Proyectos

Fondecyt Regular 2017-N°1171575. CONICYT. Proyecto titulado: Synergistic transcriptional response of Listeria monocytogenes to low temperature and copper. Instituciones patrocinantes: Universidad de Chile y Universidad de O’Higgins. Investigadora responsable Dra. Angélica Reyes. Marzo 2017-Diciembre 2019.

The main goal of this proposal is to study different particle systems interacting through a empirical mean term and the corresponding mean field descriptions when the number of individuals is going to infinity. In particular, we are interested in well posedness questions for those nonlinear SDEs, with coefficients depending on the law of the unknown process itself. Classically, the existence of those solutions can be addressed via propagation of chaos arguments. By using tightness arguments on the particle system solutions one can find weak solutions to the mean field nonlinear processes using the consistency of equations. In that sense, we are concerned in revisit some classical techniques and adapt them to some modern problems involving L ́evy, Hawkes and Asymmetric noisy components. Thus, we are interested in stating a new framework to englobe some applications that are not rigorously treated in literature (specially in theoretical biology, neuroscience and ecology). Mathematically speaking, the proposed methodology is quite standard. We start by well known problems and modify the proofs to attack perturbations of the seminal equations. In doing so, we will find out what are the key constraints and get some insights of the elements we need to develop in order to study the new questions stated. To fill the gaps between classic and modern applications, there are some very interesting prints stating general BDG inequalities for L ́evy processes and central limit theorems for Hawkes processes (see Proposed Research for more details). The proposed research was motivated by the final goal of having process with memory, but when Markov Property no longer apply there is no much we can solve directly. Thus, we will try to restraint first to the Markovian framework by using compactness assumptions in past/delay dependance and/or regular approximations of the functions involved in the SDEs. The goals and expected results are in particular: 1. Toy model 1: The justification of the mean-field limit process for a general model of interacting particles connected through Poisson measures. The main novelty of the equation is presence of the unknown in the size and intensity of the mean field jump part. By imposing a non mass creation hypothesis, solutions still upper bounded and the tightness method of Sznitman for proving the chaos propagation holds. 2. Toy model 2: The analysis of a model composed by a small world network of interacting particles and a large scale interaction in an open system approach. To use the coupling technique and prove the consistence between the particle system and the nonlinear process of a system driven by one (or in general a sequence) of compound Poisson process(es). This result can be done by using the arguments of Jourdain et al. 2008, in particular, by using the BDG type of inequality stated there. 3. Study numerically the shape of the invariant measures in some ecologically inspired models of type 1 and 2. This equations are naturally found by using logistic growth dynamics and competitive/cooperative interaction functions (Lotka-Volterra kind of interactions). To find resolvable toy models and prove stability of the steady states. ˆ 4. L ́evy particle system: to use the L ́evy-Ito decomposition theorem to write the randomness as the sum of a continuous part, a compound Poisson process and a square integrable pure jump part. To control the deterministic part of the dynamics by using a Lipschitz hypothesis. The stochastic components, can be treated by using classical arguments on diffusive particles, the toy-model 2 and again the BDG type of inequality of Jourdain et al. 2008. 5. Hawkes particle system: to prove an equivalence between Hawkes processes and a system of equations involving Poisson measures by using the method of Delattre et al. 2016. To solve the well posedness of the new particle system by using the arguments of the toy-model 1. By using the results of Chevallier et al. 2015 for the pure point process case, to solve our limit equation. With the well posedness of both systems, to use the consistence part of the chaoticity to conclude the convergence in law towards the mean field equation. 6. Asymmetric particle system: to use the regularity approximation of LeGall to justify the passage from an equation involving local times, to a classical diffusive equation. Solve the well-posedness and the chaos propagation questions for this new system which is an application of the classical version of the chaos property. To take the limit on the approximation functions for the new mean-field equation and study the convergence towards an asymmetric nonlinear process.

