Proyectos

Proyecto interno de la UOH de carácter multidisciplinario que busca crear mapas moleculares multiómicos de los cánceres prevalentes en la región, utilizando tecnologías de vanguardia y algoritmos avanzados.

TRANSFERENCIA SUPERCOMPUTACIÓN PARA INNOVACIÓN EN SALUD REGIONAL: HPC-UOH Y HRLBO

Chile is facing one of the steepest increases in Colorectal cancer (CRC) incidence and mortality in the Southern Cone, with the greatest surge occurring in adults ≤ 50 years. Incidence is lowest in the far north and increases toward the south-central regions, mirroring a gradient in Aymara-Mapuche Native-American ancestry, an axis largely absent from the European reference cohorts that guide modern precision oncology. To fill this gap, we propose a four-year project to create the first Chile-specific, multi-omic and histopathological atlas of CRC and to explore ancestry-aware, AI-assisted diagnostics. Rationale and Hypothesis. We hypothesise that Chilean CRC shows (i) unique, ancestry-driven molecular patterns that differ from European tumors; (ii) AI models can detect these patterns directly on routine whole-slide images, and (iii) they shape distinct evolutionary paths in early- versus late-onset disease. Specific objectives. Molecular landscape & heterogeneity: Produce single-gland long-read WGS, methylome, and transcriptome profiles for 100 tumors (30 early-onset, 70 late-onset; ≥30× coverage, ≥50 % purity). Ancestry impact: Phase somatic alterations by local ancestry and contrast their frequencies with European CRC genomes (TCGA, PCAWG). AI-enhanced histopathology: Train and externally validate multi-instance-learning (MIL) models that predict microsatellite instability, driver mutations, and whole-genome doubling from matched WSIs, targeting AUC ≥ 0.80 (pilot: AUC ≥ 0.85 for whole-genome doubling on TCGA WSIs). Evolutionary trajectories: Multi-region sequence early-onset and late-onset tumors, reconstruct their clonal phylogenies, and contrast the resulting evolutionary patterns between the two age groups. Team capacity & resources. Computational biologist Alex Di Genova (genomics & AI), pathologist Juan Carlos Araya (digital pathology), and gastro-immunologist Tamara Pérez-Jeldres (clinical phenotyping) have prospective access to >220 new CRC resections and >1,800 registry entries each year. A biobank already houses 100 well-annotated tumour specimens from hospitals in Santiago and the O’Higgins Region, ready for immediate sequencing and imaging. As a team we are delivering important results as (i) the generation of the first telomere-to-telomere Chilean genome, establishing a population-specific reference; (ii) sequenced >270 high-coverage whole genomes of chileans individuals (70 healthy donors, 120 hereditary-breast-cancer cases, 80 primary gallbladder tumors); iii) built the first multi-omic atlas of gallbladder cancer by integrating our data with Korean (n = 94) and Indian (n = 64) cohorts, uncovering a Chile-enriched proliferative phenotype; and (iv) developed CRAB-MIL, a weakly supervised deep-learning framework that predicts whole-genome doubling from routine H&E slides with an AUC > 0.85 and provides attention maps for interpretability. These accomplishments demonstrate our ability to generate, integrate, and clinically interpret large-scale genomic and AI datasets—capabilities directly transferable to Chilean CRC. International collaborators Anaïs Baudot (Marseille) and Luis Zapata (Institute of Cancer Research, London) further contribute multi-omic network analysis and evolutionary-genomics expertise, respectively. Interdisciplinary workflow. Clinical phenotyping, computational histopathology, PromethION sequencing, and Nextflow harmonisation feed ancestry-aware genomic analyses; attention-based models are fine-tuned on TCGA and Chilean WSIs; computational, pathology, and gastroenterology teams jointly review outputs to prioritise clinically relevant signals. All variant calls, methylomes, expression matrices, AI prediction, and metadata will be released through an open and intuitive TumorMap portal. Expected Outcomes and Impact. The project will (i) reveal population-specific drivers and mutational processes, (ii) quantify the frequency of clinically actionable biomarkers originally identified in Europeans, (iii) deliver image-based tools that offer low-cost, molecular stratification and heterogeneity scoring, and (iv) provide a high-resolution evolutionary framework for EO versus LO CRC. Collectively, these data will offer the first high-resolution portrait of the Chilean CRC and lay the groundwork for ancestry-aware screening, diagnostic, and treatment strategies.

