Abstract

In perceptual decision-making, sensory information is accumulated over time to reach a categorical decision. This accumulation process can follow different strategies, where certain moments of the stimulus are weighted more heavily in the decision than others. Such temporal weighting strategies have been linked to various underlying mechanisms, including bounded accumulation, which prioritizes early stimulus information (primacy weighting); perfect integration, which treats all stimulus information uniformly (uniform weighting); and leaky integration, which favors only the later part of the stimulus (recency weighting). Moreover, experimental evidence suggests that both primates and humans can dynamically adjust their temporal weighting strategy in response to stimulus statistics, prioritizing later or early stimulus information as the task requires. Nevertheless, the proposed mechanisms fall short in explaining this flexibility, as none can generate the full diversity of experimentally observed temporal weighting patterns. In this thesis, we combine data analysis, computational modeling, and recurrent neural networks to develop a comprehensive framework that proposes one single mechanism that can capture the full range of flexibility in the temporal weighting patterns, characterize the behavioral and neural correlates of this flexibility, and examine the optimality of these patterns. First, we present experimental evidence of how temporal weighting flexibility is represented in the behavior of both humans and macaques. We found that this flexibility is acquired through exposure to the stimulus statistics with the state of vigilance previous to the presentation of the stimulus having a significant impact on effectively performing the task. Additionally, we analyzed middle temporal (MT) recordings from behaving macaques and found that, while the average firing rate of the neural population remains stable despite changes in stimulus statistics, stimulus-related and choice-related activity depend on the stimulus statistics in a non-trivial way. Second, we developed a two-area firing rate model consisting of a sensory circuit and a decision circuit, connected by bottom up and top down, with a modulation signal that controls the attractor dynamics of the decision circuit. This model successfully replicates the experimentally observed flexibility in temporal weighting. Mechanistically, the modulation signal initiates the decision-making process by pushing the network into a competitive regime and modulates the decision dynamics by either accelerating or delaying the choice, similar to an urgency signal. By controlling the initiation of evidence integration and altering the signal’s shape and/or duration, we were able to generate a broader range of temporal weighting patterns, extending beyond what was observed experimentally. Third, we use Recurrent Neural Networks (RNNs) to explore whether the temporal weighting patterns observed in experimental evidence are optimal. Non-uniform weighting strategies emerge as optimal adaptations to internal noise when RNNs specialize in a single stimulus condition. These patterns are robust across networks, with task-specific low- or high-dimensional dynamics. When trained in a sequential setup similar to the macaques, the RNN adopted a rigid temporal weighting strategy biased towards the initial training condition, failing to adapt to new task demands. To overcome this limitation, we introducing contextual signals representing the task conditions which allowed simultaneous training across all conditions in a single RNN. This approach allowed the RNN to exhibit flexibility comparable to independently trained networks (aligning temporal weighting with stimulus statistics), faster learning, and generalization to novel tasks. These findings suggest that contextual signals may be the mechanism underlying the context switching observed in experimental data.

Thesis advisor(s): Klaus Wimmer

University: Universitat Autònoma de Barcelona

Abstract

Given a dynamic system, it is important to identify the invariant objects that organize long-term behavior, as well as their dynamic connections. Both in theory and in applications. The objective of this thesis is to advance in the development of Kolmogorov-Arnold-Moser (KAM) type techniques within the framework of the parameterization method and its application to problems of celestial mechanics. We have developed KAM iterative schemes for the calculation of partially hyperbolic invariant torus and their invariant bundles in quasiperiodic Hamiltonian systems. We look for invariant bulls and bundles under adequate time-1 maps, which allow us to reduce the dimension of the bull to be calculated by one. The computational cost of manipulating functions grows exponentially with the number of variables in the parameterization. Therefore, reduction by flow maps is computationally advantageous, although it requires numerical integration. However, this integration can be easily parallelized. If the parameterization is approximated with N Fourier coefficients, the iterative step requires O(N) of storage and O(N log N) operations, in contrast to standard Newtonian methods, which need O(N^2) of storage and O(N^3) operations. This gain in efficiency comes from the geometric properties of phase space (i.e., symplectic geometry), systems (symplectic accuracy), torus (isotropy, Lagrangianity), as well as dynamical properties (reducibility). In particular, the reducibility of the linearized dynamics around the torus to a triangular matrix by blocks is known as automatic reducibility and is an important property both in theory and in applications. The algorithms have been implemented and applied to the Three-Body Elliptic Restricted Problem (ERTBP) to compute an extensive set of non-resonant three-dimensional invariant torus along with their invariant bundles. From these results, we have obtained an a posteriori theorem for partially hyperbolic invariant bulls and their rank 1 invariant bundles in quasiperiodic Hamiltonian systems. The approach followed allows the theorem to be applied to autonomous, periodic and quasiperiodic Hamiltonian systems, and constitutes the demonstration of the convergence of methods based on flow maps. In addition, we simultaneously obtain both stable and unstable bundles, providing a clear geometric view of the tangent space to the torus. The proof is based on geometric properties of a symplectic nature, which hold approximately when the parameterizations approximately satisfy their equations of invariance. We have obtained geometric lemmas that control error in the KAM iterative process. The new error in the invariance equations is controlled with explicit constants, which requires a careful treatment of the loss of analyticity at each iterative step. The demonstration concludes by obtaining convergence conditions for the KAM iterative process. The a posteriori theorem obtained allows computer-aided proofs to be carried out. Partially hyperbolic bulls have associated stable and unstable varieties, whiskers, where dynamics converge exponentially fast in the future and in the past, respectively. The stable and unstable bundles with the linear approximations of these varieties. We have also developed KAM schemes to compute high-order Fourier-Taylor expansions of whiskers in autonomous and quasiperiodic Hamiltonian systems. Unlike order-to-order methods, which first calculate the torus and its bundles before calculating the whiskers on an order-by-order basis, the approach followed simultaneously computes both the torus and the whiskers using the same KAM iterative method. This unified framework improves the efficiency of whisker calculation by doubling the number of correct terms in expansion in each iteration. The algorithms have been applied to the calculation of high-order expansions of partially hyperbolic non-resonant invariant torus in the circular and three-body elliptical constrained problems.

Thesis advisor(s): Alejandro Haro Provinciale & José María Mondelo González

University: Universitat de Barcelona

Abstract

Finite transformation group theory investigates the finite symmetries of topological objects, such as manifolds or CW-complexes. In this thesis, we focus on actions on closed topological manifolds and adopt the following approach: instead of directly studying the action properties of a finite group G to a manifold M, we focus on the properties of action restricted to certain subgroups H of bounded index. Several problems align with this philosophy, such as determining whether the group of homeomorphisms of a manifold is Jordan, calculating the discrete degree of symmetry of a manifold, determining whether a manifold is quasi-asymmetric, and studying the number and size of isotropy subgroups for finite group actions on manifolds. In the first part of the thesis, we provide solutions to these problems for two general classes of manifolds, namely: (1) Closed, connected and aspherical manifolds, whose fundamental group has a group of external Minkowski automorphisms (a group G is Minkowski if there exists a constant C such that every finite subgroup H of G has order at most C). (2) Closed, connected and orientable manifolds that admit a non-zero degree application to a nilmanifold. We show that the group of external automorphisms of a lattice of a connected Lie group is Minkowski, which allows us to apply our results to locally homogeneous aspherical closed manifolds. In addition, we provide the earliest known examples of manifolds M and M’ with isomorphic cohomology rings such that Homeo(M) is Jordan but Homeo(M’) is not. We establish two stiffness results for the discrete degree of symmetry: if M is a closed, connected, aspherical manifold and the external automorphism group of the fundamental group of M is Minkowski, or if M admits a non-zero degree application to a nilmanifold and its fundamental group is virtually solvable, then M is homeomorphic to a torus if its discrete degree of symmetry is equal to the dimension of M. In the second part, we refine the concept of group actions to explore in greater depth the topological and cohomology rigidity of closed and connected manifolds. This framework allows us to analyze in more detail the structure of closed aspherical manifolds and those that admit a non-zero degree application to a nilmanifold. We define new invariants, such as the iterated length of a manifold, which is closely related to its self-coatings, and introduce a refined version of the discrete degree of symmetry, called the discrete degree of iterate symmetry. We show that if M is a closed, oriented manifold that admits a non-zero degree application to a nilmanifold of nilpotency class 2, and both manifolds have the same discrete degree of iterated symmetry, then the rational cohomology of M is isomorphic to that of the nilmanifold. Furthermore, if the fundamental group of M is virtually solvable, then M is homeomorphic to the nilmanifold. We also prove that if M is a locally homogeneous closed aspherical manifold with a discrete degree of iterated symmetry equal to its dimension, then M is homeomorphic to a nilmanifold of nilpotency class 2.

Thesis advisor(s): Ignasi Mundet i Riera

University: Universitat de Barcelona

Abstract

This thesis explores the boundary behavior of solutions to elliptic partial differential equations in non-smooth domains.

Thesis advisor(s): Xavier Tolsa Domènech

University: Universitat Autònoma de Barcelona

Abstract

 This thesis characterizes the removable compact sets for solutions of fractional caloric equations that satisfy certain regularity conditions. Specifically, we introduce fractional caloric capacities to measure whether certain compact sets, where the function does not a priori satisfy the equation, can still admit it as a solution. This study is conducted under the assumption that the solution satisfies regularity conditions of the BMO, Hölder, or Lipschitz type, all defined in a fractional parabolic context. Furthermore, in some cases, we succeed in identifying these capacities with parabolic Hausdorff contents. We also provide estimates and exact values for the capacities of Cantor sets, using formulas that incorporate specific information about their geometry, the dimension of the ambient space, and the fractional parameter of the diffusion equation. Finally, we address the seemingly simple problem of determining the 1/2-caloric capacity of a rectangle in a planar context (dimension 2). For a symmetric variant of this capacity, we obtain a formula that explicitly describes its anisotropic nature, revealing an essentially different behavior compared to other known capacities in the plane, such as the analytic or Newtonian ones. Additionally, in the process of studying this problem, we prove the semi-additivity of the symmetric 1/2-caloric capacity in the plane.

Thesis advisor(s): Laura Prat Baiget & Joan Mateu Bennassar

University: Universitat Autònoma de Barcelona

Abstract

This thesis focuses primarily on the study of the (harmonic and elliptic) Dirichlet boundary value problem. The final chapter treats an independent problem in geometric measure theory: the structure of the support of locally uniform and locally uniformly distributed measures. Briefly speaking, given a (sufficiently regular) domain, the (harmonic) Dirichlet boundary value problem consists of finding a harmonic extension of a given function defined on the boundary. For continuous functions on the boundary, the solution is determined by the harmonic measure: the unique Radon probability measure that reconstructs the harmonic extension through integration of the boundary function. The elliptic Dirichlet boundary value problem and elliptic measure are defined similarly, with the Laplacian replaced by a second-order operator in divergence form with uniformly elliptic matrix. The first three chapters investigate the Hausdorff dimension of elliptic measures in planar domains. Chapter I presents preliminary material and establishes fundamental results concerning elliptic measure. Chapter II focuses on two particular cases: symmetric matrices with determinant 1, and CDC domains. Through the application of quasiconformal mappings, we derive new bounds for the Hausdorff dimension that depend solely on the ellipticity constant. For matrices with additional regularity assumptions, we establish more refined dimensional estimates. These results extend the classical work of Makarov, Jones, and Wolff to the elliptic setting, while recovering their celebrated theorems in the harmonic case. In Chapter III, we extend Wolff’s theorem to elliptic measures associated with Lipschitz matrices in Reifenberg flat domains, proving that these measures have $\sigma$-finite length provided small enough flatness depending on the ellipticity of the matrix. Chapter IV addresses the $L^2$ solvability of the (harmonic) Dirichlet boundary value problem for domains with unbounded locally flat boundary. That is, we not only establish the existence of a harmonic extension for boundary data in $L^2$, but also prove that the $L^2$ norm of the nontangential maximal function of this extension is controlled by the $L^2$ norm of the boundary data. Following classical approaches, our analysis relies on layer potential techniques. We further extend these results to the $L^p$ setting for $p$ in a dimension-dependent range, and establish the $L^{p^\prime}$ solvability for the corresponding Neumann problem. The thesis concludes with Chapter V, on locally uniform and locally uniformly distributed measures. While the structure of (globally) uniform and (globally) uniformly distributed measures is well understood through work of Kirchheim, Kowalski, and Preiss (though some open problems remain), the local analogues are not yet fully characterized. We prove that for locally uniform measures, vanishing mean curvature (defined almost everywhere in the support in codimension 1) at a point forces flatness of the connected component that contains the point. Furthermore, we prove that for locally uniformly distributed measures, local flatness of a connected component implies its global flatness. These two results are presented in any codimension. These appear to be among the first results in a possible extension (if true) the Kirchheim and Preiss theorem on the real-analytic nature of globally uniformly distributed measures to this local setting. Moreover, we show that if a locally uniform measure satisfies an additional standard growth condition on balls centered in a neighborhood of a connected component of its support, then that connected component must be a linear variety.

