriza las extensiones autoadjuntas en función de operadores unitarios actuando en el espacio de datos en la frontera. Estos operadores unitarios son adecuados para representar los grupos de simetría de la variedad subyacente [36], hecho que fue usado en [8] para caracterizar las condiciones de frontera compatibles con las estructuras de grafo que consideramos aquí. Antes de poder abordar el problema de controlabilidad, es necesario afrontar el problema de existencia de soluciones de la ecuación de Schrödinger con condiciones de contorno dependientes del tiempo, problema que tiene su propia importancia. Situaciones similares a la que presentaremos han sido consideradas anteriormente para algunos sistemas cuánticos concretos [21, 23, 58]. La dependencia temporal del dominio del Hamiltoniano del sistema añade dificultades extra a las habituales complicaciones que surgen de la naturaleza discontinua de los operadores no acotados, lo que compromete la existencia de soluciones para la ecuación de Schrödinger. En este sentido, los resultados más generales de que disponemos son debidos a J. Kisyński [44]. Basándose en una idea aproximativa de K. Yosida [72], Kisyński propone condiciones suficientes para la existencia de operadores de green para problemas de Cauchy abstractos en espacios de Banach, incluyendo el caso en el que los operadores que definen la evolución tienen dominios dependientes del tiempo. Estos resultados pueden ser aplicados al caso concreto de la ecuación de Schrödinger, tal y como hace el propio Kisyński, cf. [44, Sec. 8]. Más concretamente, proporciona condiciones suficientes para la existencia de propagadores unitarios para sistemas cuánticos cuyos Hamiltonianos tienen dominio dependiente del tiempo pero no así las formas sesquilineales asociadas a éstos. Este caso también fue estudiado por B. Simon [65], cuyo enfoque se basa en el mismo método aproximativo de Yosida, pero abordado el problema de manera ligeramente distinta. Para simplificar la exposición, en lugar de aplicar los resultados más generales de Kisyński directamente a nuestros sistemas cuánticos de control en la frontera, introduciremos una familia de sistemas de control estándar relacionada que llamamos sistemas de inducción cuántica, cuyos Hamiltonianos tendrán dominio de forma constante. La relación entre estas dos familias de sistemas es usada frecuentemente en la discusión, lo que nos permite probar resultados primero para los sistemas de inducción cuántico y transferirlos a los sistemas de control en la frontera. Siguiendo esta estrategia, usaremos los resultados de Simon y de Kisyński para Hamiltonianos con dominio de forma constante para proporcionar condiciones suficientes para que ambos sistemas tengan dinámica bien definida. En el estudio de las propiedades dinámicas de los Hamiltonianos dependientes del tiempo con dominio de forma constante, la escala de espacios de Hilbert asociada a el Hamiltoniano, H+ ⊂ H ⊂ H−, aparece como una herramienta básica tanto en el desarrollo de Simon como en el de Kisyński. A través de esta tesis, usamos nociones como la existencia de dinámica o la controlabilidad tanto en H como en H−; por ello, usaremos los adjetivos fuerte y débil para referirnos a las nociones en H y en H− respectivamente. Una vez resuelto el problema de la existencia de soluciones de la ecuación de Schrödinger, podremos abordar el problema de controlabilidad. Usaremos como una herramienta útil para este fin el Teorema 4.3.4, un resultado de estabilidad que generaliza los resultados de A.D. Sloan [66] constituyendo uno de los resultados originales principales expuestos e esta tesis. Que sepamos, esta generalización es el resultado de estabilidad más general en el contexto de la ecuación de Schrödinger dependiente del tiempo. La demostración de este teorema se basa en las técnicas usadas por Simon y, en la prueba de dicho resultado, obtenemos una ligera generalización de algunos resultados en [65] (cf. Lemma 4.3.7). La importancia de este resultado de estabilidad no reside tan solo en su papel a la hora de demostrar la controlabilidad de los sistemas de control bajo estudio permitiendo demostrar la viabilidad del Control Cuántico en la Frontera; tiene consecuencias de mayor alcance para la Teoría de Control Cuántico ya que permite obtener cotas a priori del error cometido al controlar el estado del sistema. Demostramos la controlabilidad aproximada de los sistemas de inducción cuántica basándonos en un resultado de T. Chambrion et al. in [18], donde estudian la controlabilidad de una clase de sistemas de control cuántico bilineales con controles constates a trozos. La estructura que tienen los sistemas de inducción cuántica es similar, pero no exactamente igual, a la de los problemas estudiados por Chambrion et al. Por ello, definimos un sistema auxiliar para el cual el teorema de Chambrion et al. asegura controlabilidad aproximada y después, usando el resultado de estabilidad, mostramos que esta controlabilidad aproximada se extiende al sistema de inducción cuántica original. Si embargo, la clase de controles considerada por Chambrion et al. no puede ser llevada automáticamente a sistemas con Hamiltonianos cuyos dominios dependen del tiempo, ya que las soluciones de la ecuación de Schrödinger (en el sentido fuerte) podrían no existir. Por lo tanto, la controlabilidad aproximada obtenida para los sistemas de inducción cuántica usando este procedimiento es en el sentido débil (es decir, en H−). Usando de nuevo el resultado de estabilidad podemos demostrar controlabilidad aproximada en sentido fuerte para los sistemas de inducción cuántica con controles suaves, y ésta puede ser transferida a los sistemas de control cuántico en la frontera usando la relación entre ambos. Estos resultados de controlabilidad, demostrados en el Capítulo 7, junto con los resultados de estabilidad en el Capítulo 4 constituye las contribuciones originales más relevantes de esta tesis. Por último, remarcamos el hecho de que la caracterización de las extensiones autoadjuntas que usamos aquí puede ser aplicada a otros operadores diferenciales como el operador de Dirac [35, 40, 57], por lo que los resultados presentados en esta tesis podrían extenderse a tales casos. Más aún, dicha caractrerización está adaptada al cálculo numérico [39, 51], por lo que podrían desarrollarse algoritmos con aplicaciones al control óptimo de éstos sistemas a partir de los resultados aquí mostrados. This work was partially supported by the “Ministerio de Economía, Industria y Competitividad” research project MTM2017-84098-P, the QUITEMAD project P2018/TCS-4342 funded by “Comunidad Autónoma de Madrid” and by “Universidad Carlos III de Madrid” through its Ph.D. program grant PIPF UC3M 01-1819 and through its mobility grant in 2020. Programa de Doctorado en Ingeniería Matemática por la Universidad Carlos III de Madrid. Presidente: Fernando Lledó Macau.- Secretario: Paolo Facchi.- Vocal: Thomas Chambrion

Topological indices have been widely used in different fields associated with scientific research. They are recognized as useful tools in applied research in Chemistry, Ecology, Biology, Physics, among others. For many years, scientists have been trying to improve the predictive power of the famous Randi’c index. This led to the introduction and study of new topological descriptors that correlate or improve the level of prediction of the Randi’c index. Among the most commonly used descriptors are the Inverse index, the first general Zagreb index and the recently introduced Arithmetic- Geometric index. In this work we study the mathematical properties and relationships of the aforementioned topological indices. Programa de Doctorado en Ingeniería Matemática por la Universidad Carlos III de Madrid Presidente: Domingo de Guzmán Pestana Galván.- Secretaria: Ana Portilla Ferreira.- Vocal: Eva Tourís Lojo