Facultad de Matemáticas

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Almost 1-1 extensions, equicontinuous systems and residually finite groups
(2024) Gómez Ortiz, Jaime Andrés; Cortéz, María Isabel; Pontificia Universidad Católica de Chile. Facultad de Matemáticas
The purpose of this document is to present our study on the various properties of almost 1-1 extensions of G-odometers with regards to the realization of Choquet simplices, mean-equicontinuity, and the construction of specific almost 1-1 exten- sions of equicontinuous systems. These systems can be viewed as a topological generalization of equicontinuous systems with diverse behavior on some aspects as entropy, the set of probability invariant measures, and more. Each problem is addressed within a general framework without assuming any amenable property on the acting group, except for the last problem where amenability was essential for constructing a specific type of almost 1-1 extensions. This thesis is divided into three parts, with the first two chapters presenting the results of two different manuscripts that are published and submitted, respectively.
Arithmetic of Drinfeld modules
(2025) Alvarado Torres, Matías Nicolás; Pasten Vásquez, Héctor Hardy; Pontificia Universidad Católica de Chile. Facultad de Matemáticas
Drinfeld modules, introduced by Vladimir Drinfeld in the 1970s, have become acornerstone in the arithmetic of global function fields. These objects serve as thefunction field analogues of elliptic curves and abelian varieties, but with a structure thatis uniquely adapted to the arithmetic of positive characteristic. Defined over rings offunctions rather than number fields, Drinfeld modules allow for the development of arich arithmetic theory that mirrors, and in many ways extends, the classical theory ofelliptic curves. Their moduli spaces, Galois representations, and associated L-functionshave all been studied extensively, revealing deep analogies with the number field caseand offering new phenomena unique to the function field setting. From an arithmeticstandpoint, Drinfeld modules provide explicit realizations of class field theory for globalfunction fields, particularly through the theory of Hayes modules and the use of shtukas.They give rise to Galois representations whose image and ramification behavior encodesignificant arithmetic information. Moreover, the theory of heights and canonical measuresassociated with Drinfeld modules has led to important results in Diophantine geometry,such as analogues of the Mordell-Weil theorem and the Bogomolov conjecture in positivecharacteristic. Beyond their arithmetic significance, Drinfeld modules also exhibit a richdynamical structure.
Bayesian nonparametric hypothesis testing
(2019) Pereira Hoyos, Luz Adriana; Gutiérrez, Luis; Pontificia Universidad Católica de Chile. Facultad de Matemáticas
In this thesis, we propose novel Bayesian Nonparametric hypothesis testing procedures for correlated data. First, we develop and study a proposal for comparing the distributions of paired samples. Next, we propose and analyze a hypothesis testing procedure for longitudinal data analysis. Both proposals are based on a flexible model for the joint distribution of the observations. The flexibility is given by a mixture of Dirichlet processes. Besides, for setting up the hypothesis testing procedures, we use a hierarchical representation with a spike-slab prior specification for the base measure of the Dirichlet process and a prior specification on the space of models. For the paired sample test, we use an appropriate parametrization for the kernel of the mixture to facilitate the comparisons and posterior inference. Consequently, the joint model allows us to derive the marginal distributions and test whether they differ or not. The procedure exploits the correlation between samples, relaxes the parametric assumptions, and detects possible differences throughout the entire distributions. For the longitudinal data, we propose to use a mixture of Dependent Dirichlet Processes to capture the correlation between the repeated measurements. The weights of the mixture are built via a stick-breaking prior, that comes from a Markovian process evolving in time. The effect of the predictors is modeled by the underlying atoms. The proposal can provide an estimation of the density through the time for different levels of the predictors, and at the same time can identify the effect of the predictors, without assuming restrictive distributional assumptions. We show the performance throughout the document of our proposals in illustrations with simulated and real data sets. Finally, we provide concluding remarks and discuss open problems.
