Màster Oficial - Matemàtica Avançada

URI permanent per a aquesta col·leccióhttps://hdl.handle.net/2445/42661

Treballs finals del Màster en Matemàtica Avançada de la Facultat de Matemàtiques i Informàtica de la Universitat de Barcelona

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Introduction to contact topology
(2025-01-09) Velasco Soldevila, Eduard; Cardona Aguilar, Robert
This master’s thesis provides an introduction to contact topology, with the primary objective of proving Martinet’s Theorem, which asserts that every closed, connected 3-manifold admits a contact structure. The proof heavily relies on the Lickorish-Wallace Theorem, which states that any such 3-manifold can be obtained from $S^{3}$ via a finite sequence of Dehn surgeries. The thesis explores key concepts in contact topology, such as contact structures, Darboux’s Theorem, and Gray stability. A complete proof of the Lickorish-Wallace Theorem is given before focusing on the detailed proof of Martinet’s Theorem, highlighting the ubiquity of contact structures in 3-manifolds.
2D Euler system for Hölder continuous vorticities
(2025-01-09) Castanyer Bibiloni, Francesc Josep; Clop, Albert
This thesis investigates the two-dimensional incompressible Euler equations under smooth initial data, focusing on three formulations: 1. the classical Euler system, 2. the vorticity-stream forumlation, 3. the integrodifferential (particle-trajectory) formulation. In Chapter 1, we first show that these three viewpoints are mathematically equivalent. We also develop core background material on vector fields, flows, and singular integral operators. These tools, particularly the singular integral operator theory, lay the groundwork for the vorticity-stream and integrodifferential formulations used in subsequent chapters. Chapter 2 establishes the local-in-time existence and uniqueness of solutions when the initial vorticity lies in a Hölder space $C^\gamma$ (with $\gamma \in(0,1)$ ). By formulating the problem in terms of the integrodifferential formulation, we show that the relevant operator is locally Lipschitz in an appropriate Banach space. Applying a Picard-Lindelöf-type argument then yields the desired local well-posedness result. Key technical ingredients include singular integral estimates and careful composition bounds in Hölder spaces. Finally, Chapter 3 addresses the global nature of these solutions. In two dimensions, the vorticity remains constant along particle trajectories and thus stays bounded for all time, preventing finite-time blow-up. Using a method inspired by the analysis of Beale, Kato, and Majda, we show that bounding the vorticity's supremum norm also bounds the velocity gradient, which in turn guarantees global existence. Thus, for sufficiently smooth initial vorticity in 2D, the solutions to the Euler equations extend indefinitely in time and remain regular.
Carleman estimates
(2024-09-02) Caballero Luján, Guillermo; García-Ferrero, María Ángeles; Ortega Cerdà, Joaquim
This work aims to study Carleman estimates, a weighted-type of inequalities first introduced by Carleman in 1939. Such estimates are very important for proving unique continuation properties of differential and pseudo-differential operators. We first derive a Carleman estimate for the Laplacian operator as an illustrative example following the work of Jérôme Le Rousseau and Gilles Lebeau in [7] which is a summary of a much large study. We try to extend the methodology to non-local operators. In particular we aim to deal with the fractional Laplacian. The results are focused on proving unique continuation properties and showing the significance of weighted estimates and the operators involved. For this we mainly use: Fourier analysis, Symbol theory and differential and pseudodifferential analysis.
An introduction to rational homotopy theory
(2024-09) Bisbal Castañer, Onofre; Cirici, Joana
Rational homotopy theory is the study of homotopy groups modulo torsion. The idea is to consider the torsion-free part of $\pi_n(X)$ by tensoring by $\mathbb{Q}$, so computations become much more affordable. The aim of this work is to provide a fairly detailed introduction to rational homotopy theory from Sullivan's approach. We will start by introducing the concept of rationalization from both the topological and algebraic points of view. Second, we will construct the functor piece-wise linear forms, which establish the link between topology and algebra associating to each simply connected space $X$ a commutative differential graded algebra $A_{\mathrm{pl}}(X)$. However, the key part of this theory is to associate to $A_{\mathrm{pl}}(X)$ a much more simple type of cdga's: Sullivan algebras, which allows us to do computations explicitly. Finally, given a fibration $F \hookrightarrow F \rightarrow B$ we will study the relation between the Sullivan models of each space.
