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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Effect of domain growth rates on Turing patterns during embryo development
(2025-01-09) Tacoronte Hernandez, Alba; Haro, Àlex; Muñoz Romero, José Javier
This work examines reaction–diffusion dynamics for Turing pattern formation on expanding domains, with the aim of explaining how growth drives pattern selection. Under exponential growth, new analytical insights that predict mode jumping (both to neighboring modes and to higher, non-adjacent modes) consistent with prior theoretical accounts have been obtained. Under linear growth, analogous phenomena are shown using FEM where simple closed-form analysis is not possible. For exponential growth, the computational findings align with the analysis, revealing regimes where growth promotes, inhibits, or redirects Turing modes, while for linear growth, they enable a comparison with prior results that establishes analogies between the two regimes. By integrating analytical and numerical perspectives, this study delineates the mechanisms by which domain growth reorganizes the pattern spectrum and shapes the pathways through which form emerges.
The Ovals Conjecture by Benguria and Loss
(2026-01-09) Ribas Moyà, Miquel; Csató, Gyula
Reformulation of the problem: For a smooth closed curve $\Gamma \subset \mathbb{R}^2$, Benguria and Loss conjectured that the lowest possible eigenvalue of the operator $\mathcal{H}_\Gamma = -\Delta_\Gamma + \kappa_\Gamma^2$ is $\lambda = 1$. The problem of finding this eigenvalue can be transformed into the problem of finding the infimum of two geometric functionals. Three improving bounds for $\lambda$ are given, up to $\lambda \ge 0.6085$. Also, a proof of the existence of a minimizing $\Gamma$ is provided, showing that $\lambda = 1$ for the round circle and its degeneration to a two-line segment. It is stated that such $\Gamma$ is a planar, convex, analytic curve with strictly positive curvature. A particular example is given at the end, starting from its curvature.
Mathematical Modelling of Beach Litter Distributions using Drone Images
(2026-01-01) Kozić, Božidar; Cabaña Nigro, Ana Alejandra; Rieger, Niclas; Olmedo, Estrella
Given that plastic pollution has increased worldwide, accurate quantification of marine litter is essential to develop effective remediation and preventive strategies. However, conducting effective monitoring surveys is difficult due to the inherent spatial variation and laborious nature of current sampling methods. Classic survey protocols often rely on counting items in small sample areas, which may fail to accurately represent overall pollution levels given the tendency of litter to accumulate in clusters. This thesis develops a probabilistic framework using empirical data derived from drone images to quantify the uncertainty of different beach litter sampling protocols in estimating mean litter densities. Data analysis of provided beach samples demonstrated that litter counts exhibit significant overdispersion and scatteredness, requiring probabilistic distributions beyond standard Poisson models. After extensive model comparisons, a Zero-Inflated Negative Binomial Log-Gaussian Cox Process (ZINB-LGCP) model was implemented, since it best captured the overdispersion and sparsity observed in the empirical data. The model was developed within a Bayesian machine learning framework, employing Penalized Complexity (PC) priors and Hilbert Space Gaussian Process (HSGP) approximations to ensure stable convergence and computational efficiency. Three representative beaches were chosen to approximate the distinct spatial clustering patterns observed across samples. Using posterior predictive distributions from the fitted model, we generated 5000 synthetic beach realizations that replicate the statistical properties and spatial structure of the three chosen ones. These synthetic samples served as a controlled ground truth to simulate and evaluate the performance of two widely used monitoring protocols: the transect-based National Oceanic and Atmospheric Administration (NOAA) protocol and the station-based Científicos de la Basura (CdB) protocol. The bias, accuracy and precision of both protocols were quantified by comparing their estimates against known baselines of synthetic beaches. This analysis provides insights into the uncertainties of in-situ measurements and demonstrates that generative Bayesian modeling offers a rigorous validation tool for environmental sampling designs, enabling the optimization of survey efforts without the need for exhaustive physical collection.