Bilinear Form Test: Theoretical Properties and Applications
(2025) Gárate Barraza, Ángelo Fabián; Galea Rojas, Manuel Jesús; Osorio, Felipe; Pontificia Universidad Católica de Chile. Facultad de Matemáticas
The present thesis investigates the Bilinear Form Test (BF Test) as a robust statistical tool for evaluating parameter constraints across various models. It examines the test's theoretical foundations, with a particular focus on its invariance under reparameterizations and its performance in finite-sample settings. By leveraging bilinear forms, the BF Test provides an alternative to likelihood-based methods, employing an asymptotic chi-squared distribution that simplifies hypothesis testing. Monte Carlo simulations and empirical applications—including its use in financial models like the Capital Asset Pricing Model (CAPM) and in Generalized Estimating Equations (GEE) for correlated data—demonstrate the method’s efficiency, robustness, and versatility. Key contributions of this work include a detailed exploration of the BF Test's theoretical properties, validation of its invariance across different model structures, and a comprehensive comparison with traditional testing approaches, alongside proposed extensions for future research.
C*-algebric methods for transport phenomena
(2023) Polo Ojito, Danilo; De Nittis, Giuseppe; Pontificia Universidad Católica de Chile. Facultad de Matemáticas
Contributions to the singular perturbation theory of infinite-dimensional coupled systems
(2025) Arias Neira, Gonzalo Andrés; Cerpa, Eduardo; Marx, Swann; Pontificia Universidad Católica de Chile. Facultad de Matemáticas
Singular perturbation and separation of time scales methods have been used to study the stability and control design for coupled ODE systems with different time scales for many years. This important literature was motivated by the fact that systems with significantly different time scales appear in several applications, in which the constituents of a coupled system may model different physical phenomena taking place in different time scales. The singular perturbation method (SPM), roughly speaking, aims to decouple a full system into two approximated subsystems based on a suitable time-scale separation. This thesis addresses problems concerning the stability, Tikhonov's approximation, stabilization, and control of singularly perturbed coupled infinite-dimensional systems.
Equilibrium states and asymptotic variance for geometric potentials
(2025) Arévalo Hurtado, Nicolás; Iommi Echeverría, Godofredo; Pontificia Universidad Católica de Chile. Facultad de Matemáticas
En esta tesis abordamos tres problemas dentro del marco del formalismo termodinámico. En primer lugar, estudiamos el espectro de Lyapunov de las aplicaciones de Markov-Rényi-Lüroth en el intervalo, una familia de aplicaciones de intervalo de tipo Markov numerables que pueden presentar simultáneamente puntos fijos parabólicos y una discontinuidad en la presión topológica asociada al potencial geométrico. En segundo lugar, investigamos la existencia de estados de equilibrio para una familia de aplicaciones monótonas a trozos con convexidad promedio. Estas aplicaciones pueden tener puntos fijos parabólicos, particiones no Markovianas y un potencial geométrico que no es necesariamente Hölder continuo. Finalmente, analizamos la varianza asintótica para transformaciones abiertas, topológicamente transitivas y expansivas en espacios métricos compactos. Proporcionamos nuevas cotas para las diferencias en desigualdades de media potencia para potenciales Hölder continuos, expresadas en términos de la varianza asintótica.
Essential minimum in families
(2023) Morales Inostroza, Marcos; Kiwi Krauskopf, Jan Beno; Sombra, Martín; Pontificia Universidad Católica de Chile. Facultad de Matemáticas
Fibred non-hyperbolic quadratic families
(2024) Domínguez Calderón, Igsyl; Ponce Acevedo, Mario; Pontificia Universidad Católica de Chile. Facultad de Matemáticas
The aim of this thesis is two-folding. In the initial instance, we have made signifIcant progress in the problem of density of hyperbolic components within the context of fibred quadratic polynomial dynamics by demonstrating the existence of robust non- hyperbolic fibred quadratic polynomials. Secondly, we present a more complex class of invariant sets that are distinct from the invariant curves for fibred polynomial dynamics, called multi-curves. Furthermore, a construction for multi-curves in quadratic polynomial dynamics is shown, resulting in the attainment of not only invariant multi-curves, but also with the characteristic of being attracting.