Auxiliary polynomials for transcendence results
(2024-09-02) Valcarce Dalmau, Eduard; Sombra, Martín
The main goal of this work is to prove several transcendence results using auxiliary functions, and in doing so showcase their effectiveness in various contexts. The main theorems covered will be Hermite-Lindemann, Gelfond-Schneider, Schneider-Lang, and Baker’s theorem. We will employ two different proof strategies with auxiliary polynomials: two similar ones for Hermite-Lindemann and Schneider-Lang, and a noticeably different one for Baker’s theorem. Gelfond-Schneider will come as a corollary to Schneider-Lang. We will ease into these theorems however, by first delving into the preliminary results and background knowledge requiered to understand their proofs. This includes but is not limited to derivations over number fields, valuation theory and height functions, and complex analysis. Furthermore, we will take a detour into ellipitic functions after proving the Schneider-Lang theorem due to independent interest, and to present a few applications of the Schneider-Lang theorem, as it is the most general one we will present.
Fundamental principles of Binary Latent Diffusion
(2024-09-02) Pujol Vidal, Àlex; Casacuberta, Carles; Escalera Guerrero, Sergio
In this thesis we explore the fundamental principles of Binary Latent Diffusion Models (BLDM), a novel class of generative models that leverage probabilistic deep latent variable models and diffusion processes to approximate complex data distributions. The research delves into probability theory, generative models, and latent space representations, with a focus on Variational Autoencoders (VAE) that lead to Bernoulli Variational Autoencoders (BVAE). The study provides a comprehensive overview of the foundations of Diffusion Models, leading to the formal definition of Discrete Bernoulli Diffusion Models (DBDM) and its training objective. Both, BVAE and DBDM, are the building blocks of the BLDM. Additionally, a practical application is presented. This exploration highlights the mathematical formalization and implementation strategies for BLDMs, paving the way for future advancements in generative modeling.
Deformationts of Galois representations
(2024-09-02) Abdul Parveen, Habib Ullah; Dieulefait, L. V. (Luis Victor)
Inspired by a Galois representation, we will set up the theory of deformations with more generality and try to give two proofs, one using the Schlesinger criteria and another explicitly about the existence of an object called the universal deformation. That’s why we give a brief introduction about Galois deformations.
Schauder estimates for linear elliptic PDEs
(2024-09-02) Fierro González, Antoni; Ros, Xavier
Partial differential equations (PDEs) are fundamental tools in mathematics. They serve as the backbone for modeling a vast array of physical phenomena and have many applications in a very wide range of mathematical subjects. In this project, we will focus on the study of second-order linear elliptic partial differential equations, though we will not address them in their most general form for simplicity. Our primary focus will be on regularity. However, before diving into the mathematical aspects, Chapter 2 will provide motivation by presenting examples from physics and probability where the equations we are interested in appear. For this chapter, we will primarily follow the work in [3] (Chapter 12) and [6] (Chapter 1). In Chapter 3, we will explore the key properties of harmonic functions, primarily following the approach in [4]. These properties will be fundamental for proving the project’s main theorems. In Chapter 4, we will introduce the concept of Hölder continuity and establish important results concerning Hölder spaces, drawing from [4] and [6]. We will see how Hölder continuity is particularly well-suited for the study of partial differential equations. In Chapter 5, following [6], we will establish interior Schauder estimates for equations in both divergence form (providing one proof) and non-divergence form (offering two different proofs). Following that, in Chapter 6, we will prove global Schauder estimates for non-divergence form equations and state the corresponding results for divergence form equations. We will also examine the critical role that boundary regularity plays in this context. Finally, in Chapter 7, we will explore how Schauder estimates, in conjunction with the Continuity Method, can be utilized to prove the regularity and existence of solutions for linear elliptic PDEs.
Minimal energy on the circle
(2024-09-02) Arribas Viera, David; Marzo Sánchez, Jordi
We find minimizing configurations for most of the Riesz-$s$ energies on the unit circle $S^{1}$ . We also provide a complete asymptotic expansion of the Riesz-$s$ energy associated to $N$ equally spaced points on the $S^{1}$. Finally, we present Chui's conjecture, prove a partial result and show how it leads to an interesting consequence about function approximation in the Bergman space.
On the basins of attraction of root-finding algorithms
(2024-06) Rosado Rodríguez, David; Jarque i Ribera, Xavier
Root-finding algorithms have historically been employed to solve numerically nonlinear equations of the form $f(x)=0$. Newton's method, one of the most well-known techniques, started being analyzed as a dynamical system in the complex plane during the late 19th century. This thesis explores the dynamics of damped Traub's methods $T_{p, \delta}$ when applied to polynomials. These methods encompass a range from Newton's method $(\delta=0)$ to Traub's method $(\delta=1)$. Our focus lies in investigating various topological properties of the basins of attraction, particularly their simple connectivity and unboundedness, which are crucial in identifying a universal set of initial conditions that ensure convergence to all roots of $p$. While the former topological properties are already proven for Newton's method $(\delta=0)$, they remain open for $\delta \neq 0$. We present results that contribute to addressing this gap, including a proof for cases where $\delta$ is close to 0 and for the polynomial family $p_d(z)=z\left(z^d-1\right)$.
