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

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