Thesis advisor(s): Martí Prats Soler & Xavier Tolsa Domènech

University: Universitat Autònoma de Barcelona

Abstract

The planar circular restricted three body problem (PCRTBP) models the motion of a massless body under the attraction of other two bodies, the primaries, which describe circular orbits around their common center of mass. In a suitable system of coordinates, this is a two degrees of freedom Hamiltonian system. The orbits of this system are either defined for all (future or past) time or eventually go to collision with one of the primaries. For orbits defined for all time, Chazy provided a classification of all possible asymptotic behaviors, usually called final motions. By considering a sufficiently small mass ratio between the primaries, we analyze the interplay between collision orbits and various final motions and construct several types of dynamics. We show that orbits corresponding to any combination of past and future final motions can be created to pass arbitrarily close to either one of the primaries. In particular, we also establish oscillatory motions accumulating to collisions. That is, oscillatory motions in both position and velocity, meaning that as time tends to infinity, the superior limit of the position and velocity is infinity while the inferior limit of the distance to one of the primaries is zero. Additionally, we construct arbitrarily large ejection-collision orbits (orbits which experience collision in both past and future times) and periodic orbits that are arbitrarily large and get arbitrarily close to either one of the primaries. Combining these results, we construct ejection-collision orbits connecting both primaries.

Thesis advisor(s): M. Teresa Martínez-Seara Alonso & Marcel Guàrida Munarriz

University: Universitat Politècnica de Catalunya

Abstract

This thesis presents contributions to the areas of exponentially small splitting of separatrices and Arnold diffusion in the context of theory of perturbation in Hamiltonian systems In the first chapter que study the problem of exponentially small splitting in planar Hamiltonian systems of one and a half degrees of freedom with a chomoclinic connections under the effect of a fast periodic perturbation. The literature provides different asymptotic formulas to measure the distance between the invariant manifolds in this context. In many cases, these formulas consist in a validation of the classical Melnikov approximation. In others, an alternative more sophisticated first order of the splitting based on what is referred to as inner equation has to be given. In both cases, the validity of the formulas depends on certain non-degeneracy conditions that ensures that the leading term does not vanish. The question we address is how we can describe the splitting when the non-degeneracy conditions are not met in a family of planar systems. Our result shows that the order of the splitting changes under the degeneracy conditions, but the inner equation is still valid a a tool to describe the splitting. After establishing the general results, we study with more depth the case of the classical pendulum, which is, in turn, motivated by a possible relation to Arnold’s model of diffusion. For the pendulum we introduce an explicit formula for the splitting distance in a simple case of the perturbation and we illustrate a more complex case by a numerical exploration. The second chapter centers on the generalized Arnold model. The original model was used to give, for the first time, an example of global instability in a nearly integrable system under the hypothesis that one of the perturbative parameters is exponentially small with respect to the other. In particular, Arnold proved that there exist a orbits that drift in one of the action following a chain of invariant tori at the resonance I_1=0. In a posterior result, D. Sauzin proposed a generalized version that extends the number of dimensions and generalizes the temporal dependence of the perturbation. The author studied the exponentially small character of the splitting for the tori of the aforementioned family and he gave precise exponentially small bounds. In this work we revisit the generalized Arnold model —restricted to two and a half degrees of freedom— with the aim of centering on the heteroclinic connections between invariant tori around double resonances. Our result gives explicit conditions that ensure the existence of a family of heteroclinic connections of algebraig length in the perturbative parameters. In particular, we show that, for a double resonance (0,p/q), the existence of two families of heteroclinic connections of length d ~ ε q exp(-q) under the restrictionn μ ~ √ε. The constant that determine exactly the restrictions on the parameters and the length of the connections depend on the double strongly on the resonance under consideration. We also include a comment on the possible application of the result to the construction of transition chains, even though, for the moment, we have not obtained any significant result in this direction. Our methods consist in techniques of analysis of exponentially small phenomena based on the Hamilton-Jacobi formalism. Besides, we exploit the advantage posed by usinng double ressonances, as the process of establishing the dominance of certain harmonics in exponentially small functions of two periodic functions is greatly simplified.

Thesis advisor(s): M. Teresa Martínez-Seara Alonso & Inmaculada Baldomá Barraca

University: Universitat Politècnica de Catalunya

Abstract

This thesis addresses two specific problems associated with count data: excess of zeros and their absence in capture-recapture experiment data. Counts with excess of zeros are analysed using zero-inflated models, which allow for situations where the dependent variable has more zeros than would be expected under a classical model. The basic concept is that the generation of zeros may follow a different process than the one generating positive values. The zero-inflated model is a mixture distribution combining a reference count distribution – typically Poisson or negative binomial – and a degenerate mass at zero. In this thesis, we first present a new statistic to estimate excess of zeros counts in two-parameter distributions such as the negative binomial. We also establish a classification of the different types of zeros that may be encountered, identify the various sources that give rise to them, and provide a set of recommendations for researchers to choose the best statistical model in the presence of excess of zeros counts in an ecological context. Additionally, a new statistical test (Score test) has been developed to answer the question of whether the data come from a zero-inflated Poisson distribution or a mixture of two Poisson distributions. It has been shown that the distribution of the test statistic is independent of the parameter values, making it a robust alternative to traditional tests with asymptotic distributions. The performance of the test has been evaluated using datasets from the field of dosimetry. On the other hand, regarding counts obtained through capture-recapture methods, these are used to estimate the size of a population of interest that is only partially observed. In these designs, a count is kept of the number of times an individual has been captured/observed during the observation period. In real-world applications, only positive counts are recorded, resulting in a zero-truncated distribution. Estimating the number of unobserved individuals – that is, estimating the proportion of zeros – is key to estimating the population size. In this thesis, capture-recapture data are analysed by restricting the case to situations where it is only recorded whether a subject has been observed once or multiple times (recidivist). For this type of data, a new Bayesian methodology is proposed to estimate population size, given a carefully selected prior distribution. Subsequently, another new population size estimation method is presented, this time based on non-parametric methods and with broader applicability. In this case, it is applied to right-censored data, that is, individuals observed/captured once, twice, three times, or up to “r” or more times, where “r” is the censoring value. If “r = 2”, we encounter the case of recidivist data.

Thesis advisor(s): Pere Puig Casado

University: Universitat Autònoma de Barcelona

Abstract

This thesis has two chapters, one on CaTT, a type theory for (∞, ∞)-categories, and one on decomposition spaces, which is joint work with Joachim Kock. The first chapter contains two parts. The first part begins with the fact that the dimension of an operation is equal to that of the underlying pasting diagram being composed, whereas the dimension of a coherence is strictly larger than that of the underlying pasting diagram. Based on this observation we propose a new set of rules describing an (∞, ∞)-category, in which the free-variable side conditions of the original rules in CaTT are replaced by a dimension side condition. The new rules have the advantage of being more geometric. The main result is then that the new rules and the original rules are mutually admissible. Building up to the main result are a number of technical results, which are of independent interest. A key result states that the free variables of a term in a pasting context form themselves a pasting context. In the second part we introduce and study a purely syntactic notion of lax cones and (∞, ∞)-limits on finite computads in CaTT. Conveniently, finite computads are precisely the contexts in CaTT. We define a cone over a context to be a context, which is obtained by induction over the list of variables of the underlying context. In the case where the underlying context is globular we give an explicit description of the cone and conjecture that an analogous description continues to hold also for general contexts. We use the cone to control the types of the term constructors for the universal cone. The implementation of the universal property follows a similar line of ideas. Starting with a cone as a context, a set of context extension rules produce a context with the shape of a transfor between cones, i.e. a higher morphism between cones. As in the case of cones, we use this context as a template to control the types of the term constructor required for universal property. The second chapter on decomposition spaces relates to a theorem of Bergner, Osorno, Ozornova, Rovelli, and Scheimbauer, which states an equivalence between 2-Segal spaces and certain augmented stable doubleSegal spaces, and the work of Carlier, who introduced the notion of bicomodule configuration. We establish more general equivalences, involving simplicial maps of 2-Segal spaces and abacus bicomodule configurations, extending results of Carlier. The BOORS equivalence is recovered from the special case of the identity map. One main ingredient is an analysis of the relationship between the BOORS and Carlier notions of augmentation, hitherto considered unrelated.

Thesis advisor(s): Joachim Kock

University: Universitat Autònoma de Barcelona

Abstract

The Madden-Julian Oscillation (MJO) is a fundamental component of tropical atmospheric variability, influencing significantly on global weather patterns. It modulates tropical convection, drives sub-seasonal variability, and interacts with large-scale climate phenomena such as the El Niño–Southern Oscillation (ENSO). Despite its importance, key aspects of its lifecycle, particularly its initiation, duration, and termination, remain poorly understood. Understanding these dynamics is essential for improving weather predictions and climate modeling. A statistical analysis of MJO events reveals that their durations and sizes follow a double power-law distribution, suggesting a distinction between small and long events but without a characteristic scale in this two categories. A critical duration of approximately 27 days marks a sharp increase in the probability of an event’s termination, independent of the starting and ending phases. This suggests an internal exhaustion mechanism rather than external forcing, which may have important implications for the predictability of the MJO on sub-seasonal timescales. Under this framework, this PhD thesis examines the evolution of extreme MJO events and their specific impacts on South America during the austral summer. Extreme MJO events, defined by a threshold-based classification, are distinguished from weak events through composite analyses of outgoing longwave radiation (OLR), eddy streamfunction, and velocity potential. MJO events with initiation phases 2-3 and 6-7 are analyzed in detail to determine which phases lead to the most significant regional impacts and how the associated anomalies evolve over time. Results indicate that extreme MJO events produce stronger convection anomalies in the equatorial region globally, with enhanced or suppressed convection centers exhibiting a southeastern displacement compared to weaker events. Furthermore, these MJO extreme events exert a notable influence on the South American Dipole (SAD), a key feature of regional climate variability. The study also explores how different ENSO phases modulate extreme MJO events, affecting their frequency, intensity, and subsequent impacts on South American weather patterns.