Introduction to Berkovich spaces
(2024-06-04) Reig Fité, Oriol; Sombra, Martín
In this Master Final Project I have studied Berkovich spaces, which is one of the existing approaches to non-Archimedean geometry, a branch that deals with analytic spaces over non-Archimedean fields. Let us first give some context on $p$-adic geometry and the necessity to develop such a theory of analytic spaces. Any norm gives rise to a metric space by setting the distance between two elements as the norm of their difference. In the case of a metric space induced by a non-Archimedean norm, the topological space is totally disconnected. For this reason, when we try to develop a theory of analytic functions similar that for the complex case (i.e., the Archimedean case), we encounter some notorious problems.
Gromov's’ theorem on groups of polynomial growth
(2024-06-28) Pozuelo Terradas, Eduard; Mundet i Riera, Ignasi
The aim of this project is to prove Gromov’s theorem on groups of polynomial growth. In order to do so, we will follow the original proof from Mikhail Gromov [Gro81], in which he introduced a convergence for metric spaces, called the Gromov-Hausdorff convergence, that is now widely used in geometry. With this in mind, one of the objectives of the project will also be to study this convergence. It is worth noting that an alternative simpler proof has been found by Bruce Kleiner [Kle10], though it still relies on Tits alternative, a theorem that Gromov’s proof uses too. Later, Terence Tao and Yehuda Shalom [ST10] provided a more fundamental proof based on the work of Kleiner. However, for this project we will not study such proofs.
Del Pezzo surfaces
(2024-06-28) Maristany Sala, Pau; Naranjo del Val, Juan Carlos
Del Pezzo surfaces, named after the Italian mathematician Pasquale Del Pezzo, are a central object of study in algebraic geometry. These surfaces exhibit rich geometric properties and have numerous applications in various areas of mathematics, including complex geometry and theoretical physics. In this project, we will explore three different definitions of Del Pezzo surfaces, analyze their equivalences and differences, and delve into some of their geometric properties. We start this project by introducing concepts that will be needed to work and understand the different definitons. And so forth the first two chapters are presented as a continuation of the Algebraic Geometry course in the Advanced Mathematics Master, UB. It is not until the third chapter that the first properties of Del Pezzo surfaces are presented. The foruth and final chapter is based on using the results previously found to give a geometrical structure of some Del Pezzo surfaces.
Classification of artin algebras
(2024-06-28) Ken, Nikhil; Elías García, Joan
The aim of this project is to study Artin rings which are fundamental structures which arise in broad areas of mathematics including algebraic geometry number theory and representation theory and therefore studying and classifying them can give new and deep perspectives for solving problems in many different areas. In this thesis we start by reviewing the preliminaries to establish the Matlis duality which was introduced in [11] which was closely related to the work of Francis Sowerby Macaulay. Macaulay established a correspondence between Gorenstein Artin algebras $A=R / I$ and cyclic submodule $\langle F\rangle$ of the polynomial where $R$ is the power series ring in n variable and $S$ is polynomial ring with the module structure of $S$ depending on the characteristic of the given field. This correspondence can be seen as special case of the Matlis duality because the injective hull of $\mathbf{k}$ as $R$ module is isomorphic to $S$.
Concentration of analytic functions
(2024-06-27) James Cano, Joaquı́n; Ortega Cerdà, Joaquim
In this work we study different problems concerning the characterization of those measurable sets that, among all sets having a prescribed measure, can capture the largest possible energy fraction of an analytic function in both the Euclidean and hyperbolic settings. In other terms, considering as spaces of analytic functions the Fock space $\mathcal{F}^2\left(\mathbb{C}^n\right)$, with $n \geq 1$, and the Bergman space $\mathcal{A}_\alpha^2(\mathbb{D})$, with $\alpha>1$, we show that given some measurable sets $\Omega \subset \mathbb{C}$ and $\Omega^{\prime} \subset \mathbb{D}$, with some fixed measure $c>0$, the concentration quantities and $$ & \max _{F \in \mathcal{F}^2\left(\mathbb{C}^n\right) \backslash\{0\}}\left\{\frac{\int_{\Omega}|F(z)|^2 e^{-\pi|z|^2} d m_{2 n}(z)}{\left.\int_{\mathbb{C}^n}|F(z)|^2 e^{-\pi|z|^2 d m_{2 n}(z)}\right\}}\right. \\ & \max _{f \in \mathcal{A}_\alpha^2(\mathbb{D}) \backslash\{0\}}\left\{\frac{\int_{\Omega^{\prime}}(\alpha-1)|f(z)|^2\left(1-|z|^2\right)^\alpha d m_h(z)}{\int_{\mathbb{D}}(\alpha-1)|f(z)|^2\left(1-|z|^2\right)^\alpha d m_h(z)}\right\} $$ are maximized when considering the sets to be a ball (in each respective geometry) with the same measure $c>0$. Specifically, we give a sharp upper bound for each of the previous problems and characterize not only the subsets but also the functions where the maxima are attained.