Fitting time-varying parameters to astronomical time series
(2022) Soto Vásquez, Darlin Macarena; Motta, Giovanni; Galea Rojas, Manuel Jesús; Pontificia Universidad Católica de Chile. Facultad de Matemáticas
Flexible spatio-temporal strategies for modeling mosquito-borne diseases
(2024) Pavani, Jessica Letícia; Quintana Quintana, Fernando; Pontificia Universidad Católica de Chile. Facultad de Matemáticas
Growing awareness of environmental threats has encouraged researchers to increasingly focus on analyzing spatial and temporal patterns of diseases, including vector-borne diseases. A byproduct of this is the also increased interest in cluster analysis. Over the last few decades, the frequency and magnitude of disease outbreaks caused by insects have increased dramatically. In addition to areas that are recurrently affected, outbreaks are spreading into regions that were previously unaffected. Faced with such a scenario, clustering analysis is essential for recognizing areas and times with high disease incidence, thus aiding in intervention planning. Moreover, the increasing availability of large datasets of high quality has culminated in the emergence of more sophisticated statistical models and methods. In response to this need, we have developed some flexible Bayesian approaches whose main goal is to identify and cluster neighboring regions where the infection behaves similarly, and to evaluate how the spatial clustering pattern changes over time. To begin with, we develop a technique for recognizing and grouping regions that display similar time-based patterns for a specific disease. Our method employs product partition models that take into account the influence of neighboring regions to cluster geographical data. This prior is tied to temporal modeling, as it aligns the classification of regions with their time trends. Consequently, the temporal coefficients are common among areas within the same cluster. Furthermore, we introduce a directed acyclic graph structure to manage the spatial dependencies among these regions. As a contribution to the literature on multivariate data, we extend the first approach to jointly modeling multiple diseases, explicitly accounting for potential space-time correlations between them. In this case, we employ a multivariate directed acyclic graph autoregressive framework to capture both spatial and inter-disease dependencies. In the initial two models, the spatial cluster stays unchanged throughout time. However, the challenge of modeling intensifies when we attempt to examine temporal changes across different spatial partitions. To address this, we introduce a model for time-dependent sequences of spatial random partitions, establishing a prior based on product partition models that correlate spatial configurations. By utilizing random spanning trees as a methodological tool, we ease the exploration of the complex partition search space. We validate the properties of all models through simulation studies, demonstrating its competitive performance against alternative approaches. Furthermore, we apply them to mosquito-borne diseases dataset in the Brazilian Southeast region.
Inference from RDS data over Directed Networks
(2023) Sepúlveda Peñaloza, Alejandro Adrián; Beaudry, Isabelle; Pontificia Universidad Católica de Chile. Facultad de Matemática
El muestreo dirigido por los encuestados (Respondent-Driven Sampling, RDS) es una técnica utilizada para recolectar datos de poblaciones humanas socialmente conectadas que no tienen un marco de muestreo definido. Un paso fundamental para realizar inferencias basadas en el diseño de datos RDS es estimar las probabilidades de muestreo. Tradicionalmente, se ha asumido que una cadena de Markov de primer orden sobre una red completamente conectada y no dirigida representa adecuadamente el RDS. Sin embargo, este modelo simplificado no tiene en cuenta que la red puede ser dirigida y homofílica. Este trabajo propone métodos para abordar estos problemas y estimar la prevalencia de un estado de infección en redes de este tipo.Las principales contribuciones metodológicas de esta tesis son tres: primero, la introducción de un modelo de configuración de red parcialmente dirigida y homofílica; segundo, el desarrollo de dos representaciones matemáticas del proceso de muestreo RDS en el modelo propuesto; y tercero, la propuesta de un modelo bayesiano que considera una red dirigida y el número de conexiones entre nodos infectados y no infectados para estimar la prevalencia del estado de infección.Se realizaron estudios de simulación para demostrar que las probabilidades de muestreo resultantes con nuestras propuestas son similares a las del RDS tradicional, mejorando la estimación de prevalencia bajo diversos escenarios realistas, asumiendo que dichas probabilidades son conocidas. La estimación de la prevalencia del estado de infección se realiza bajo fuertes suposiciones sobre la red, como la ausencia de homofilia o la dirección de los bordes.Para la aplicación del modelo, se utilizó la teoría de copulas, el modelamiento de distribuciones marginales y un modelo de superpoblación para estimar información a partir de datos no observados de la red. Las simulaciones realizadas mostraron una mejora en la estimación de la prevalencia del estado de infección en términos de sesgo y variabilidad utilizando datos de RDS.