In general, machine learning aims to learn a model from the input data in order to make reliable and repeatable decisions. The learning of a model is either done automatically or semiautomatically. While deep learning can be used to automatically learn a model from arbitrary raw data, the number of successful application domains is still very restricted. This proposal is concerned with supervised learning - a machine-learning technique that aims at learning a model from input-output examples. A crucial task in supervised learning is the engineering of the features. Features are used to extract the relevant information from the raw data in order to learn a classifier that is based on the extracted data. A classifier is a function that partitions the input data into different categories. Feature engineering is a time-consuming process that includes a lot of trial and error, and stepwise addition or deletion of features. We aim at automating that process and learn a classifier based on some automatically generated features.

Problems that cannot be solved by classical computers in reasonable time due to their high computational cost arise in many research areas. In general, the evaluation of conjunctive queries over relational databases belongs to those problems. Conjunctive queries form the core of the Structured Query Language (SQL) which became a de facto standard for querying and maintaining relational databases. This work is about developing new approximation techniques for conjunctive queries which cannot be evaluated in reasonable time. Our new approximation techniques should lead to significant improvements for data aided decision making, e.g., for early warning system which are based on the analysis of big data or to make business-critical decisions by analyzing big data. In the last decades, a very good understanding of the classes of conjunctive queries which can be evaluated in reasonable time has been gained and it has been proven that an under-approximation of a query always exists within each of those classes. However this approach is rather strict and some of the under-approximations can be rather uninformative, i.e., the under-approximation might return the empty result set while the original query would not. over-approximations might be helpful when this happens, as they return all answers to a query. One of our goals is to study the foundational aspects of over-approximations, including the existence problem and the problem of computing an approximation. Unfortunately, over-approximations do not always exist (within a class of queries which can be evaluated in reasonable time), and it is not even known to be decidable whether a conjunctive query admits an over-approximation. Therefore, another goal of the proposed work is the development of more liberal approximation techniques that yield some kind of quantitative guarantees. This means that they should guarantee that the result of the approximation is not too “far” from the result of the original query over a set of databases of interest. Therefore we need to define a measure of disagreement between queries and/or results. For conjunctive query evaluation, such measures do not exist up until now. Based on that measure, we study approximations whose disagreement with the result of the query they approximate is below a certain threshold. Furthermore, we investigate how the underlying data of a database can help us to find better approximations. It has been shown that there are close relations between the approximation of conjunctive queries over relational databases and some classes of Semantic Web queries over semi-structured data. We also study possible connections between our approximation techniques and approximating Semantic Web queries.