Thesis advisor(s): Marcelo Barreiro Parrillo & Álvaro Corral Cano

University: Universitat Autònoma de Barcelona

Abstract

This thesis is framed in the field of Complex Dynamics, which studies discrete dynamical systems generated by the iteration of holomorphic functions. More precisely, given a transcendent, integer or meromorphic function, we consider the discrete dynamical system generated by it. Then the complex plane is divided into two totally invariant sets: the Fatou set, where the dynamics are stable; and Julia’s ensemble, its complement, where the dynamics are chaotic. The Fatou set is open and generally has infinite related components, called Fatou components, and they are periodic, pre-periodic, or wandering. One of the basic results in Complex Dynamics (demonstrated by Fatou and Julia for rational functions) is that the Julia set is the closure of the repulsing periodic points of the function. This result was generalized by Baker by integer functions, and by Baker, Kotus, and Lü by transcendent meromorphic functions. We note that, given an invariant Fatou component, then its boundary is an invariant closed subset of the Julia set. So, the next question arises naturally: given a meromorphic function, and it is a periodic Fatou component, are the periodic points dense at their boundary? Note that, although the periodic points are dense in the Julia set, a priori they could accumulate at the boundary from its complement, without being at the boundary For example, if the Fatou component is a rotation domain with a locally connected boundary, then there is no periodic point. However, F. Przytycki and A. Zdunik showed that, by rational functions, rotation domains (i.e. Siegel’s disks and Herman’s rings) are the only exceptions for which the periodic points are not dense at the boundary. In particular, they gave a positive answer to the previous question for attraction or parabolic basins of rational functions. The work of F. Przytycki and A. Zdunik already shows us that the answer to such an elementary question is far from simple. Indeed, an exhaustive study of the boundaries of such Fatou components (which may not be locally connected) is necessary, combining tools of dynamics, measurement theory and conformal analysis. In the particular case of simply connected attraction basins, the proof is based on the properties of the function at the boundary from the point of view of the theory of measurement and Lyapunov’s exponents, as well as precise estimates of the distortion of the Riemann application and the finite Blaschke products in the unit circle, and Pesin’s theory conforms. For components of Fatou not limited to transcendent functions, the situation is even more delicate, due to the presence of the essential singularity, and most of the above techniques cannot be applied. Moreover, since the boundary of the Fatou component is not compact, it is not compact, nor is the existence of periodic points on the boundary evident. In view of the above questions, and the existing previous work to understand the boundaries of transcendent Fatou components, the following conjecture naturally arises, which is a large open problem in transcendent dynamics. Let it be a meromorphic function, and let it be a simply connected periodic Fatou component, such that it is not univalent. Then, there is a periodic point on the border of such a component of Fatou. In addition, if it is an attractor or parabolic basin, or a doubly parabolic Baker domain, then the periodic points are dense at the boundary. This thesis should be understood as significant progress in proving the above conjecture. Indeed, we demonstrate the existence and density of periodic points at the boundary of Fatou components under very weak hypotheses in the postsingular set, together with additional results in relation to boundary dynamics, escape points and accessibility. During the thesis, new techniques have been demonstrated, such as estimates in the distortion of internal functions and Pesin’s theory for transcendent functions.

Thesis advisor(s): Núria Fagella Rabionet

University: Universitat de Barcelona

Abstract

The ability of the immune system to recognize tumor cells as a threat and destroy them is a crucial defense mechanism against neoplastic growth and metastatic spread of cancer. However, this recognition alone is often insufficient to prevent cancer development. To bolster these defenses, modern oncology relies on immunotherapy, which aims to enhance the efficiency of the immune response. The efficacy of these therapies can be compromised by various factors, including the tumor microenvironment and the metabolic state of cancer cells. Consequently, weakening cancer cell metabolism to hinder their proliferation and render them more susceptible to immune attacks holds great promise for improving treatment outcomes. Understanding the role of cancer cells metabolic states in determining their interactions with the immune system is crucial for advancing and broadening cancer immunotherapy. In this regard, mathematical models that consider cancer cell-intrinsic metabolic traits and are validated with experimental data are paramount in cancer immunotherapy research. This thesis focuses on developing phenotype-structured mathematical models to elucidate the dynamic interplay between cancer cells and T cells. Specifically, we examine how inactivating gene mutations in key metabolic drivers within cancer cells influence the cytolytic potency of the immune response. We constructed an ordinary differential equation model depicting the competition between cancer cells and T cells using real-time cytolysis data from killing potency experiments involving the co-culture of metabolically engineered cancer cells with T cells. In this model, T cells can exist in one of two states: active or exhausted, which reflect their functionality. We calibrated the model using experimental data obtained under various scenarios, including pairs of parental, metabolically-competent cancer cells and isogenic counterparts with targeted disruption of one or more metabolic enzymes and transporters, through extensive numerical simulations and evolutionary computation methods (artificial intelligence). Estimating key parameters revealed the diverse impacts of metabolic pathway inhibition on tumor proliferation, metabolic fitness, and immune response efficacy. Our original approach highlights the importance of understanding the intricate roles of metabolic pathways in cancer cells in shaping the clinical outcomes of T cell-based immunotherapy. Our finding also provide valuable insights for future metabolically guided cancer immunotherapy strategies.

Thesis advisor(s): Tomás Alarcón Cor, Josep Sardanyes Cayuela & Javier Abel Menéndez Menéndez

University: Universitat Autònoma de Barcelona

Abstract

 Development is the process by which the complex anatomy of multicellular organisms is built in each generation and it constitutes one of the very few natural processes capable of generating so much complexity in such a relatively short period of time. From a broad and phenomenological point of view, development can be described as the sequence of transformations from one developmental pattern (i.e., a specific distribution of cell types along the developing embryo) to another, that begins with the fertilized egg and finishes with the complete functional adult individual. Although there are many intertwined mechanisms responsible for pattern transformation during development, in our work we focus only in what developmental biologists call inductive mechanisms, that is, we will only take into consideration those transformations that occur as a response to biochemical signals sent from one cell to another. In this sense, the main question we want to address is: ”which gene regulatory networks governing cell signaling (i.e, which network topologies of activating and inhibiting interactions between gene products) can actually lead to proper pattern transformations?”. Our theoretical analysis and numerical simulations, performed in the framework of reaction-diffusion equations, show that, regardless of the immense number of patterning gene networks that one can think of (specially when the number of gene products at hand is large), they can all be sorted out into just three fundamental classes of gene networks capable of pattern transformation, and their combinations. Gene networks within each of these three classes share the same topological properties, exhibit similar reaction-diffusion dynamics and lead to analogous final patterns.

Thesis advisor(s): Isaac Salazar Ciudad

University: Universitat Autònoma de Barcelona

Abstract

This thesis consists of two parts. In the first part, we propose an algorithm to calculate rotation intervals of dynamical systems defined by maps on the unit circle. This algorithm allows for the exact computation of the rotation interval for a wide family of maps when their endpoints are rational. This algorithm is general and does not require the target function to be differentiable. The second part of the thesis focuses on obtaining a semianalytic method to calculate the truncated wavelet expansion for an attractor on a quasi-periodically forced skew product. To obtain the wavelet expansion, we had to refine and enhance the capability to evaluate a wavelet at a point, developing applications of the Daubechies-Lagarias Algorithm in the process. Finally, with the obtained truncated series, we applied results from functional analysis that allows us to determine the regularity of functions based on their wavelet coefficients. This has enabled us to measure (imperfectly) the strangeness of the attractors. From this, we have been able to study some cases of non-chaotic strange attractors.

Thesis advisor(s): Lluís Alsedà i Soler

University: Universitat Autònoma de Barcelona

Abstract

In this thesis we study several mathematical objects that are essential to formulate and model physical systems. Applying the tools provided by differential geometry, we develop and analyze different mathematical structures that are used in three physical contexts: dissipative dynamics, integrable systems and geometric quantization. To do it, we mainly employ the setting of b-symplectic geometry, a natural extension of symplectic geometry which is specifically designed to address manifolds with boundary. It is based on the concept of b-forms introduced by Melrose and was initiated by Guillemin, Miranda and Pires. Firstly, in the context of dissipative dynamics, we introduce and discuss a variety of twisted b-cotangent models. In these models, defined on the cotangent bundle of a smooth manifold, the fundamental structure is a b-symplectic form that is singular within the fibers of the bundle. Our models give rise to dynamical systems governed by the standard Hamiltonian of a free particle, accompanied by a positiondependent potential. After examining different types of potentials and finding that all of them induce dissipation of energy in the system, we prove that these twisted bcotangent models offer a suitable Hamiltonian formulation for dissipative systems. Consequently, they expand the scope of Hamiltonian dynamics and bring a new approach to the study of non-conservative systems. Secondly, in the context of integrable systems, we introduce and investigate bsemitoric systems, a family of systems that generalizes simultaneously semitoric systems and b-toric systems, and which is tailored for b-symplectic manifolds. We provide a comprehensive definition of b-semitoric systems, that adapts the characteristics of semitoric systems to the framework of b-symplectic manifolds, and we construct three examples of this type of system. The three examples are based on modifications of the coupled angular momenta system, a classical semitoric system that represents the coupling of two rigid rotors. Our examination of the examples, which includes the classification of the singular points and the study of the global dynamics, allows us to highlight the unique characteristics of b-semitoric systems. Thirdly, in the context of geometric quantization, we introduce a Bohr-Sommerfeld quantization method for b-symplectic toric manifolds. We establish that the dimension of this quantization method depends on a signed count of the integer points in the image of the moment map of the toric action. Additionally, we demonstrate its equivalence with the formal geometric quantization of such manifolds. Furthermore, we present a geometric quantization model based on sheaf cohomology, suitable for integrable systems with non-degenerate singularities, that also relies on the count of the integer points in the image of the moment map.

Thesis advisor(s): Eva Miranda Galcerán

University: Universitat Politècnica de Catalunya

Abstract

In this thesis we investigate various aspects of the dynamics of Hamiltonian vector fields in singular symplectic manifolds.

We concentrate on two questions: first, we investigate a generalization of the Arnold conjecture in the setting of singular symplectic geometry. Second, we explore constructions for integrable systems in this context.

In Chapter 2 we provide the background material required for this thesis. We start by delving into the theory of symplectic geometry. Then, we present the Arnold conjecture, which asserts that there is a lower bound on the number of 1-periodic orbits for a non-degenerate Hamiltonian system, and that this lower bound can be formulated strictly in topological terms. We also present a tool used in the investigation of this conjecture: Floer theory.

Then, we explain some notions of Poisson geometry before we explore a notion fundamental to this thesis: that of a b m–symplectic manifold. These are manifolds with a structure that is symplectic almost everywhere but “blows up” at a hypersurface, which we call the singular hypersurface. We lay out some techniques used in the study of b m–symplectic manifolds, with an emphasis on a procedure called desingularization.

Finally, we give a summary of the theory of integrable systems and the study of their singular points.

In Chapter 3 we investigate the dynamical behaviour of certain vector fields in b m–symplectic geometry, coming from b m–Hamiltonians. We focus on the study of their dynamics in a neighbourhood of the singular hypersurface, and find a family of b m–Hamiltonians where a version of the Arnold conjecture can be formulated. Then, we explore new aspects of the desingularization procedure in relation to the b m–Hamiltonian dynamics, and provide some techniques that allow us to relate these dynamics to those of classical symplectic geometry. We conclude with two results yielding partial versions of the Arnold conjecture for b m–Hamiltonian vector fields.

In Chapter 4 we show the existence of a Floer homology for b m–symplectic manifolds. This we manage through an investigation of the Floer equation for the family of b m–Hamiltonians presented in Chapter 3. In Chapter 5 we introduce the notion of the classes of b-integrable and b- semitoric systems. We study the features of b-semitoric systems using some interesting examples and the investigation of their singular points.