Lower bounds for the relative regulator
(2021) Castillo Gárate, Víctor; Friedman R., Eduardo; Pontificia Universidad Católica de Chile. Facultad de Matemáticas
El regulador relativo Reg(L/K) de una extensión de cuerpos de números L/K está estrechamente relacionada con el cuociente Reg(L)/Reg(K) de reguladores clásicos de L y K. En 1999 Friedman y Skoruppa [FS99] demostraron que Reg(L/K) posee cotas inferiores que crecen exponencialmente con el grado absoluto [L : Q], siempre que el grado relativo [L : K] sea suficientemente grande. Friedman y Skoruppa partieron de una desigualdad analítica que involucra Reg(L/K) y desarrollaron un análisis asintótico que funciona bien para grados relativos [L : K] ≥ 40. En esta tesis, partimos de la misma desigualdad, pero para grados [L : K] ≤ 40 usamos técnicas numéricas y asintóticas para demostrar el crecimiento exponencial de las cotas inferiores cuando [L : K] ≥ 12. Imponiendo algunas hipótesis sobre la descomposición en L/K de los lugares arquimedianos, obtenemos también buenas cotas inferiores para Reg(L/K) para algunos grados [L : K] < 12. Por ejemplo, si K es totalmente complejo obtenemos buenas cotas inferiores para el regulador relativo si [L : K] ≥ 5.
Mathematical analysis and applications of neural networks, with applications to image reconstruction
(2025) Molina Mejía, Juan José; Courdurier, Matías; Pontificia Universidad Católica de Chile. Facultad de Matemáticas
This thesis explores two fundamental aspects of neural networks: their frequency learning behavior and their application to quantitative Magnetic Resonance Imaging (MRI) reconstruction. The first part investigates the phenomenon of frequency bias, the empirical observation that neural networks tend to learn low-frequency components of a target function more rapidly than high-frequency ones. To provide a rigorous understanding of this behavior, we develop a theoretical framework based on Fourier analysis. Specifically, we derive a partial differential equation that governs the evolution of the error spectrum during training in the Neural Tangent Kernel regime, focusing on two-layer neural networks. Our analysis centers on Fourier Feature networks, a class of architectures where the first layer applies sine and cosine activations using pre-defined frequency distributions. We demonstrate that the network's initialization, particularly the initial density distribution of first-layer weights, plays a crucial role in shaping the frequency learning dynamics. This insight provides a principled way to control or even eliminate frequency bias during training. Theoretical predictions are validated through numerical experiments, which further illustrate the impact of initialization on the inductive biases of neural networks.The second part of the thesis applies neural network techniques to the reconstruction of quantitative MRI data. Quantitative MRI enables the estimation of tissue-specific parameters (e.g., T1, T2, and T2*) that are vital for clinical diagnosis and disease monitoring. However, these methods typically require long acquisition times, which are often mitigated through aggressive undersampling of k-space data. Undersampling, in turn, introduces reconstruction artifacts that must be addressed through regularization. To this end, we propose CConnect, a novel iterative reconstruction method that incorporates convolutional neural networks into the regularization term. CConnect connects multiple CNNs through a shared latent space, allowing the model to capture common structures across different image contrasts. This design enables the effective suppression of aliasing artifacts and improves image quality, even in highly undersampled scenarios. We evaluate CConnect on in-vivo brain T2*-weighted MRI data, demonstrating its superiority over classical low-rank and total variation methods, as well as standard deep learning baselines.