Determinantal point processes
(2025-09-28) Jiménez Lumbreras, Rubén; Ortega Cerdà, Joaquim
Linearization problems for circle diffeomorphisms and generalized interval exchange transformations
(2026-01-09) Baumeister, Maximilian; Fagella Rabionet, Núria; Drach, Kostiantyn
The goal of this thesis is to study the linearization problem for circle diffeomorphisms and their natural extensions, generalized interval exchange transformations (GIETs). Linearization questions for these systems concern the existence and regularity of conjugacies to their corresponding linear models, rigid rotations in the circle case and piecewise isometries in the GIET setting. This problem lies at the core of modern one-dimensional dynamics and remains an active area of research, with inffuential contributions made by A. Avila, V. Arnold, M. Herman, S. Marmi, P. Moussa, C. Ulcigrai, M. Viana, J.-C. Yoccoz, among many others. The thesis is developed along two complementary directions. First, we investigate obstructions to linearization. For circle diffeomorphisms, this is exempliffed by the construction of the classical Denjoy counterexample, which shows that topological conjugacy to a rotation may fail in low regularity. We generalize this counterexample to GIETs, which to the best of our knowledge is done explicitly for the first time in the literature for this setting. The second direction concerns positive linearization and rigidity results. Here the contrast between the two settings becomes apparent: for example, while suficient smoothness governs the existence of topological conjugacy for circle diffeomorphisms, the situation for GIETs differs and smoothness stops playing a role in the existence of a conjugacy to the linear model. We present a survey of recent developments and discuss the extent to which linearization results persist beyond the classical circle setting.
Toward logarithmic sheaves on singular varieties: marked points in line arrangements
(2025-06-12) Vilageliu Giró, Júlia; Marchesi, Simone
[en] The main objective of this project is to take a first step toward extending the classical definition of the logarithmic tangent sheaf to the setting of singular ambient schemes. The first part of the work is dedicated to developing the necessary background to rigorously establish and motivate the framework in which our definition is formulated. In the second, and main part of the text, we propose a definition for the logarithmic tangent sheaf of a set of marked points in line arrangements in $\mathbb{P}^{2}_{\mathbb{C}}$. [ca] L’objectiu principal d’aquest treball és fer un primer pas a la generalització de la definició del feix tangent logarítmic en el context d’un esquema ambient singular. La primera part d’aquest treball està dedicada a l’estudi de la teoria necessària per tal de fixar i motivar el context de la nostra definició. En la segona part del text, i la principal, proposem una definició pel feix tangent logarítmic d’un conjunt de punts marcats en un arranjament de rectes a $\mathbb{P}^{2}_{\mathbb{C}}$.
Density of hyperbolicity in families of complex rational maps
(2025-06-13) Timoner Vaquer, Francesc; Drach, Kostiantyn
In this work, we address the fundamental open problem of whether hyperbolic rational maps, the ones for which every critical point lies in the basin of an attracting cycle, are dense in the space of rational maps of the same degree. By this, we mean if any such map can be uniformly approximated on compact sets by hyperbolic ones. Conjecturally, the answer is ’yes’, and this is known as the Density of Hyperbolicity Conjecture. After reviewing key tools from complex dynamics such as puzzle pieces constructions, quasi-conformal conjugacies, Böttcher coordinates and holomorphic motions, we introduce complex box mappings as a natural extension of polynomiallike maps and discuss their rigidity under combinatorial equivalence. Focusing on non-renormalisable polynomials without neutral periodic points, we reproduce, clarify and check the Kozlovski–van Strien result that such polynomials admit approximating hyperbolic maps by constructing dynamically natural box mappings and applying topological and rigidity results. In conclusion, we outline how this framework, with a careful setting, promises to extend beyond the polynomial case to prove density of hyperbolicity in broader families such as Newton and McMullen maps, thereby sketching a clear path for future advances in complex dynamics.
The inverse Calderón’s problem in the plane
(2025-06-11) Riu Pont, Jordi; Clop, Albert
[en] This work establishes a positive answer to the planar inverse Calderón problemby combining the theory of quasiconformal mappings, Beltrami equations, and singular integral operators. Key to the approach is the construction and study of complex geometric optics solutions and their properties, which enable the unique determination of conductivity from boundary data. [ca] Aquest treball dóna una resposta positiva al problema invers de Calderón al pla tot combinant la teoria de les aplicacions quasiconformes, les equacions de Beltrami i els operadors d’integrals singulars. La clau de l’enfocament és la construcció i l’estudi de les solucions d’òptiques geomètriques complexes i les seves propietats, que permeten determinar la conductivitat a partir de les dades a la vora.