Thesis advisor(s): Sonja Verena Hohloch and Eva Miranda Galcerán

University: Universiteit Antwerpen and Universitat Politècnica de Catalunya

Abstract

Episodic memory depends crucially on the capacity of neuronal circuits to store information in a one-shot fashion about events that unfold over a time-scale of seconds. Standard Hebbian plasticity rules, such as STDP that require repeated pairing of pre- and post-synaptic activation, are inadequate as physiological mechanisms underlying this type of rapid learning. Contrary to this, Behavioral Timescale Synaptic Plasticity (BTSP), a newly discovered form of plasticity in the hippocampus, operates on a timescale of seconds. This mechanism induces long-lasting synaptic changes after a single experience, driven by dendritic plateau potentials, making it ideally suited for encoding episodic memories. After just one trial, BTSP’s ability to rapidly form place fields in CA1 neurons underscores its critical role in memory formation. This thesis investigates the role of BTSP in memory storage within the hippocampal network. We derive a simplified BTSP model that lends itself to rigorous mathematical analysis, extending this framework to recurrent networks such as the CA3 region of the hippocampus to explore its memory storage properties. Through a detailed examination of recall dynamics, our results demonstrate that BTSP facilitates the encoding and retrieval of a large number of memories, with variability enhancing both storage and recall. Additionally, we explore the non-Hebbian aspect of BTSP, showing that it supports homogeneous representations in CA3. Consequently, we conclude that BTSP is a viable candidate mechanism underlying episodic memory.

Thesis advisor(s): Alexander Charles Roxin

University: Universitat Autònoma de Barcelona

Abstract

In this work, we explore the existence of tubular formulas for valuations on Riemannian manifolds. Specifically, we calculate these formulas for invariant valuations in real, complex, and quaternionic space forms. We introduce the tubular operator, both for valuations and curvature measures, which gives the value of these functionals over the tubes around submanifolds. In isotropic spaces, the operator is expressed in a simplified form. For complex space forms, we derive explicit tubular formulas. This approach is then extended to quaternionic space forms, where we focus on Federer valuations. Finally, we apply our results to calculate the push-forward of valuations via the quaternionic Hopf fibration.

Thesis advisor(s): Gil Solanes Farrés

University: Universitat Autònoma de Barcelona

Abstract

This thesis focuses on utilising mathematical models to analyse health data, aiming to enhance analytical processes through practical and accessible frameworks. Organised as a collection of publications, it explores two primary themes. The first theme centres on analysing population health data using hierarchical models for survival and spatial analysis within extensive datasets. Bayesian statistical models, specifically employing Integrated Nested Laplace Approximation (INLA) as an alternative to the conventional Markov Chain Monte Carlo (MCMC) methods, are applied to examine stroke prevalence in Poland. This approach is selected for its computational efficiency, addressing the challenges posed by the significant computational complexity of MCMC methods in large-scale population-based datasets. Consequently, the proposed methodology for analysing stroke data in Poland demonstrates potential for broader application in similar health research contexts. The second theme focuses on statistical models in biological dosimetry, which involves estimating levels of ionising radiation exposure based on biomarkers that quantify radiation-induced damage in human blood cells. Bayesian methods are particularly valuable here for their practicality and efficiency in developing and applying models. The thesis first discusses Bayesian approaches to refining models for the gamma-H2AX assay exploring a finite-mixture Poisson model. Given the biomarker’s sensitivity to time, a novel approach incorporating time as a covariate in the model is presented. Secondly, this thesis includes a study comparing dicentric and translocation biomarkers, examining whether these biomarkers provide consistent inferences and how the scoring method used to obtain the biomarker counts affects the results. It also addresses challenges in partial body irradiation scenarios and provides guidance on biomarker selection for specific radiation exposures.

Thesis advisor(s): Pere Puig Casado, Virgilio Gómez Rubio i Carmen Armero i Cervera

University: Universitat Autònoma de Barcelona

Abstract

This thesis is dedicated to the study of elliptic and parabolic Partial Differential Equations, both local and nonlocal. More specifically, this work concerns the regularity properties of some obstacle problems. Obstacle problems are prototypical examples of free boundary problems, that is, PDE problems where the unknowns are not only a function, but also a subdivision of the domain into different regions, and the PDE satisfied in each region is different. Free boundary problems are a very active field of research. On the one hand, free boundaries are a good model for interfaces in real-world settings, with applications in Physics, Biology, Finance and Engineering. On the other hand, they have been a source of interesting mathematical challenges, motivating the fine analysis of solutions to elliptic and parabolic equations. This Thesis is divided into two Parts. Part I is devoted to the study of several different obstacle problems. In Chapter 1, we study the obstacle problem for parabolic nonlocal operators, in the supercritical regime s < 1/2. We establish the optimal C^{1,1} regularity of solutions, which is surprisingly better than in the elliptic problem, and we also show that the free boundary is globally C^{1,α}. Our main difficulties are the lack of monotonicity formulas, and the supercritical scaling of the equation, that is, the fact that the highest order of differentiation corresponds to the time derivative. Chapter 2 is devoted to the generic regularity properties of the free boundary in the thin obstacle problem. Since there are many pathological examples of solutions to free boundary problems, often the goal is instead of proving regularity for all solutions, proving regularity for most solutions in an appropriate sense. In our work, we show that, for one-parameter monotonous families of solutions, for almost every solution, the free boundary is smooth outside of a set of codimension 2 + α (in the free boundary). In particular, this means that in R^3 and R^4, the free boundary is generically smooth. We conclude Part I with Chapter 3, where we use a nonlocal analogue of the Bernstein technique to establish semiconvexity estimates for a wide class of nonlinear nonlocal elliptic and parabolic equations, including obstacle problems. As a consequence, we extend the known regularity theory for nonlocal obstacle problems in the full space to problems in bounded domains. In Part II, we extend the boundary Harnack inequality to (local) elliptic and parabolic equations with a right-hand side. The boundary Harnack is a classical result that states that if u and v are positive harmonic functions that vanish on part of the boundary of a regular enough domain, then u/v is bounded and Hölder continuous up to the boundary. Boundary Harnack inequalities are used in the proof of the smoothness of free boundaries in several obstacle problems, in the key step of seeing that if a free boundary is flat Lipschitz, then it is C^{1,α}. The goal of our work was to extend the regularity theory of obstacle problems to the fully nonlinear setting. To do so, we developed boundary Harnack inequalities for equations in non-divergence form with a right-hand side. Chapter 4 concerns elliptic equations and Chapter 5 is about parabolic equations. The techniques used are different. In the elliptic setting, it is enough to use barriers, scaling arguments and a standard iteration to deduce the Hölder regularity of the quotient. However, in the parabolic world, the proofs are much more involved and they are based on a delicate contradiction-compactness argument.

Thesis advisor(s): Xavier Ros Oton

University: Universitat de Barcelona

Abstract

This thesis concerns three problems of geometric realizations of combinatorial structures via polytopes and polyhedral subdivisions. A polytope is the convex hull of a finite set of points in a Euclidean space Rd. It is endowed with a combinatorial structure coming from its faces. A subdivision is a collection of polytopes whose faces intersect properly and such that their union is convex. It is regular if it can be obtained by taking the lower faces of a lifting of its vertices in one dimension higher. We first present a new geometric construction of many combinatorially different polytopes of fixed dimension and number of vertices. This construction relies on showing that certain polytopes admit many regular triangulations. It allows us to improve the best known lower bound on the number of combinatorial types of polytopes. We then study the projections of permutahedra, that we call sweep polytopes because they model the possible orderings of a fixed point configuration by hyperplanes that sweep the space in a constant direction. We also introduce and study a combinatorial abstraction of these structures: the sweep oriented matroids, that generalize Goodman and Pollack’s theory of allowable sequences to dimensions higher than 2. Finally, we provide geometric realizations of the s-weak order, a combinatorial structure that generalizes the weak order on permutations, parameterized by a vector s ∈ (Z>0)n. In particular, we answer Ceballos and Pons’s conjecture that the s-weak order can be realized as the edge-graph of a polytopal complex that is moreover a subdivision of a permutahedron.

Thesis advisor(s): Arnau Padrol Sureda, Francisco Santos Leal i Ilia Itenberg

University: Universitat de Barcelona

Abstract

In this thesis we adapted the problem of continuous congested optimal transport to the Heisenberg group, equipped with a sub-Riemannian metric: we restricted the set of admissible paths to the horizontal curves. We obtained the existence of equilibrium configurations, known as Wardrop Equilibria, through the minimization of a convex functional, over a suitable set of measures on the horizontal curves. Moreover, such equilibria induce trans­ port plans that solve a Monge-Kantorovic problem associated with a cost, depending on the congestion itself, which we rigorously defined. We also proved the equivalence between this problem and a minimization problem defined over the set of p-summable horizontal vector fields with prescribed divergence. We showed that this new problem admits a dual formulation as a classical minimization problem of Calculus of Variations. In addition, even the Monge-Kantorovich problem associated with the sub-Riemannian distance turns out to be equivalent to a minimization problem over measures on horizontal curves. Passing through the notion of horizontal transport density, we proved that the Monge-Kantorovich problem can also be formulated as a minimization problem with a divergence-type constraint. Its dual formulation is the well-known Kantorovich duality theorem. In the end, we treated the continuous congested optimal transport problem with orthotropic cost function: we proved the Lipschitz regularity for solutions to a pseudo q-Laplacian-type equation arising from it.

Thesis advisor(s): Albert Clop i Giovanna Citti

University: Universitat de Barcelona

Abstract

L’objectiu d’aquesta tesi ha estat representar la valoració p-àdica dels valors especials de funcions-L p-àdiques mitjançant característiques d’Euler utilitzant diferents teories de cohomologia en diversos contextos. Més específicament, s’han utilitzat invariants homotòpics com l’espectre K(1)-local de K-teoria per descriure aquests valors en el context de cossos reals totalment commutatius i no-commutatius, i també en alguns casos per a cossos quadràtics imaginaris. El cas més exitós ha estat el commutatiu i totalment real, on ara es pot entendre la valoració p-àdica de la funció-L p-àdica de Deligne-Ribet utilitzant cohomologia ètale, K-teoria K(1)-local, i la fibra del mapa de traça ciclotòmica (K(1)-local), estenent així el treball de Hesselholt, que només cobreix el cas F = Q. Finalment, s’ha establert la base per estendre aquests resultats en altres direccions, com ara en el cas de cossos de funcions, o més concretament, en el cas de varietats abelianes amb reducció semiestable sobre cossos de funcions.

Thesis advisor(s): Victor Rotger Cerdà

University: Universitat Politècnica de Catalunya

Abstract

Working memory, the capacity to maintain and manipulate information in our minds when it is no longer available in the environment, is a central function of cognition. One of the most important neuronal correlates of this cognitive function are the so-called persistent neurons, which respond selectively to sensory stimulation and sustain their increased activity even after removing the stimulus. This phenomenon, most frequently observed in the prefrontal cortex, has been successfully described by neural network models with attractor dynamics. However, only a few of the neurons engaged in working memory tasks have persistent activity. Moreover, analysis of the experimental recordings at the population level reveals that the code undergoes a change between the stimulus presentation and the maintenance epochs, which is not compatible with a working memory code that would only rely on stably active persistent cells. The prevalence of this finding has motivated the proposal of alternative mechanisms, but current computational models that explain dynamics fail to include stable epochs or do not provide a clear mechanistic interpretation. In this thesis, we use statistical data analysis and neural network modeling to investigate whether specialized neuronal subpopulations underlie the stable and dynamic working memory codes. First, we investigated the connection between the observed dynamics in the working memory code and the functional structure of the prefrontal circuits. We analyzed prefrontal recordings from behaving macaque monkeys and observed that feature selectivity is non-randomly distributed across the neurons. This non-random or structured feature selectivity distribution is related to functional distinct subpopulations whose contrasting activity explains the dynamic to stable transition in the working memory code. Second, we developed a computational model that represents three functional subpopulations as attractor networks working on different dynamic regimes. The model illustrates how the population structure, which implies different neurons active at different task epochs, is directly related to the dynamic transition in the code. Furthermore, we show how the three-network architecture can be easily extended to account for additional features, such as ramping activity and variable maintenance periods. Third, our subpopulation-based networks have the functional advantage of being robust against distracting stimuli. The model captures the experimentally observed vulnerability to distractors presented shortly after stimulus removal. Moreover, it predicts that top-down feedback enhances the overall network’s robustness. In summary, we show how the presence of functional subpopulations in the prefrontal cortex can be related to the dynamic to stable transition in the working memory code and to an enhanced capacity to filter out distracting stimuli. In conclusion, our work reconciles attractor dynamics with the observed dynamic changes in the code, still suggesting that attractor dynamics are essential for working memory maintenance.