The main goal of this proposal is to study different particle systems interacting through a empirical mean term and the corresponding mean field descriptions when the number of individuals is going to infinity. In particular, we are interested in well posedness questions for those nonlinear SDEs, with coefficients depending on the law of the unknown process itself. Classically, the existence of those solutions can be addressed via propagation of chaos arguments. By using tightness arguments on the particle system solutions one can find weak solutions to the mean field nonlinear processes using the consistency of equations. In that sense, we are concerned in revisit some classical techniques and adapt them to some modern problems involving L ́evy, Hawkes and Asymmetric noisy components. Thus, we are interested in stating a new framework to englobe some applications that are not rigorously treated in literature (specially in theoretical biology, neuroscience and ecology). Mathematically speaking, the proposed methodology is quite standard. We start by well known problems and modify the proofs to attack perturbations of the seminal equations. In doing so, we will find out what are the key constraints and get some insights of the elements we need to develop in order to study the new questions stated. To fill the gaps between classic and modern applications, there are some very interesting prints stating general BDG inequalities for L ́evy processes and central limit theorems for Hawkes processes (see Proposed Research for more details). The proposed research was motivated by the final goal of having process with memory, but when Markov Property no longer apply there is no much we can solve directly. Thus, we will try to restraint first to the Markovian framework by using compactness assumptions in past/delay dependance and/or regular approximations of the functions involved in the SDEs. The goals and expected results are in particular: 1. Toy model 1: The justification of the mean-field limit process for a general model of interacting particles connected through Poisson measures. The main novelty of the equation is presence of the unknown in the size and intensity of the mean field jump part. By imposing a non mass creation hypothesis, solutions still upper bounded and the tightness method of Sznitman for proving the chaos propagation holds. 2. Toy model 2: The analysis of a model composed by a small world network of interacting particles and a large scale interaction in an open system approach. To use the coupling technique and prove the consistence between the particle system and the nonlinear process of a system driven by one (or in general a sequence) of compound Poisson process(es). This result can be done by using the arguments of Jourdain et al. 2008, in particular, by using the BDG type of inequality stated there. 3. Study numerically the shape of the invariant measures in some ecologically inspired models of type 1 and 2. This equations are naturally found by using logistic growth dynamics and competitive/cooperative interaction functions (Lotka-Volterra kind of interactions). To find resolvable toy models and prove stability of the steady states. ˆ 4. L ́evy particle system: to use the L ́evy-Ito decomposition theorem to write the randomness as the sum of a continuous part, a compound Poisson process and a square integrable pure jump part. To control the deterministic part of the dynamics by using a Lipschitz hypothesis. The stochastic components, can be treated by using classical arguments on diffusive particles, the toy-model 2 and again the BDG type of inequality of Jourdain et al. 2008. 5. Hawkes particle system: to prove an equivalence between Hawkes processes and a system of equations involving Poisson measures by using the method of Delattre et al. 2016. To solve the well posedness of the new particle system by using the arguments of the toy-model 1. By using the results of Chevallier et al. 2015 for the pure point process case, to solve our limit equation. With the well posedness of both systems, to use the consistence part of the chaoticity to conclude the convergence in law towards the mean field equation. 6. Asymmetric particle system: to use the regularity approximation of LeGall to justify the passage from an equation involving local times, to a classical diffusive equation. Solve the well-posedness and the chaos propagation questions for this new system which is an application of the classical version of the chaos property. To take the limit on the approximation functions for the new mean-field equation and study the convergence towards an asymmetric nonlinear process.