Muckenhoupt weights and doubling measures
(2025-06-12) Rams Domenech, Roger; Prats Soler, Martí
The main goal of this project is to study Muckenhoupt weights in a context of integration with respect to doubling measures. The first part is an introduction to measure theory, covering theorems and differentiation of measures. In the second part, we introduce the Calderón–Zygmund decomposition and the Hardy–Littlewood maximal operator, concepts that will be important to study $A_p$ spaces. Finally, the third section is dedicated to the study of the Muckenhoupt weights, characterizing them through inequalities and proving relations between different $A_p$ spaces.
Introduction to non-linear least squares
(2025-06-13) Parcerisas Vela, Francesc; Alabert, Aureli
Non-linear least squares (NLLS) problems occur whenever a smooth model $r : \mathbb{R}^n \to \mathbb{R}^m$ must be fitted to data by minimizing $f(x) = \tfrac{1}{2} \| r(x) \|_2^2$. Although NLLS is a special case of unconstrained optimization, its Jacobian structure allows algorithms that are faster and more reliable than generic methods. This thesis reviews, and compares two mainstream approaches as stated by Nocedal \& Wright \cite{4}: (i) Gauss--Newton line-search methods, and (ii) Levenberg--Marquardt trust-region methods. After summarizing the required analysis (first- and second-order conditions, convergence proofs, and regularity assumptions), we study a special case of non-linear least squares when the model involves exponential functions.
Statistical methods for spatial transcriptomics data
(2025-06-13) Moles Seró, Pere; Cabaña, Alejandra
Spatial transcriptomics is a set of techniques that enables the quantification of gene expression within intact tissue sections, preserving the spatial context of where specific genes are active. The main goal of this master thesis is to review state-of-the-art statistical methods for analyzing spatial transcriptomics data and to draw parallels with traditional spatial statistical analysis, providing a robust theoretical and mathematical foundation for these methodologies. Section 1 provides an introduction to basic concepts in genomics and an overview of spatial transcriptomics technologies, which are divided between sequencing-based and imaging-based technologies. Section 2 presents various spatial statistical concepts such as spatial autocorrelation, kriging, Gaussian-Markov random fields and point processes. Section 3 discusses normalization methods for different types of RNAseq data: bulk RNA-seq, single-cell RNA-seq and spatial transcriptomics, highlighting its similiarities and differences. Section 4 covers dimensionality reduction methods, ranging from nonspatial approaches such as PCA to spatially-aware methods that extend PCA by incorporating spatial information. Section 5 explores the identification and analysis of spatial domains, distinguishing between non-spatial approaches, such as the Louvain and Leiden methods, and spatially-aware methods specifically designed for spatial transcriptomics data. Section 6 examines various approaches for detecting spatially variable genes, based on different strategies, including methods that use Gaussian process regression, marked point process theory, and other nonparametric techniques. Section 7 addresses cell type deconvolution strategies.
Exchange rings, their extensions, and a K-theoretic characterization
(2025-06-13) Modenes Montero, Víctor; Perera Domènech, Francesc
Almost isometric maps of the hyperbolic disc
(2025-06-13) Lorenzo Martínez, Angel; Nicolau Nos, Artur
Our main goal is to understand the paper [GP91] of J. Garnett and M. Papadimitrakis on almost isometries. An analytic self-map of the unit disc is called an almost isometry if there exists $c > 0$ such that $\operatorname{diam}(f(B)) \ge \operatorname{diam}(B) - c$ for any hyperbolic disc $B \subset \mathbb{D}$. Here, $\operatorname{diam}(E)$ means the hyperbolic diameter of $E$. Three equivalent characterizations of almost isometries will be considered: one based on geodesics, one involving angular derivatives, and another one using the distribution of zeros. These results combine notions from hyperbolic geometry, complex analysis, and measure theory. The project reveals that almost isometries form a rich subclass of Blaschke products. Furthermore, an explicit example of an almost isometry whose angular derivative diverges almost everywhere is constructed, demonstrating the subtle boundary behaviour of these functions.
Cayley partial cubes
(2025-06-01) Jaén Guedes, Daniel; Knauer, Kolja
This thesis explores partial cubes, a well-studied class of graphs that can be isometrically embedded into hypercubes, with a particular focus on those that are also Cayley graphs. A central result is a modern and mostly self-contained reconstruction of the proof that Cayley graphs of finite Coxeter groups are partial cubes, clarifying a classical but often fragmented argument. The second major contribution is the construction of non-Coxeter Cayley partial cubes, built from perfect codes in finite fields. These examples demonstrate that not all Cayley partial cubes arise from Coxeter groups, refuting a long-standing conjecture.