Thesis advisor(s): Klaus Wimmer

University: Universitat Autònoma de Barcelona

Abstract

This thesis is dedicated to the complex wave structure arising in hydrodynamics of relativistic scenarios when considering realistic fluids with a rich thermodynamics. The equation of state is a constitutive relation encoding the thermodynamic properties of a fluid and, in compressible fluid dynamics, it is needed to close the evolution equations. A nonconvex equation of state is a candidate for inducing complex wave dynamics. With the purpose of studying nonconvex Special Relativistic Hydrodynamics (SRHD), the thesis is divided in two parts. The first one is devoted to the study of nonconvex SRHD from the point of view of the solution of the evolution equations, which consist of a nonlinear hyperbolic system of conservation laws. The second part put the stress on the modeling of realistic fluids taking into account the implications on the dynamics studied in the first part. On the one hand, we present an exact Riemann solver for nonconvex SRHD, extending its applicability to the case of nonzero tangential velocities. The Riemann problem is an initial condition for the system, the fundamental test in hydrodynamics. Its solution contains all the elements present in more complicated scenarios and allows to understand the wave dynamics that may arise. By providing the exact solution, we enhance the understanding of the intricate dynamics at play in nonconvex relativistic systems. We particularize the solver for a phenomenological nonconvex equation of state and provide the exact solution for a series of standard problems including relativistic blast waves. We employ the exact solutions obtained to validate numerical methods used to solve SRHD equations initialized with complex initial conditions. We measure the accuracy of two of the most commonly used methods in the field and analyze their performance in the presence of complex wave structure. We continue our analysis focusing on neutron stars as astrophysical objects composed by a fluid that undergo relativistic hydrodynamics evolution. Realistic models for this matter lead to tabulated equations of state, comprising detailed mycrophysical effects but representing a computationally inefficient option for numerical simulations. We concentrate on the modeling of this tabulated data with a simple analytic expression that gives special consideration to phase transitions, a phenomena of the matter with the potential to make the equation of state nonconvex. We analyze the implications of our model in the stellar properties of the neutron star and its hydrodynamic evolution, comparing the results with current analytic models employed in simulations.

Thesis advisor(s): Susana Serna Salichs

University: Universitat Autònoma de Barcelona

Abstract

We study discrete energy minimization problems on two-point homogeneous manifolds. Since finding N-point configurations with optimal energy is highly challenging, recent approaches have involved examining random point processes with low expected energy to obtain good N- point configurations. In Chapter 2, we compute the second joint intensity of the random point process given by the zeros of elliptic polynomials, which enables us to recover the expected logarithmic energy on the 2-dimensional sphere previously computed by Armentano, Beltrán, and Shub. Moreover, we obtain the expected Riesz s-energy, which is remarkably close to the conjectured optimal energy. The expected energy serves as a bound for the extremal s-energy, thereby improving upon the bounds derived from the study of the spherical ensemble by Alishahi and Zamani. Among other additional results, we get a closed expression for the expected separation distance between points sampled from the zeros of elliptic polynomials. In Chapter 3, we explore the average discrepancies and worst-case errors of random point configurations on the d-dimensional sphere. We find that the points drawn from the so called spherical ensemble and the zeros of elliptic polynomials achieve optimal spherical L^2 cap discrepancy on average. Additionally, we provide an upper bound for the L^intiy discrepancy for N-point configurations drawn from the harmonic ensemble on any two-point homogeneous space, thereby generalizing the previous findings for the sphere by Beltrán, Marzo and Ortega- Cerdà. We introduce a nondeterministic version of the Quasi Monte Carlo (QMC) strength for random sequences of points and compute its value for the spherical ensemble, the zeros of elliptic polynomials, and the harmonic ensemble. Finally, we compare our results with the conjectured QMC strengths of certain deterministic distributions associated with these random point processes. In Chapter 4, our focus hits to the Green energy minimization problem. Firstly, we extend the work by Beltrán and Lizarte on spheres to establish a close to sharp lower bound for the minimal Green energy on any two-point homogeneous manifold, improving on the existing lower bounds on projective spaces. Secondly, by adapting a method introduced by Wolff, we deduce an upper bound for the L^intiy discrepancy of N-point sets that minimize the Green energy.

Thesis advisor(s): Jordi Marzo Sánchez

University: Universitat de Barcelona

Abstract

This PhD dissertation deals with qualitative questions from the theory of elliptic Partial Differential Equations (PDE) and integro-differential equations. We are primarily interested in a distinguished class of solutions satisfying appropriate minimality conditions. The first part of the thesis provides a regularity theory for stable solutions to semilinear problems involving variable coefficients. Here, stability refers to the nonnegativity of the principal eigenvalue of the linearized equation. For variational problems, this amounts to the nonnegativity of the second variation, a necessary condition for minimality. Our main achievement is to show the boundedness of stable solutions in C11 domains in the optimal range of dimensions n < 10. This result is new even for the Laplacian, for which a C3 assumption on the domain was needed. The second part furnishes natural sufficient conditions for the minimality of critical points in a general nonlocal framework. Namely, we construct a calibration for nonlocal energy functionals, under the assumption that the critical point is embedded in a family of sub/supersolutions whose graphs produce a foliation. As a consequence, we deduce that the solution is a minimizer with respect to competitors taking values in the foliated region. Our result extends, for the first time, the classical Weierstrass extremal field theory in the Calculus of Variations to a nonlocal setting. To find a calibration for the most basic fractional functional, the Gagliardo-Sobolev seminorm, was an important open problem that we have solved.

Thesis advisor(s): Xavier Cabré Vilagu

University: Univeristat Politècnica de Catalunya

Abstract

This work is dedicated to the study several models of random structures from the perspective of first-order logic. We prove that the asymptotic probabilities of first-order statements converge in a general model of random structures with linear density, extending previous results by Lynch. Additionally, we give an application of this result to the random SAT problem. We also inspect the set of limiting probabilities of first-order properties in sparse binomial graphs, binomial d-uniform hypergraphs and graphs with given degree sequences. In particular, we characterize the conditions under which this set of asymptotic probabilities is dense in the interval [0, 1]. Finally, we introduce the question of whether preservation theorems, namely Los-Tarski Theorem and Lyndon’s Theorem, hold in a probabilistic sense in various models of random graphs. We obtain several positive results in different regimes of the binomial random graph and uniform graphs from addable minor-closed classes.

Thesis advisor(s): Marc Noy Serrano

University: Univeristat Politècnica de Catalunya

Abstract

In the case of partially ordered categories (posets for short), it is shown that pseudo-projective property is equivalent to cofibrant in the covariant functors category described in this work. A notion of Mackey functor for posets is also introduced, inspired by the classical notion of Mackey functor for orbit categories. In this case, it is proven that Mackey functors with an additional notion of quasi-unit are cofibrant; therefore, their higher colimits vanish in positive degrees. Using the combinatorial structure of the replacement and the presented computation tools, explicit vanishing bounds for the higher limits are proven. Using different strategies, these are described based on the geometry of the poset, local bounds of higher limits, and filtrations from atomic functors. Finally, the case of higher limits of functors indexed on CL-shellable posets is studied in detail. These posets have the homotopy type of a wedge sum of spheres of the same dimension, so the higher limits in strictly positive degrees of a constant functor are concentrated in a single degree. Motivated by this particular case, a sufficient property for a functor is abstracted, which guarantees that its higher limits vanish for dimensions lower than the length of the poset. As an example of application, the case of the family of n-linear forms functors in hyperplane arrangements is described.

Thesis advisor(s): Natàlia Castellana and Antonio Díaz

University: Universitat Autònoma de Barcelona

Abstract

In the present thesis, we study the theory of decomposition spaces, focusing on the interval construction for decomposition spaces and the decomposition space of subdivided intervals U, which was constructed by Gálvez, Kock, and Tonks as a recipient of Lawvere’s interval construction. Our interest in U is due to the Gálvez–Kock–Tonks conjecture, which states that U enjoys a certain universal property: for every complete decomposition space X, the space of culf functors from X to U is contractible. The first main contribution, developed in collaboration with Alex Cebrian, is to introduce the concept of connected and non-connected directed hereditary species and show that they have associated monoidal decomposition spaces, comodule bialgebras, and operadic categories. The second main contribution is to prove the Gálvez–Kock–Tonks conjecture. First, we proved the conjecture for the discrete case. For the general case of the conjecture, we impose cardinal bounds through the Möbius condition for decomposition spaces. This is a certain finiteness condition ensuring that the general Möbius inversion principle admits a homotopy cardinality. From this perspective proving the conjecture is equivalent to proving that the decomposition space of subdivided Möbius intervals is a terminal object in the ∞-category of Möbius decomposition spaces and culf maps. The proof is given by combining the theory of (∞,2)-colimits, the interval construction, and the straightening-unstraightening equivalence of ∞-categories. The Möbius case, together with the fact that the ∞-category of decomposition spaces and culf maps is locally an ∞-topos imply that the ∞-category of Möbius decomposition spaces and culf maps is an ∞-topos.

Thesis advisor(s): Joachim Kock

University: Universitat Autònoma de Barcelona

Abstract

In this work we study various mathematical problems arising from the robotic manipulation of cloth. First, we develop a locking-free continuous model for the physical simulation of inextensible textiles. We present a novel ‘finite element’ discretization of our inextensibility constraints which results in a unified treatment of triangle and quadrilateral meshings of the cloth. Next, we explain how to incorporate contacts, self-collisions and friction into the equations of motion, so that frictional forces and inextensibility and collision constraints may be integrated implicitly and without any decoupling. We develop an efficient ‘active-set’ solver tailored to our non-linear problem which takes into account past active constraints to accelerate the resolution of unresolved contacts and moreover can be initialized from any non-necessarily feasible point. Then, we embark ourselves in the empirical validation of the developed model. We record in a laboratory setting –with depth cameras and motion capture systems– the motions of seven types of textiles (including e.g. cotton, denim and polyester) of various sizes and at different speeds and end up with more than 80 recordings. The scenarios considered are all dynamic and involve rapid shaking and twisting of the textiles, collisions with frictional objects and even strong hits with a long stick. We then, compare the recorded textiles with the simulations given by our inextensible model, and find that on average the mean error is of the order of 1 cm even for the largest sizes (DIN A2) and the most challenging scenarios. Furthermore, we also tackle other problems relevant to robotic cloth manipulation, such as cloth perception and classification of its states. We present a reconstruction algorithm based on Morse theory that proceeds directly from a point-cloud to obtain a cellular decomposition of a surface with or without boundary: the results are a piecewise parametrization of the cloth surface as a union of Morse cells. From the cellular decomposition the topology of the surface can be then deduced immediately. Finally, we study the configuration space of a piece of cloth: since the original state of a piece of cloth is flat, the set of possible states under the inextensible assumption is the set of developable surfaces isometric to a fixed one. We prove that a generic simple, closed, piecewise regular curve in space can be the boundary of only finitely many developable surfaces with nonvanishing mean curvature. Inspired on this result we introduce the dGLI cloth coordinates, a low-dimensional representation of the state of a piece of cloth based on a directional derivative of the Gauss Linking Integral. These coordinates –computed from the position of the cloth’s boundary– allow to distinguish key qualitative changes in folding sequences.