En los campos de investigación de Análisis, Ecuaciones en Derivadas Parciales y Geometría, así como en otras áreas como la Física Experimental, Física Teórica y Ciencias de los Materiales, un gran foco de atención se pone en explorar las cuestiones de existencia y descripción cualitativa de puntos críticos a problemas variacionales, los cuales dependen de parámetros que surgen naturalmente en el modelamiento de fenómenos en los campos antes mencionados. Igualmente importante es el entendimiento de las propiedades de rigidez de estos modelos, a saber, decidir si la existencia y propiedades cualitativas de tales puntos críticos se mantienen, o cambian abruptamente, cuando los valores de los parámetros varían. En las últimas décadas, se ha logrado un gran progreso en estas direcciones - desde el punto de vista del análisis matemático - en dos clases de problemas variacionales que comparten ``características de isotropía sin pesos": los funcionales son construídos de manera que, a grandes rasgos, las funciones/conjuntos son penalizados independientemente de la posición, y no hay direcciones preferidas o distinguidas. El primer problema corresponde al modelo de gotas líquidas de Gamow, el cual es un problema isoperimétrico con un término adicional no-local de tipo Riesz, propuesto originalmente para describir la existencia/no-existencia y características geométricas del núcleo atómico, en física nuclear, dependiendo de un parámetro de masa. El segundo, es sobre el estudio de existencia y propiedades cualitativas de vórtices para la energía de Ginzburg-Landau (GLE), usada para describir defectos en superfluídos y física de materia condensada. En los años recientes, los investigadores se han dedicado a proponer y explorar variantes del modelo variacional antes descrito, donde el funcional de energía asociado adquiere pesos o anisotropías, en un intento por capturar fenómenos más finos y delicados encontrados en experimentos, así como para extender la teoría matemática a contextos más generales. Los objetivos principales del proyecto son el estudio de existencia, y propiedades cualitativas así como de rigidez para: (1) versiones con pesos del modelo de gotas líquidas, donde un funcional de perímetro reemplaza al perímetro usual en el sentido de De Giorgi, y (2) una version anisotrópica de la energía de Ginzburg-Landau derivada del modelamiento de defectos umbilicales en un límite 3D a 2D de cristales líquidos nemáticos, en un régimen físico que favorece el comportamiento anisotrópico. Concretamente, la primera parte del proyecto buscar establecer existencia, acotamiento y regularidad optimal para conjuntos que minimizan el funcional isoperimétrico con pesos más un potencial no-local, para una clase de pesos continuos que se asumen degenerados (valen cero en una cantidad finita de puntos) y coercivos en infinito. Además, esperamos probar una propiedad de rigidez en cuanto a la forma del minimizantes (unicidad): debe ser una bola, para un rango adecuado del parámetro de masa. Adicionalmente, planeamos argumentar rigurosamente el fenómeno de fragmentación infinita, cuando los valores del parámetro de masa tienden a cero. Esto requiere del entendimiento y adaptación de multiples heramientas profundas y sofisticadas en la teoría de medida geométrica y el cálculo de variaciones, que incluye: una versión geométrica del principio de concentración compacidad de Frank y Lieb; la teoría de regularidad estándar para conjuntos quasi-minimizantes en el contexto sin pesos, así como la adaptación de resultados de regularidad muy recientes de Pratelli y Saracco para conjuntos isoperimétricos con pesos (sin término no-local); la celebrada desigualdad isoperimétrica cuantitativa ajustada de Fusco, Maggi and Pratelli y la estrategia de Acerbi, Fusco and Morini para mostrar rigidez de minimizadores (en el marco isotrópico), entre otros. A este respecto, resaltamos un resultado reciente obtenido por el Investigador Principal sobre la existencia y regularidad de funciones con menor gradiente con pesos, un problema que está muy relacionado con el problema isoperimétrico con pesos. Concerniendo la segunda parte del proyecto, nuestro punto de partida es el trabajo de Clerc-Dávila-Vidal Henríquez-Kowalczyk quienes derivaron una version anistrópica de la energía GLE y estudiaron como la anisotropía quiebra algunas invarianzas de la asociada ecuación de GL. Las soluciones tipo-vórtice ahora consisten de un conjunto discreto (restricción en la fase), y para cada una ellos argumentan heurísticamente su estabilidad/inestabilidad, como una función del parámetro de anisotropía. El objectivo principal de este parte del proyecto es analizar rigurosamente la propiedades de estabilidad/inestabilidad lineal de soluciones tipo-vórtice simétricas (defectos puntuales) para la GLE anisotrópica, dependiendo del tamaño del núcleo de la solución. Para lograr estas metas se propone adaptar herramientas clave en el análisis de estabilidad de la energía, tales como "Fourier-splitting" y ortogonalidad de formas cuadráticas, de Mironescu; y el approach de del Pino-Felmer-Kowalczyk basado en decomposiciones tipo-Hardy, usado para argumentar la no-degeneracia de la forma cuadrática asociado al vórtice de grado-uno de GL; entre otros. Al IP le gustaría extender el análisis de estabilidad para soluciones de tipo-vórtice para la GLE anisotrópica que no son simétricas (vórtices con ``carga topológica" negativa). Este es un problema mucho más desafiante, ya que la representación polar de la solución no se desacopla en una parte radial y otra angular, de manera uni-dimensional. Finalmente, otro resultado de interés independente consiste en establecer existencia de soluciones tipo-vórtice generales para la GLE anisotrópica sin restricción en la fase; esto causa que el perfil radial sea a valores complejos, una característica fundamentalmente distinta comparada con aquella del caso isotrópico. En resumen, este proyecto buscar dar una contribución sustancial a la teoría de problemas variacionales geométricos con pesos y a la teoría de estabilidad de soluciones tipo-vórtice para versiones anisotrópicas de energías tipo Ginzburg-Landau, las cuales se estan convirtiendo en una foco de investigación profusa en la comunidad de Análisis No-Lineal. El desarrollo de herramientas y la adaptación de nuevas técnicas del Análisis nos permitirá obtener un mayor entendimiento sobre los mecanismos que influencian la existencia, propiedades cualitativas y de rigidez, en la presencia de pesos y anisotropías para los tipos de problemas variacionales propuestos. Esperamos esto llevaré a nuevas direcciones de investigación en problemas relacionados dentro del Cálculo de Variaciones, Teoría de Medida Geométrica, y Ecuaciones en Derivadas Parciales.