Decomposability bundle of measures and differentiability of Lipschitz functions
(2025-06-13) Horas Marcos, David; Puliatti, Carmelo
Topology and dynamics of the escaping set
(2025-06-13) Hernández Antón, Sergio; Núria Fagella Rabionet
The set of points that escape to infinity under iteration is a fundamental object in complex dynamics, often providing valuable insight into the global behavior of iterates and their relationship with the Fatou and Julia sets. The aim of this thesis is to study the topological and dynamical properties of the escaping set for polynomials, as well as transcendental entire and meromorphic functions. As an original contribution, we show that for the function $f(z) = z - \tan(z)$, the Julia set becomes connected upon adjoining infinity and contains the escaping set, which is totally disconnected. This is the first known nontrivial example exhibiting such behavior.
Regularity theory for the obstacle problem
(2025-06-01) Guerrero Rey, Jacobo; Weidner, Marvin
In this work, we introduce and study a classical free boundary problem known as the obstacle problem. This problem serves as a foundational example in the broader theory of free boundary problems, where part of the solution involves determining an unknown interface or region. We begin by presenting the classical formulation of the obstacle problem, which arises naturally in various physical and geometric contexts, such as elasticity, fluid dynamics, and potential theory. We then explore the key theoretical aspects of the problem, focusing on the existence, uniqueness, and regularity of solutions. Special attention is given to the structure and behavior of the free boundary, the interface separating the active and inactive regions which plays a central role in understanding the qualitative features of the solution. To analyze the behavior of the solution near the free boundary, we employ the method of blow-ups, a powerful technique that allows for the study of local properties by rescaling the problem around singular points. This approach provides deep insights into the regularity and classification of free boundary points, distinguishing between regular and singular behavior and leading to a better understanding of the geometry of the free boundary. Overall, this work offers a rigorous introduction to the obstacle problem, combining classical theory with modern analytical tools to examine one of the most important and illustrative problems in the study of partial differential equations and variational inequalities.
Structure theorems for non-Noetherian modules aimed at non-Noetherian Iwasawa theory
(2025-06-13) Gonzalo Calbetó, Gerard; Bars Cortina, Francesc
We explore several structure theorems for modules over non-Noetherian rings, with a particular interest in finitely presented torsion modules. Our main objective is to prove structure theorems for admissible modules, a new class introduced by Burns et al. in [7]. This work intends to be mostly self-contained: a first chapter is devoted to fixing preliminaries, a second chapter is devoted to presenting some key homological results due to Endo [10] and to the proof of structure theorems for finitely presented torsion modules over EDR domains, valuation rings and Prüfer rings, the last two due to Warfield [28]. In the third and last chapter, we prove the classical structure theorem for Noetherian Krull domains, introduce admissible modules, and prove theorems describing their structure. Finally, we present an application to classical and non-Noetherian Iwasawa theory.
An approach to topos theory related to neural networks
(2025-06-13) Garrido Garcia, Diego; Casacuberta, Carles
This work describes an application of topos theory and stacks to the semantic modeling of deep neural networks. Following the work of by Jean-Claude Belfiore and Daniel Bennequin, we present tools from topos theory to capture the learning process and semantic diffusion in a deep neural network architecture. We introduce the concepts of Grothendieck topoi, stacks and classifying topoi. Within this framework we establish conditions under which the semantic flow of information in a network can be transported across its layers.
Okounkov bodies and giansiracusa-giansiracusa valuations
(2025-06-13) Garcés Paniagua, Daniel; Roé Vellvé, Joaquim
Okounkov bodies are convex sets associated with divisors on algebraic varieties that allow for the study of asymptotic properties of both the divisor and the variety. This theory, developed independently by Lazarsfeld–Mustata [5], and Kaveh–Khovanskii [4], is based on tools from discrete semi-groups and valuations associated with divisors. It can be seen as a generalization of the Newton polytope in the context of projective toric varieties. On the other hand, prevaluations, also known by their author’s names as Giansiracusa–Giansiracusa valuations [16], are a more relaxed notion of valuation, replacing the value group with an idempotent semi-ring. This generalization allows for the definition of a natural product of prevaluations, which opens new paths for reinterpreting and/or extending the construction of Okounkov bodies. We will discuss the fundamental concepts and present some preliminary developments towards developing a theory of Okounkov bodies with respect to prevaluations.

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