Thesis advisor(s): Jaume Amorós Torrent and Maria Alberich Carramiñana

University: Universitat Politècnica de Catalunya

Abstract

The thesis is mainly divided into two parts. In essence, the first one is an extension of the paper [Ter22]. Using the regularized Siege-Weil formula of [GQT14] we obtain an explicit expression for the truncated integral of the Siegel theta function. The main application of this result is an explicit formula for the integral of the logarithm of the Borcherds forms. The result involves different zeta values and coefficients of Eisenstein series. It completes the work of [Kud03]. Besides the aforementioned formula for the integral of the theta function, a detailed analysis of the Siegel theta function near the infinity is required. Chapter two is an extension of the work with Antonio Cauchi in [CT]. The purpose of this part is twofold. On the one hand, under some conditions, we show that the multiplicity of the Shalika model of unramified representations for the group GU(2, 2) is one. Using this result and following the ideas of [Sak06], we are able to find an expression of the Shalika functional in terms of the Satake parameter of a representation in GSp4. On the other hand, we use this result and to establish a relationship between a zeta integral for a group GU(2,2) and a twisted standard L-function of GSp4, where the relation between the involved automorphic representations is given by the theta correspondence.

Thesis advisor(s): Victor Rotger Cerdà and Gerard Freixas i Montplet

University: Universitat Politècnica de Catalunya

Abstract

This thesis concerns the study of invariant manifolds and periodic orbits of discrete and continuous dynamical systems. The memoir is divided into two parts that can be read independently. The first part (Chapters 1-6) is dedicated to the study of invariant manifolds associated with parabolic points and parabolic invariant tori. The second part (Chapters 7-9) concerns the study of periodic orbits of dynamical systems on manifolds. In Chapters 2 and 3 we study the existence and regularity of invariant manifolds of planar maps having a parabolic fixed point with nilpotent part using the parameterization method. The study is done for analytic maps and for finitely differentiable maps. In the analytic case, we prove the existence of an analytic one-dimensional invariant manifold under suitable conditions on the coefficients of the nonlinear terms of the map. In the differentiable case, we prove that if the regularity of the map is bigger than some value, then there exists an invariant manifold of the same regularity, away from the fixed point. In Chapter 4 we consider an analogous problem as in Chapters 2 and 3, but for planar vector fields. We present the results of existence of invariant curves of such vector fields using the results from the previous chapters and the fact that, under suitable conditions, the invariant manifolds of a vector field are the same ones as the invariant manifolds of its time-t flow. In Chapters 5 and 6 we consider maps and vector fields having a d-dimensional parabolic invariant torus with nilpotent part. In this context, we give conditions on the coefficients of the nonlinear terms of the map (resp. vector field) under which the invariant torus possesses stable and unstable invariant manifolds. We also consider the same problem for non-autonomous vector fields that depend quasiperiodically on time, and we present some applications of our results. All the results of existence of invariant manifolds are stated in two steps. In the first step we present an algorithm to compute an approximation of a parameterization of the invariant manifold. In the second step, we present an «a posteriori» result, which ensures that there exists a true invariant manifold close to that approximation. Combining the two results we obtain the existence of an invariant manifold which is well approximated by the parameterization provided in the first step. In Chapter 8 we use the Lefschetz numbers and the Lefschetz zeta function to obtain information on the set of periods of certain diffeomorphisms on compact manifolds. We consider the class of Morse-Smale diffeomorphisms defined on the n-dimensional sphere, on products of two spheres of arbitrary dimension, on the n-dimensional complex projective space, and on the n-dimensional quaternion projective space. Then, we describe the minimal sets of Lefschetz periods for such Morse-Smale diffeomorphisms, which is a subset of the set of periods that are preserved under homotopy equivalence. Finally, in Chapter 9 we study the existence of limit cycles of linear vector fields on manifolds. It is well known that linear vector fields in R^n can not have limit cycles, because either they do not have periodic orbits or their periodic orbits form a continuum. In that chapter, we show that linear vector fields defined in some manifolds different from R^n can have limit cycles and we consider the question of how many limit cycles can they have at most.

Thesis advisor(s): Ernest Fontich Julià and Jaume Llibre

University: Universitat Autònoma de Barcelona

Abstract

Rational iteration is the study of the asymptotic behaviour of the sequences given by the iterates of a rational map on the Riemann sphere. According to Montel’s theory on normal families, the phase space (also called the dynamical plane) is divided in two completely in­ variant sets known as the Fatou set (an open set where the dynamics is tame) and the Julia set (a closed set where the dynamics is chaotic). The main topic of this thesis is the study of the connectivity of the Fatou components for certain families of rational maps. On the one hand, we consider a family of singular perturbation and extend previous results on singular perturbations of Blaschke products. The main result is to show that the dynamical planes for the corresponding maps present Fatou components of arbitrarily large connectivity and determine precisely these connectivities. On the other hand, we consider a different problem related to root-finding algorithms. More precisely, we study the Chebyshev-Halley methods applied to a symmetric family of polynomials of arbitrary degree. The main goal is to show the existence of parameters such that the immediate basins of attraction corresponding to the roots of unity are infinitely connected. Moreover, we also prove that the corresponding dynamical plane contains a connected component of the Julia set, which is a quasiconforrnal deformation of the Julia set of the map obtained by applying Newton’s method.

Thesis advisor(s): Xavier Jarque Ribera and Jordi Canela Sánchez

University: Universitat de Barcelona

Abstract

The thesis develops an incipient methodology to study bifurcations of invariant curves in one-dimensional and quasiperiodic discrete systems, based on translated curve theorems and KAM theory.The (extended) phase space is a bundle whose base is a torus of dimension 1, and the real-line is the fiber but both the methodology and the results can be easily adapted to higher dimensional tori (the dimension being the number of external frequencies). The systems themselves are maps of bundles over translations in the torus with d frequencies. over translations on the torus with d frequencies. The methodology involves KAM theory, bifurcation theory, and translated curve theorems (in the spirit of Moser, Rüßmann, Herman, Delshams and Ortega). In the project, rigorous results are obtained in a posteriori format on the existence of families of translated tori in the analytical framework, establishing a methodology to study the bifurcations of translated tori. The a posteriori format is suitable to develop rigorous numerical calculations. Complementarily, the algorithms derived from the iterative process associated with this methodology have been implemented on the computer.

Thesis advisor(s): Àlex Haro Provinciale and Ernest Fontich Julià

University: Universitat de Barcelona

Abstract

The 3 Body Problem (3BP) models the motion of three bodies interacting via Newtonian gravitation. It is called restricted when one body has zero mass and the other two, the primaries, have strictly positive masses. In the region of the phase space where one body is far from the other two (the primaries for the restricted case) both models can be studied as a nearly integrable Hamiltonian system. This is the so-called hierarchical regime. The present thesis deals with the existence of unstable motions, in the 3BP and/or its restricted versions. More concretely, we analyze the existence of topological instability, non trivial hyperbolic sets and oscillatory motions (complete orbits which are unbounded but return infinitely often to some bounded region). On one hand, the existence of (a strong form of) topological instability in the N Body Problem was coined by Herman to be “the oldest question in dynamical systems”. On the other hand, oscillatory motions are the unique type of complete motions for the 3BP which are not present in the integrable approximation. Their connection with the existence of non trivial hyperbolic sets have lead to the formulation of fundamental, yet unsolved, conjectures about their abundance.We establish the existence of Arnold diffusion, a robust mechanism leading to topological instability, in the Restricted 3BP for any value of the masses of the primaries. The transition chain leading to Arnold diffusion is built in the hierarchical region. We extend a previous result by Kaloshin, Delshams, De la Rosa and Seara, which applied to arbitrarily small mass ratio. Their setting, which exploits the trick, used by Arnold in his original paper, of making use of two perturbative parameters, lead to an a priori unstable model. In our setting, we face some of the challenges present in a priori stable systems.We present several results concerning the existence of oscillatory motions and non trivial hyperbolic sets in the restricted and non restricted 3 Body Problem. First, we develop new tools which blend geometric ideas with variational techniques to prove that there exist oscillatory motions in the restricted 3BP in a non nearly integrable regime. Second we show the existence of non trivial hyperbolic sets and oscillatory motions in the 3BP for all values of the masses. The non trivial hyperbolic set, contained in a subset of the hierarchical region where the inner bodies perform approximately circular motions, is associated to a transverse intersection between the stable and unstable manifolds of a Normally Hyperbolic Invariant Manifold. The existence of center directions complicates heavily both the analysis of existence of transverse intersections between these invariant manifolds and the construction of the horseshoe. The contribution of the author focuses on completing the first of these two steps.Finally, we study the existence of Arnold diffusion in the 3BP for all values of the masses. The robustness of the mechanism which we use to prove the existence of Arnold Diffusion in the Restricted 3BP implies that the obtained transition chain admits a continuation in the 3BP if one mass is sufficiently small. The substantial difference when the masses are fixed is that one can construct a transition chain along which there is a significant exchange of momentum between the inner and outer bodies, resulting in a large change of the eccentricity of the inner bodies. This requires considerably more work than in our construction of the transition chain in the Restricted 3BP and our construction of hyperbolic sets for the 3BP. The first step towards establishing this result, which constitutes the subject of the last chapter of this thesis, is the analysis of the so called Melnikov approximation associated to the aforementioned transition chain.

Thesis advisor(s): Marcel Guàrdia Munarriz and Teresa Martínez-Seara Alonso

University: Universitat Politècnica de Catalunya

Abstract

The Poisson distribution represents a reference point for modeling count data, either in the case of independent observations, with repeated measurements or in the presence of random factors. But in practice, limitations appear in the analysis of this type of data in complex experimental designs. On the one hand, the distribution has the restriction that the adjusted data must be equidispersed, which is not common and requires the consideration of more complex distributions. On the other hand, it is challenging to compare alternative proposals, quantify the goodness of fit, or validate the model assumptions, due to the nature of the modeling tools. The general objective of this doctoral thesis is to describe the main strategies for the analysis of count data with repeated measures, focusing on their practical limitations, and in addition, to introduce new complementary proposals. First, the main modeling techniques used in statistical practice are reviewed: linear models, generalized linear models, mixed models, and generalized linear mixed models, with special emphasis on the case of count data. The corresponding formulation is presented along with details on the fitting procedures, validation, and inferential tasks. In relation to mixed linear models and generalized linear mixed models, two opposing views of modeling are emphasized: the conditional model and the marginal model, which give rise to some controversy. In this sense, different practical cases are presented to exemplify these modeling strategies and their limitations: the study of car accidents at different intersections in the city of Barcelona under certain preventive intervention, by means of conditional generalized linear mixed models and the analysis of the impact of red cards on the number of goals scored in different soccer matches using marginal generalized linear mixed models. Finally, alternative strategies for modeling count data in experiments with sub-replicates using order statistics from discrete distributions are presented.