Aplicación de técnicas variacionales y de Ecuaciones Diferenciales Parciales en grafos para problemas de agrupación de datos y para segmentación de imágenes

Esta propuesta está dedicada al estudio de problemas locales y no locales, elípticos y parabólicos, en Ecuaciones en Derivadas Parciales (EDP). Se espera obtener resultados de existencia para los problemas planteados, son 3 los problemas que se quieren estudiar: El primer modelo involucra al Laplaciano Fraccionario con singularidades no lineales (Ecuación Fraccionaria de Burger ver (3)). En este problema queremos probar la existencia y la no unicidad de soluciones débiles de (3), teniendo en cuenta que las soluciones de entropía son soluciones débiles, el primer paso será probar existencia de soluciones de entropía, para esto usaremos el método de sub y supersoluciones, donde probaremos los resultados de los principios de Comparación y L1-Contracción. Una vez teniendo la existencia de solución de entropía, pasaremos a construir una solución débil que no sea solución de entropía, donde esta solución débil se obtendrá como límite de soluciones a problemas regularizados estacionarios, en donde usaremos métodos variacionales para resolver el problema regularizado. El segundo problema, es una extensión al caso no local del problema estudiado en [L. Jeanjean and V. Radulescu, Nonhomogeneous quasilinear elliptic problems: linear and sublinear cases, Journal d'Analyse Mathématique (2021)]. Para probar existencia de soluciones aplicaremos estudios de mínimos locales y el teorema del paso de montaña para el funcional de energía asociado. Para finalizar nuestra propuesta, queremos encontrar soluciones periódicas para sistemas de EDP de cuarto orden (tipo Fisher-Kolmogorov generalizado, ver (9). Siguiendo las técnicas en [P. Smyrnelis, Connecting orbits in Hilbert Spaces and application to PDEs, Comm. Pure Appl Anal 19 (5): 2897--2818 (2020)], usaremos métodos variacionales para probar la existencia de órbitas conectadas en espacios de Hilbert. Construiremos órbitas periódicas usando la construcción desarrollada por Alessio, Montecchiari y Zúñiga en [F. Alessio and P. Montecchiari and A. Zuniga, Prescribed energy connecting orbits for gradient systems, Discr. Cont. Dyn. Systems 39 (8): 4895--4928 (2019].