Topological approaches to Euler characteristics in odd-dimensional orbifolds
(2024-06-28) Gallardo Campos, Ramon; Porti, Joan; Mundet i Riera, Ignasi
This master thesis deals with orbifolds, a generalization of manifolds. On one hand, for compact manifolds of odd dimension one has a pretty interesting formula: the Euler characteristic of the manifold is half the characteristic of its boundary. On the other hand, Ichiro Satake stated and proved in 1957 that the Euler characteristic of an odd-dimensional compact Riemannian orbifold without boundary is 0. From this last result it can be proven a generalisation of the formula for odd-dimensional compact smooth orbifolds. Nevertheless, the proof given by Satake uses the Chern-Gauss-Bonnet formula, so the objective of this Master Thesis is to give a purely topological proof of the formula described. For this, the idea is to dissect an orbifold into smaller parts where the study of this formula becomes easier. In the following we define the main characteristics and properties of orbifolds, as well as some of their topological and geometrical features.
Regularity of Lipschitz free boundaries in the alt-Caffarelli problem
(2024-06-27) Domingo Pasarin, Joan; Ros, Xavier
In this work we study the regularity of Lipschitz free boundaries in the Alt-Caffarelli problem. We prove that Lipschitz free boundaries are $C^{1, \alpha}$ by exploiting the rescaling invariance of the problem and the initial Lipschitz regularity of the boundary. Moreover, we also show that $C^{1, \alpha}$ boundaries are smooth, which combined with the previous result implies that Lipschitz free boundaries are smooth.
Markov chains for non abelian gauge theories
(2024-06-28) Díaz Rodríguez, Juan Carlos; Iblisdir, Sofyan; Juliá-Díaz, Bruno; Gonchenko, Marina
[en] We present a Metropolis-Hastings Markov chain for correlated systems of continuous variables ruled by the Boltzmann distribution. An initial introduction to the algorithm with discrete variables is given and then extended to variables in a compact group. We initially work with the SU (2) Higgs model with frozen matter term on a 2D lattice. The algorithm is then extended to a 3D lattice. The configuration updates are done though tensor network renormalization schemes that approximate the partition function of the system. We evaluate the performance of the chains based on equilibration and autocorrelation times. The mean value of different observables are compared to existing literature and additional computations are done with the groups SU (3), SU (5) and SU (10).
Classifying spaces for equivariant $\mathbb{Z} /(2)$-bundles
(2024-06-28) Codina Broto, Sergi; Pitsch, Wolfgang; Mundet i Riera, Ignasi
[en] The aim of this project is to study classifying spaces for $\mathbb{Z} / 2$-equivariant principal $G$-bundles, where $G$ denotes a topological group. In the first chapter, we will study the category of principal $G$-bundles with some important results, including its motivation through the theory of real vector bundles, and the construction of their classifying spaces; reference for this study will be taken from [Die08], [MS74] and [Hus94]. In the second chapter, we introduce the notion of $\Gamma$-equivariant principal $G$ bundles for a topological group $\Gamma$, and follow the work done by Lück and Uribe [LU14] while interested in the specific case $\Gamma=(\mathbb{Z} / 2)$, which allows for simplifications in the proofs of some results which lead to the construction of a model for the classifying space for $\mathbb{Z} / 2$-equivariant principal $G$-bundles, and the subsequent study of the properties of such classifying spaces.
Stochastic differential equations driven by a fractional brownian motion
(2024-05-28) Burés Mogollón, Òscar; Rovira Escofet, Carles
[en] This project is a general study of Stochastic Differential equations driven by a fractional Brownian motion of Hurst parameter $H>1 / 2$. Sections 3,4 and 5 follow the lines of [16] in order to define a stochastic integral with respect to the fractional Brownian motion and then, discussing the existence and uniqueness of solutions. The sixth section is a general discussion about Malliavin calculus with respect to the fractional Brownian motion that will be useful in sections 7 and 8 . Moreover, in section 6 we prove that by reinforcing the conditions on the coefficients, we obtain absolute continuity of the law of the solution in the same way as it is done in [14]. Section 7 is the application of the Malliavin calculus in order to bound the density function of the solution to a specific type of equations by using a general method constructed in [12]. Finally, section 8 is devoted to show all the work we weren't able to finish during the elaboration of this thesis. We decided to attack the problem of bounding the density of a general family of stochastic delay differential equations. The approach given in [12] turned out to be inefficient, so we decided to follow the same approach as in [1], [10] and [15].

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