Thesis advisor(s): Pere Puig Casado

University: Universitat Autònoma de Barcelona

Abstract

A solar sail is a method of spacecraft propulsion that uses only the solar radiation pressure (SRP). The main research object of this thesis is a solar sail spacecraft in the artificially created libration point orbits. It proposes a strategy to accomplish impulsive maneuvers by changing the parameters of the sail. The main new results are the following: 1. Computation of artificial libration points as a function of the parameters of a solar sail (cone angle α, clock angle δ, and lightness number β). The SRP is an additional repulsive acceleration in the CR3BP. As a result, the CR3BP equilibrium points L1, L2…L5 are shifted from their original positions. The new points SL1, SL2…SL5 correspond to positions in the rotating system where the gravitational, centrifugal, and SRP forces are balanced. These points can be represented as functions of the sail parameters α, δ, and β. Determination and adjustment of the solar sail parameters, computation of impulse maneuvers and their application to heteroclinic orbit transfers between Lissajous orbits plus a sensitivity analysis of the parameters of the maneuver for orbit transfers. The dynamics of solar sail maneuvers is conceptually different from classical control maneuvers, which rely only on impulsive changes to the velocity of a spacecraft. Solar sail orbits are continuous in both position and velocity in a varying vector field, which opens up the possibility for the existence of heteroclinic connections by changing the vector field with a sail maneuver. Based on a careful analysis of the geometry of the phase space of the linearized equations of motion around the equilibrium points, the key points are the identification of the main dynamic parameters and the representation of the solutions using the action-angle variables. The basic dynamic properties of the connecting families have been identified, presenting systematic new options for mission analysis in the libration point regime. Based on the proposed method for making impulse maneuvers, this thesis has carried out extensive research: (1) By applying a single-impulse maneuver, two spacecraft can reach the same final Lissajous orbit despite starting from different initial phases. (2) A transfer strategy is proposed that uses multi-impulse maneuvers. The initial and final solar sail parameters are fixed. (3) A spacecraft can use multi-impulse maneuvers to make back-and-forth jumps between the initial and final artificial libration point orbits. 2. Avoidance of forbidden zones considering impulsive maneuvers with the sail. There is a cylinder-like zone around the Sun–Earth axis where solar electromagnetic radiation is especially strong. The L1 libration point lies on this axis and is between the two bodies. The Earth half-shadow in the L2 region can also prevent a spacecraft from obtaining solar energy. Both problems can be modeled by placing a forbidden or exclusion zone in the YZ plane (around the libration point), which should not be crossed. To simplify and visualize the avoidance of forbidden zones, this thesis projects the 3D forbidden zones into the so-called effective phase plane (EPP), which has dimension 2. 3. Station-keeping of a solar sail moving along a Lissajous orbit. The designed station-keeping procedure periodically performs a maneuver to prevent the spacecraft to escape from a certain Lissajous orbit. The maneuver is computed so that it cancels out the unstable component of the state. Moreover, it is assumed that there is a random error in the execution of the maneuver. Considering the maneuvers performed every month, we show that the spacecraft can remain near the artificial libration points for at least 5 years, which demonstrates that station-keeping using sail reorientations to produce multiple impulses can be effective.

Thesis advisor(s): Josep Masdemont Soler, Yue Xiokui and Gerard Gómez Muntané

University: Universitat Politècnica de Catalunya

Abstract

In this work, we study the period function for fixed families of piecewise differential vector fields with a line of discontinuities. These systems, indistinctly called piecewise or nonsmooth, appear in several applications, including among others optimal control, nonsmooth mechanics, and robotic manipulation. For one family, by using a method based upon Picard-Fuchs equations for algebraic curves, we characterize the global behavior of the period function. That is, we determine regions in the parameter space for which the corresponding period function is monotonous or it has critical periods. Furthermore, in one of these families we study the bifurcation of critical periods in the interior of the period annulus from the weak center and from the isochronous center by using the calculation of the Taylor developments of the periods constants near the center. We further present the beginning of the study of the global behavior of the period function for the planar piecewise linear system that contains a period annulus at infinity.

Thesis advisor(s): Alex Carlucci Rezende and Joan Torregrosa Arús

University: Universidade Federal de São Carlos

Abstract

The idea of Terraformation comes from the science fiction literature, where humans have the capability of changing a non-habitable planet to an Earth-like one. Nowadays, Nature is changing rapidly, the poles are melting, oceans biodiversity is vanishing due to plastic pollution, and the deserts are advancing at an unstoppable rhythm. This thesis is a first step towards the exploration of new strategies that could serve to change this pernicious tendencies jeopardising ecosystems. We suggest it may not only be possible by adding new species (alien species), but also engineering autochthonous microbial species that are already adapted to the environment. Such engineering may improve their functions and capabilities allowing them to recover the (host) ecosystem upon their re-introduction. These new functionalities should make the microbes be able to induce a bottom-up change in the ecosystem: from the micro-scale (microenvironment) to the macro-scale (even changing the composition of species in the entire the ecosystem). To make this possible, the so-called Terraformation strategy needs to fuse many different fields of knowledge. The focus of this thesis relies on studying the outcome of the interactions between species and their environment (Ecology), on making the desired modifications by means of genetic engineering of the wild-type species (Synthetic Biology), and on monitoring the evaluation of the current ecosystems’ states, testing the possible changes, and predicting the future development of possible interventions (Dynamical Systems). In order to do so, in this thesis, we have gathered the tools provided by these different fields of knowledge. The methodology is based on loops between observation, designing, and prediction. For example, if there is a lack of humidity in semiarid ecosystems, we then propose to engineer e.g. Nostoc sp. to enhace its capability to produce extracellular matrix (increasing water retention). With this framework, we perform a model to understand the different possible dynamics, by means of dynamical equations to evaluate e.g. when a synthetic strain will remain in the ecosystem and the effects it will produce. We have also studied spatial models to predict their ability to modify the spatial organization of vegetation. Transient dynamics depend on the kind of transition underlying the occurring tipping point. For this reason, we have studied different systems: vegetation dynamics with facilitation (typical from drylands), a cooperator-parasite system, and a trophic chain model where different human interventions can be tested (i.e. hunting, deforestation, soil degradation, habitat destruction). All of these systems are shown to promote different types of transitions (i.e. smooth and catastrophic transitions). Each transition has its own dynamical fingerprint and thus knowing them can help monitoring and anticipating these transitions even before they happen, taking advantage of the so-called early warning signals. In this travel, we have found that transients can be an important phenomena in the current changing world. The ecosystems that we observe can be trapped into a seemingly stable regime, but be indeed in an unstable situation driving to a future sudden collapse (Fig 1) For this reason, we need to investigate novel intervention methods able to sustain the current ecosystems, for instance: Terraformation.

Thesis advisor(s): Ricard Solé, Josep Sardanyés and Núria Conde

University: Universitat Pompeu Fabra

Abstract

The G matrix is a statistical summary of the genetic basis of a set of traits and a central pillar of quantitative genetics. A persistent controversy is whether G changes slowly or quickly over time. The evolution of G is important because it affects the ability to predict, or reconstruct, evolution by selection. Empirical studies have found mixed results on how fast G evolves. Theoretical work has largely been developed under the assumption that the relationship between genetic variation and phenotypic variation—the genotype-phenotype map (GPM)—is linear. Under this assumption, G is expected to remain constant over long periods of time. However, according to developmental biology, the GPM is typically complex and nonlinear. Here, we use a GPM model based on the development of a multicellular organ to study how G evolves. We find that G can change relatively fast and in qualitative different ways, which we describe in detail. Changes can be particularly large when the population crosses between regions of the GPM that have different properties. This can result in the additive genetic variance in the direction of selection fluctuating over time and even increasing despite the eroding effect of selection.

Thesis advisor(s): Isaac Salazar

University: University of Helsinki

Abstract

Angiogenesis, the formation of new blood vessels from pre-existing ones, is essential for normal development and plays a crucial role in such pathologies as cancer, diabetes and atherosclerosis. In spite of extensive research, many aspects of how new vessels sprout from existing vasculature remain unclear. Recent experimental results indicate that endothelial cells, lining the inner walls of blood vessels, rearrange within growing vessels and that sprout elongation is dominated by cell mixing during the early stages of angiogenesis. Cell rearrangements have been shown to be regulated by dynamic adaptation of cell gene expression, or cell phenotype. However, most theoretical models of angiogenesis do not account for these phenomena and instead assume that cell positions are fixed and cell phenotype is irreversible during sprouting. In this thesis, we formulate a multiscale model of angiogenic sprouting driven by dynamic cell rearrangements. Our model accounts for cell mixing which is regulated by a stochastic model of subcellular signalling linked to phenotype switching. We initially focus on early angiogenic sprouting when the effects of cell proliferation are negligible. We validate our model against available experimental data. We then use it to develop a measure to quantify the amount of cell rearrangement that occurs during sprouting and investigate how the branching structure of vascular networks changes as the level of cell mixing varies. Our results suggest that cell shuffling directly affects the morphology of growing vasculatures. In particular, rearrangements of endothelial cells with distinct phenotypes can drive changes in the network structure since cell phenotype adaptation is slower than cell migration. Cell mixing also contributes to remodelling of the extracellular matrix which, in turn, guides vascular growth. In order to investigate the effects of cell proliferation, which operates on longer timescales than cell migration, we first develop a method, based on large deviation theory, which allows us to reduce the computational complexity of our hybrid multiscale model by coarse-graining the internal dynamics of its cell-agents. The coarse-graining (CG) method is applicable to systems in which agent behaviour is described by stochastic systems with multiple stable steady states. The CG technique reduces the original stochastic system to a Markov jump process on the space of its stable steady states. Our CG method preserves the original description of agent states (instead of converting them to discrete ones) and stochastic transitions between them, while considerably reducing the computational complexity of model simulations. After formulating the CG method for a general class of hybrid models, we illustrate its potential by applying it to our model of angiogenesis. We coarse-grain the subcellular model, which determines cell phenotype specification. This substantially reduces the computational cost of simulations. We then extend our model to account for cell proliferation and validate it using available experimental data. This framework allows us to study network growth on timescales associated with angiogenesis in vivo and to investigate how varying the cell proliferation rate affects network growth. Summarising, this work provides new insight into the complex cell behaviours that drive angiogenic sprouting. At the same time, it advances the field of theoretical modelling by formulating a coarse-graining method, which paves the way for a systematic reduction of hybrid multiscale models.

Thesis advisor(s): Tomás Alarcon , Helen M. Byrne and Philip K. Maini

University: Universitat Autònoma de Barcelona

Abstract

In the thesis we consider higher regularity of the free boundaries in different variations of the obstacle problem, that is, when the Laplace operator b. is replaced with another elliptic or parabolic operator. In the fractional obstacle problem with drift (L = (-‘6.)8 + b · v’), we prove that for constant b, and irrational s > ½ the free boundary is C00 near regular points as long as the obstacle is C00. To do so we establish higher order boundary Harnack inequalities for linear equations. This gives a bootstrap argument, as the normal of the free boundary can be expressed with quotients of derivatives of solution to the obstacle problem. Furthermore we establish the boundary Harnack estímate for linear parabolic operators (L = Ot – b.) in parabolic C1 and C1•°’ domains and give a new proof of the higher order boundary Harnack estímate in ck,a domains. In the similar way as in the fractional obstacle problem with drift this implies that the free boundary in the parabolic obstacle problem is C00 near regular points. We also study the regularity of the free boundary in the parabolic fractional obstacle problem (L = Ot + (-b.)8) in the cases > ½- We are able to provea boundary Harnack estímate in C1•°’ domains, which improves the regularity of the free boundary from C1•°’ to C2•°’. Finally, we establish the full regularity theory for free boundaries in fully non-linear parabolic obstacle problem. Concretely we find the splitting of the free boundary into regular and singular points, we show that near regular points the free boundary is locally a graph of a C00 function, and that the singular points are ” rare” – they can be covered with a Lipschitz manifold of co-dimension 2, which is arbitrarily flat in space.

Thesis advisor(s): Xavier Ros-Oton

University: Universitat de Barcelona

Abstract

The results in this thesis concern extremal and probabilistic topics in number theoretic settings. We prove sufficient conditions on when certain types of integer solutions to linear systems of equations in binomial random sets are distributed normally, results on the typical approximate structure of pairs of integer subsets with a given sumset cardinality, as well as upper bounds on how large a family of integer sets defining pairwise distinct sumsets can be. In order to prove the typical structural result on pairs of integer sets, we also establish a new multipartite version of the method of hypergraph containers, generalizing earlier work by Morris, Saxton and Samotij.

