## PhD Theses

List of PhD theses supervised by CRM researchers.

# 2022

##### Vidiella Rocamora, Blai | Terraforming Earth's Ecosystems

**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

##### Milocco, Lisandro | Evolution of the G Matrix under Nonlinear Genotype-Phenotype Maps

**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

##### Stepanova, Daria | Mathematical modelling of angiogenesis as an integrated multicellular process

**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

##### Kukuljan, Teo | Higher regularity of free boundaries in obstacle problems

**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

##### Wötzel, Maximilian | Probabilistic and Extremal studies in Additive Combinatorics

**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

##### Hakoniemi, Tuomas | Size bounds for algebraic and semialgebraic proof systems.

**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

##### Sánchez Madrigal, Álvaro | On the relation between homology and K-theory of étale groupoids

**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

##### Hernandez, Víctor | p-adic L-functions, p-adic Gross-Zagier formulas and plectic points.

**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

##### Matveeva, Anastasiia | Poisson structures on moduli spaces and group actions

**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

##### Salat Moltó, Martí | Vector bundles and sheaves on toric varieties

**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

##### Vilalta, Eduard | The range problem and dimension theory for the Cuntz semigroup

**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

##### Giralt Miron, Mar | Homoclinic and chaotic phenomena around L3 in the restricted 3-Body Problem

**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

##### Nicolás Ávila, Begoña | Invariant manifolds and transport in a Sun-perturbed EarthMoon system

**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