Medidas de acuerdo bajo el modelo estructural multivariado
(2022) Ávila Albornoz, Julio Cesar; Galea Rojas, Manuel Jesús; Pontificia Universidad Católica de Chile. Facultad de Matemáticas
En biometría, ingeniería, medicina y otras áreas es común disponer de distintos instrumentos que midan alguna característica en una unidad experimental. En ocasiones, un nuevo instrumento es propuesto como una alternativa más económica o práctica respecto al instrumento estándar. Si ambos trabajan en una misma escala, es deseable medir el grado de acuerdo o de concordancia que alcanzan. En este contexto, existen varias propuestas para medir el acuerdo entre instrumentos de las cuales se profundizaría en: El Coeficiente de Correlación de Concordancia (CCC) y la Probabilidad de Acuerdo (PA). Las mediciones de los instrumentos pueden estar sujetas a error en la medición. Si estos errores de medición no fueran considerados, las inferencias realizadas podrían estar comprometidas o ser incorrectas. Los Modelos con Error de Medición (MEM) permiten incorporar la incertidumbre que el proceso de medición pueda tener. Una aplicación de los MEM es el modelo de calibración en su versión estructural. Para modelar el error de medición, los MEM asumen una distribución multivariante, siendo la distribución Normal multivariada de gran utilidad en varias aplicaciones. Sin embargo, en presencia de colas pesadas o de datos atípicos, la suposición de normalidad puede ser poco adecuada llevando a comprometer los resultados. Una manera de afrontar este problema es emplear la distribución t multivariada considerada como una extensión de la distribución Normal multivariada. El objetivo de este trabajo es desarrollar bajo el Modelo Estructural Multivariado, herramientas de inferencia estadística para las medidas de acuerdo: CCC y PA. El Modelo Estructural considera el uso de la distribución Normal Multivariada y la t multivariada. Las herramientas estadísticas fueron aplicadas a conjuntos de datos clásicos en la comparación de instrumentos y además, en aplicaciones financieras como el retorno de acciones y las proyecciones del tipo de cambio por parte de operadores financieros.
Métodos DPG para el problema quad-curl
(2024) Herrera Ortiz, Pablo; Heuer, Norbert; Führer, Thomas; Pontificia Universidad Católica de Chile. Facultad de Matemáticas
Los problemas relevantes de la magnetohidrodinámica y la dispersión electromagnética utilizan operadores diferenciales de cuarto orden de tipo rotacional, generalmente denominados operadores quad-curl. Su uso requiere métodos de aproximación numérica. En el caso de los operadores quad-curl, la literatura correspondiente es escasa. La discretización de operadores de cuarto orden es difícil debido al requisito de regularidad para las aproximaciones conformes y la presencia de kernels no triviales. Proponemos emplear el método de Petrov-Galerkin discontinuo (método DPG) con funciones de test óptimas. Este es un marco propuesto por Demkowicz y Gopalakrishnan que tiene como objetivo la estabilidad discreta automática de los esquemas de aproximación.El trabajo está dividido en tres partes. La primera parte examina el problema quad-div en dos y tres dimensiones, mostrando su relación con el operador quad-curl $Curl^4$ en el caso 2D. Presentamos el problema como sistemas de primer y segundo orden. Adicionalmente, proporcionamos un método completamente discreto y realizamos un experimento numérico para el caso adaptativo. En la segunda parte, escribimos el operador quad-curl como $-\Curl\Delta\Curl$, formulamos el problema como un sistema de segundo orden y proporcionamos una formulación variacional ultra-débil. Utilizamos los operadores de Fortin del método DPG para el problema de Kirchhoff--Love en 2D para analizar el esquema completamente discreto. Mostramos una aplicación al problema de Stokes en 2D con cargas en $L_2$ y $H^{-1}$. En la tercera parte, estudiamos directamente el operador $\Curl^4$ en 3D como un sistema de segundo orden y proporcionamos una formulación variacional ultra-débil. En este caso, la existencia de un operador de Fortin es un problema abierto.A lo largo de la tesis, empleamos el marco teórico DPG con formulaciones ultra-débiles. La mayor parte de nuestro análisis se centra en estudiar los operadores de traza, los espacios de traza y los saltos. Estos son claves para caracterizar la regularidad, la conformidad y las condiciones de contorno. Desarrollamos operadores de Fortin los cuales son necesarios para la estabilidad de las formulaciones mixtas. Para todos los casos definimos y analizamos los operadores de traza y espacios necesarios, demostramos el buen planteamiento de las formulaciones variacionales y su discretización, y derivamos estimaciones de error a priori.También examinamos técnicas para la inclusión de condiciones de contorno no homogéneas.Proporcionamos experimentos numéricos para todos los problemas y formulaciones. Estos confirman las propiedades de convergencia esperadas.