Thesis advisor(s): Oriol Serra i Juan Jose Rue

University: Univeristat Politècnica de Catalunya

Abstract

This thesis concerns the proof complexity of algebraic and semialgebraic proof systems Polynomial Calculus, Sums-of-Squares and Sherali-Adams. The most studied complexity measure for these systems is the degree of the proofs. This thesis concentrates on other possible complexity measures of interest to proof complexity, monomial-size and bit-complexity. We aim to showcase that there is a reasonably well-behaved theory for these measures also. Firstly we tie the complexity measures of degree and monomial size together by proving a size-degree trade-off for Sums-of-Squares and Sherali-Adams. We show that if there is a refutation with at most s many monomials, then there is a refutation whose degree is of order square root of n log s plus k, where k is the maximum degree of the constraints and n is the number of variables. For Polynomial Calculus similar trade-off was obtained earlier by Impagliazzo, Pudlák and Sgall. Secondly we prove a feasible interpolation property for all three systems. We show that for each system there is a polynomial time algorithm that given two sets P(x,z) and Q(y,z) of polynomial constraints in disjoint sequences x,y and z of variables, a refutation of the union of P(x,z) and Q(y,z), and an assignment a to the z-variables, finds either a refutation of P(x,a) or a refutation of Q(y,a). Finally we consider the relation between monomial-size and bit-complexity in Polynomial Calculus and Sums-of-Squares. We show that there is an unsatisfiable set of polynomial constraints that has both Polynomial Calculus and Sums-of-Squares refutations of polynomial monomial-size, but for which any Polynomial Calculus or Sums-of-Squares refutation requires exponential bit-complexity. Besides the emphasis on complexity measures other than degree, another unifying theme in all the three results is the use of semantic characterizations of resource-bounded proofs and refutations. All results make heavy use of the completeness properties of such characterizations. All in all, the work on these semantic characterizations presents itself as the fourth central contribution of this thesis.

Thesis advisor(s): Albert Atserias

University: Univeristat Politècnica de Catalunya

Abstract

In 2016 H. Matui conjectured that the K-groups of the C*-algebra associated to an effective minimal étale groupoid, with a Cantor set as unit space, could be computed as the infinite direct sum of the homology groups of given groupoid. Although a counterexample was found by E. Scarparo in 2020, the study of sufficient and/or necessary conditions for the conjecture to hold remains relevant. The main goal of this thesis is to further deepen the knowledge of this conjecture, providing some examples and counterexamples for it and, more importantly, developing new techniques for the computation of groupoids invariants. The two main classes of groupoids involved in our work are Deaconu-Renault groupoids, and self-similar groupoids

Thesis advisor(s): Pere Ara and Joan Bosa Puigredon

University: Universitat Autònoma de Barcelona

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 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 aluation at the character ¿. This generalizes a result of Bertolini and Darmon. With a similar argument, after discovering the work of Fornea and ehrmann on plectic 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.

Thesis advisor(s): Santiago Molina and Víctor Rotger

University: Universitat Politècnica de Catalunya

Abstract

In this thesis, Poisson structures are studied in moduli countries and in group actions. In particular, the focus is on b^m-simplèctiques structures, which can be seen as simplèctiques structures with singularities or also with a particular type of Poisson structures. I also study Poisson structures in varieties of characters associated with fuchsian differential equations and the behavior of these Poisson structures under the confluence of singularities. In the case of b^m-simplèctiques varieties, consider various classes of group actions, starting with Hamiltonian b^m-actions, a natural generalization of Hamiltonian moment functions in singular simplèctic context. Afterwards, Generalitzem faced more than this, he noticed singular quasi-Hamiltonian group actions. This daring generalization is motivated by those group actions that preserve a b^m-symplèctic structure to the variety but do not admit a conventional moment function. We use both moment functions (b^m-Hamiltonian and quasi-Hamiltonian singular) to demonstrate a corresponding generalization of the Marsden-Weinstein reduction theorem, demonstrating that in the singular environment, the reduction procedure eliminates the singularity. We prove a singular slice theorem as the first step for the proof of the reduction. We show that the Marsden-Weinstein singular reduction admits the reduction “per stages” and commutes with the desingularity procedure. for the Riemann-Hilbert correspondence. Firstly, let us consider various cases in which the Riemann-Hilbert correspondence can be explicitly resolved into an elliptic curve. Next, we turn to the case of Painlevé’s transcendents on the Riemann sphere. In particular, the Hamiltonian d’Okamoto for the second equation of Painlevé tea a natural b-symplectic structure. For the rest of the equations, the structure is more complicated. We begin by considering the structures of Poisson in the space of moduli of connection planes and varieties of characters corresponding to Fuchsian equations, all the singularities are simple pols (in particular, Painlevé VI). Consider Poisson structures for which the Riemann-Hilbert correspondence is a Poisson map. I also studied Poisson structures related to the Painlevé V equation (3 pols: un d’ordre 2 i two simple pols)

Thesis advisor(s): Eva Miranda

University: Universitat Politècnica de Catalunya

Abstract

Framed within the areas of algebraic geometry and commutative algebra, this thesis contributes to the study of sheaves and vector bundles on toric varieties. From different perspectives, we take advantage of the theory on toric varieties to address two main problems: a better understanding of the structure of equivariant sheaves on a toric variety, and the EinLazarsfeld-Mustopa conjecture concerning the stability of syzygy bundles on projective varieties. After a preliminary Chapter 1, the core of this dissertation is developed along three main chapters. The plot line begins with the study of equivariant torsion-free sheaves, and evolves to the study of equivariant reflexive sheaves with an application towards the problem finding equivariant Ulrich bundles on a projective toric variety. Finally, we end this dissertation by addressing the stability of syzygy bundles on certain smooth complete toric varieties, and their moduli space, contributing to the Ein-Lazarsfeld-Mustopa conjecture. More precisely, Chapter 1 contains the preliminary definitions and notions used in the main body of this work. We introduce the notion of a toric variety and its main features, highlighting the notion of a Cox ring and the algebraic-correspondence between modules and sheaves. Particularly, we focus our attention on equivariant sheaves on a toric variety. We recall the Klyachko construction describing torsion-free equivariant sheaves by means of a family of filtered vector spaces, and we illustrate it with many examples. In Chapter 2, we focus our attention on the study of equivariant torsion-free sheaves, connected in a very natural way to the theory of monomial ideals. We introduce the notion of a Klyachko diagram, which generalizes the classical stair-case diagram of a monomial ideal. We pro- vide many examples to illustrate the results throughout the two main sections of this chapter. After describing methods to compute the Klyachko diagram of a monomial ideal, we use it to describe the first local cohomology module, which measures the saturatedness of a monomial ideal. Finally, we apply the notion of a Klyachko diagram to the computation of the Hilbert function and the Hilbert polynomial of a monomial ideal. As a consequence, we characterize all monomial ideals having constant Hilbert polynomial, in terms of the shape of the Klyachko diagram. Chapter 3 is devoted to the study of equivariant reflexive sheaves on a smooth complete toric variety. We describe a family of lattice polytopes encoding how the global sections of an equivariant reflexive sheaf change as we twist it by a line bundle. In particular, this gives a method to compute the Hilbert polynomial of an equivariant reflexive sheaf. We study in detail the case of smooth toric varieties with splitting fan. We are able to give bounds for the multigraded initial degree and for the multigraded regularity index of an equivariant reflexive sheaf on a smooth toric variety with splitting fan. From the latter result we give a method to compute explicitly the Hilbert polynomial of an equivariant reflexive sheaf on a smooth toric variety with splitting fan. Finally, we apply these tools to present a method aimed to find equivariant Ulrich bundles on a Hirzebruch surface, and we give an example of a rank 3 equivariant Ulrich bundle in the first Hirzebruch surface. Chapter 4 treats the stability of syzygy bundles on a certain toric variety. We contribute to the Ein-Lazarsfeld-Mustopa conjecture, by proving the stability of the syzygy bundle of any polarization of a blow-up of a projective space along a linear subspace. Finally, we study the rigidness of the syzygy bundles in this setting, all of which correspond to smooth points in their associated moduli space.

Thesis advisor(s): Rosa Maria Miró Roig

University: Universitat de Barcelona

Abstract

In this thesis various aspects of the Cuntz semigroup associated with a C*-algebra are studied, as well as the so-called abstract Cuntz semigroups. In particular, we analyze the rank problem by the class of separable AI algebras, obtaining a complete characterization. A notion of dimension for abstract Cuntz semigroups is also introduced, which in the case of continuous functions on a topological space coincides with the usual Lebesgue dimension. This dimension is also related to the nuclear dimension of a C*-algebra, and it is proved that both coincide in significant cases. Special attention is paid to the zero dimensional case, where a characterization of these semigroups can be given in terms of density conditions of some privileged elements. Finally, the notion of nowhere scattered C*-algebras is introduced, and it is shown that it is a very broad class, including all infinite-dimensional simple algebras. Various characterizations of this concept are given, including a description in terms of divisibility properties of the Cuntz semigroup. This notion is intimately linked to the so-called Global Glimm Problem, which is also analyzed in the thesis, giving a reformulation through conditions of the Cuntz semigroup.

Thesis advisor(s): Francesc Perera

University: Universitat Autònoma de Barcelona

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.

Thesis advisor(s): Marcel Guardia and Inmaculada Baldomá

University: Universitat Politècnica de Catalunya

Abstract

This dissertation is devoted to the analysis of the motion of small bodies, like asteroids, in the neighbourhood of the Earth-Moon system from a celestial mechanics approach. This is an extensive area of research where probably, the most extended simplified mathematical model is the well-known autonomous Hamiltonian system the Restricted Three-Body Problem (RTBP). Many modifications to this model have been proposed, looking for a more accurate description of the system. One of the simplest ways of introducing additional physical effects is through time-periodic perturbations, such that such that the new non-autonomous system is close to the autonomous one, and it has many periodic or quasi-periodic solutions. If these solutions are hyperbolic, they have stable/unstable invariant manifolds, such that stable manifolds approach the quasi-periodic solutions forward in time, meanwhile unstable manifolds do it backward in time, constituting the skeleton for the dynamical transport phenomena we are interested in. Notice that one dimension can be reduced by defining a suitable temporal Poincar´e map. Therefore, our aim is to compute the quasi-periodic solutions and their manifolds in this map. Most of the effort of this dissertation is addressed to the Bicircular Problem (BCP), in which the Earth and Moon are treated as the primaries in the RTBP and the gravitational field of the Sun is introduced as a time-periodic forcing of the RTBP. In particular, we have extensively analysed the horizontal family of two dimensional quasi-periodic solutions in the neighbourhood of the collinear unstable equilibrium point L3. We found that diverse trajectories connecting the Earth, the Moon and the outside Earth-Moon system are governed by L3 dynamics. Big attention is paid to the trajectories coming from the Moon towards the Earth, since they may give an insight of the travel that lunar meteorites perform before landing in our planet. These results have been translated and compared with those of a realistic model based on JPL (Jet Propulsion Laboratory) ephemeris, showing a good agreement between the results obtained. We also have proposed and carried out a strategy for capturing a Near Earth Asteroid (NEA) using the stable invariant manifolds of the horizontal family of quasi-periodic orbits around L3 in the BCP. To this aim the high order parametrization of the stable/unstable invariant manifolds is introduced, for which computation we have employed the jet transport technique. Finally, the strategy is applied to the NEA 2006 RH120. The contributions to the BCP presented in this dissertation include two other applications. The first one is devoted to the study of the unstable behaviour near the triangular points, meanwhile the second is devoted to a family of stable invariant curves around the Moon that are close to a resonance, promoting the appearance of chaotic motion. The last part of the dissertation is focused on the effective computation of the high or- der parametrization of the stable and unstable invariant manifolds associated with reducible invariant tori of any high dimension. To this aim, we resort on the reducible system, that offers a high degree of parallelization of the computations. Besides, we explain how to com- bine the presented methods with multiple shooting techniques to accurately compute highly unstable invariant objects. Finally, we apply the developed algorithms to compute the high order parametrization of the manifolds associated to L1 and L2 in an Earth-Moon system that includes five time-periodic forcings regarded to four physical features of the system, besides the solar gravitational field.

Thesis advisor(s): Àngel Jorba

University: Universitat Autònoma de Barcelona