This research proposal aims to study the long-term behavior of solutions to partial differential equations arising from dispersive dynamics, kinetic models, and integro-differential dynamics in ecology; and to study extremals of functional inequalities in connection to the ground states of partial differential equations arising from quantum mechanics and diffusion phenomena. Five major topics are proposed: Relativistic quantum mechanics, Dirac operators and functional inequalities; Symmetry breaking in weighted functional inequalities and weighted diffusions; long time dynamics in dispersive PDEs in one space dimension; long-term dynamics in nonlocal models from ecology; and hypocoercivity and decay to equilibrium in kinetic models with heavy tails. The first topic focuses on establishing connections between spectral problems and functional inequalities for Dirac operators. The aim is to analyse the symmetry of optimal spinors in inequalities of Keller-Lieb-Thirring type, and to obtain the solitary waves of Soler-type nonlinear Dirac equations as optimizers of a nonlinear inequality. The second topic aims to characterize a symmetry range in which optimal functions are radially symmetric for weighted logarithmic Sobolev inequalities and a new family of Caffarelli-Kohn-Nirenberg inequalities. A nonlinear carré-du-champ method will be adapted to prove entropy-type estimates. Rigidity, perturbation, and stability issues will be addressed. The third topic seeks to study the asymptotic stability of topological and non-topological solitions for a class of dispersive PDEs in dimension one. A new method is proposed, based on perturbations in weighted spaces with exponential weights, on the so-called virial identities, and on the study of existence of breathers. The fourth topic concerns the description of evolutionary stable strategies of long-term dynamics of integro-differential models that arise in the modeling of structured populations, and to obtain qualitative and quantitative insights on the concentration dynamics. In the fifth topic, the aim is to extend the Dolbeault-Mouhot-Schmeiser method to study the large-time behavior of solutions for a broad family of kinetic equations in which the confinement potential exhibits heavy tails. The goals of this project are multiple: to strengthen and to create new collaborative research networks between France and Chile in the field of nonlinear partial differential equations and applications, to publish co-authored articles in top-tier journals and disseminate the results in international meetings, and to promote the formation of advanced human capital. In order to achieve these goals, yearly workshops will be organized in France and Chile to account for the progress of the investigations as well as to encourage the participation of students and young researchers. International training of doctoral and postdoctoral researchers will be ensured by allocating resources from this project for exchanges. Considering the history of successful collaboration amongst the members of this project, and their expertise in their research fields, we are confident about the successful termination of the project. In particular, we expect to pave the way for new research avenues. The main scientific contribution of this proposal involves adapting state-of-the-art techniques from PDEs and nonlinear analysis to obtain qualitative and quantitative results for variational problems and partial differential equations, in which the setting plays a crucial role: complex-valued matrices (first topic), nonlinear and weighted (second topic), strongly nonlinear and dispersive (third topic), nonlocal (fourth topic), general assumption on the tail of the confining potential (fifth topic). This proposed research will provide insights into spectral theory, stability theory of equilibria of differential equations, optimal rates of convergence to equilibria, and their relation to optimal constants in functional inequalities. The expected results will help improving the understanding of various real-life phenomena, including population-dynamics, relativistic quantum mechanics, and diffussion processes. The viability of the project is sustained on the expertise of the members of the Chilean and French research teams, including experts in partial differential equations, nonlinear analysis, calculus of variations, and mathematical physics. Their successful collaboration record and significant contributions to these fields only strengthen the potential of this proposal. In conclusion, the present research project will not only foster the scientific cooperation between Chile and France but it will also provide meaningful advancements in the aforementioned fields and their application to various physical phenomena.

Estudio e implementación de métodos híbridos para la resolución computacional de EDP aplicadas a Ciencia de Datos.