Modelling predictive validity problems : a partial identification approach
(2021) Alarcón Bustamante, Eduardo Sebastián; San Martín, Ernesto; Pontificia Universidad Católica de Chile. Facultad de Matemáticas
New contributions to joint models of longitudinal and survival outcomes : two-stage approaches
(2021) Leiva Yamaguchi, Valeria; Silva, Danilo Alvares da; Pontificia Universidad Católica de Chile. Facultad de Matemáticas
Joint models of longitudinal and survival outcomes have gained much popularity over the last three decades. This type of modeling consists of two submodels, one longitudinal and one survival, which are connected by some common term. Unsurprisingly, sharing information makes the inferential process highly time-consuming. This problem can be overcome by estimating the parameters of each submodel separately, leading to a natural reduction in the complexity of joint models, but often producing biased estimates. Hence, we propose different two-stage strategies that first fits the longitudinal submodel and then plug the shared information into the survival submodel. Our proposals are developed for both the frequentist and Bayesian paradigms. Specifically, our frequentist two-stage approach is based on the simulation-extrapolation algorithm. On the other hand, we propose two Bayesian approaches, one inspired by frailty models and another that uses maximum a posteriori estimations and longitudinal likelihood to calculate posterior distributions of random effects and survival parameters. Based on simulation studies and real applications, we empirically compare our two-stage approaches with their main competitors. The results show that our methodologies are very promising, since they reduce the estimation bias compared to other two-stage methods and require less processing time than joint specification approaches.
On accumulation points of volumes of stable surfaces with one cyclic quotient singularity
(2021) Torres Valencia, Diana Carolina; Urzúa Elia, Giancarlo A.; Pontificia Universidad Católica de Chile. Facultad de Matemáticas
The set of volumes of stable surfaces does have accumulation points. In this paper, we study this phenomenon for surfaces with one cyclic quotient singularity, towards answering the question under which conditions we can still have boundedness. Effective bounds allow listing singularities that might appear on a stable surface after fixing its invariants. We find optimal inequalities for stable surfaces with one cyclic quotient singularity, which can be used to prove boundedness under certain conditions. We also introduce the notion of generalized T-singularity, which is a natural generalization of the well-known T-singularities. By using our inequalities, we show how the accumulation points of volumes of stable surfaces with one generalized T-singularity are formed.
On the geography of 3-folds via asymptotic behavior of invariants
(2023) Torres Nova, Yerko Alejandro; Urzúa Elia, Giancarlo A.; Pontificia Universidad Católica de Chile. Facultad de Matemáticas
Roughly speaking, the problem of geography asks for the existence of varieties of general type after we fix some invariants. In dimension 1, where we fix the genus, the geography question is trivial, but already in dimension 2 it becomes a hard problem in general. In higher dimensions, this problem is essentially wide open. In this paper, we focus on geography in dimension 3. We generalize the techniques which compare the geography of surfaces with the geography of arrangements of curves via asymptotic constructions. In dimension 2 this involves resolutions of cyclic quotient singularities and a certain asymptotic behavior of the associated Dedekind sums and continued fractions. We discuss the general situation with emphasis in dimension 3, analyzing the singularities and various resolutions that show up, and proving results about the asymptotic behavior of the invariants we fix.

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