La región de O´Higgins, cuya principal actividad económica corresponde a la agricultura, está siendo severamente afectada por la sequía como consecuencia del cambio climático, generando impactos negativos para este sector productivo . Así, la búsqueda de especies y/o cultivares con mayor tolerancia al estrés hídrico es clave para diversificar l a matriz productiva de la región. El almendro es una opción viable para la región debido a su ampliamente reportada tolerancia al estrés hídrico, con gran éxito en el cultivo de esta especie en áreas con clima Mediterráneo, incluyendo Chile, por lo que la identificación de cultivares con mayor tolerancia a este estrés es de importancia. A nivel fisiológico, el principal efecto del estrés hídrico es el cierre estomático, con el objetivo de evitar una disminución drástica en el estado hídrico de la planta (p otencial hídrico, Ψ L ), esto disminuye la asimilación de carbono y por consiguiente la producción. Sin embargo, especies y cultivares han desarrollado distintas estrategias para responder frente al estrés hídrico, lo que se ha descrito a través de los compo rtamientos isohídricos y anisohídricos. En breve, especies/cultivares con comportamiento isohídrico se caracterizan por un rápido cierre estomático frente al estrés hídrico previniendo así caídas drásticas en Ψ L . Por el contrario, especies/cultivares aniso hídricos mantienen la apertura estomática durante el estrés, permitiendo una caída en Ψ L y, por lo tanto, sosteniendo la asimilación de carbono. En consecuencia, el comportamiento anisohídrico se asocia a mayores tasas fotosintéticas y eficiencia de uso de agua a pesar de la escasez hídrica. La caracterización e identificación de genotipos anisohídricos es importante para optimizar la búsqueda de nuevos cultivares a ser producidos en la región de O´Higgins, especialmente en el secano interior donde los efec tos del déficit hídrico son más severos. Como parte del desarrollo del proyecto Fondecyt de iniciación de la Dra. Alvarez, se han identificado y caracterizado a nivel fisiológico tres cultivares de almendro con comportamiento an/isohídricos contrastantes: Avijor, Isabelona y Soleta. Si bien el proyecto anteriormente descrito incluye el análisis transcripcional de algunos genes, que hipotéticamente tendrían un rol en la diferenciación entre ambos comportamientos, este análisis puntual no permitiría comprende r en profundidad los mecanismos moleculares y rutas que subyacen procesos adaptativos y respuestas a señales ambientales a nivel de genoma completo. Para ahondar en esto último, hemos planteado el siguiente objetivo general: “ Determinar las diferencias gen ómicas, epigenéticas y transcriptómicas que determinan el fenotipo anisohídrico e isohídrico entre distintos cultivares de almendro durante el estrés de déficit hídrico” hídrico”. Sin embargo, a la fecha solamente tres variedades de almendro han sido secuenciadas : Nonpareil, Texas y Lauranne. Si bien el genoma de esta especie es pequeño (~250Mb) su alto nivel de heterocigosidad dificulta su ensamble. Específicamente para esta especie, abordaremos la secuenciación del genoma con una estrategia híbrida que integra d atos de Oxford Nanopore (para secuencias largas) y MGI (lecturas cortas) y algoritmos desarrollados por el Dr. Di Genova que permitirán realizar construcciones a escala cromosómica para las tres variedades propuestas. Los genomas ensamblados permitirán det erminar variantes genéticas y patrones epigenéticos diferenciales entre los tres cultivares propuestos. Adicionalmente, secuenciaremos y analizaremos datos de expresión génica de las tres variedades enfrentadas a estrés hídrico, lo que nos permitirá estudi ar por primera vez el impacto funcional de variantes genéticas y epigenéticas asociados al estrés hídrico. En resumen, nuestro proyecto multidisciplinario generará el genoma de referencia y epigenoma para los tres cultivares: Avijor, Isabelona y Soleta. A dicionalmente, realizaremos un primer mapeo a nivel genómico, epigenético y de expresión de los mecanismos y programas moleculares durante el estrés hídrico de los tres cultivares de almendro.

Objetivos: Caso de estudio continental 1) Caracterizar y reconocer las áreas de mayor biodiversidad de megafauna extinta (mamíferos de >44 kg de masa corporal) para el periodo de transición Pleistoceno-Holoceno en Sudamérica. 2) Identificar si dichas áreas corresponden a posibles corredores o rutas utilizadas por los primeros cazadores-recolectores que colonizaron el continente durante el Pleistoceno tardío. Caso de estudio particular: Desierto de Atacama 3) Caracterizar la distribución de especies de megafauna para para el Desierto de Atacama (y para los Andes Centrales), tomando en cuenta el rico registro paleoclimático de la región como variable explicativa de la presencia de ciertas especies de megafauna en este sector. 4) Analizar si dicha distribución podría explicar el uso por parte de los primeros humanos de distintas rutas de migración en el sector (costeras, alto andinas y a través del desierto). Metodología: Generar Modelos de Distribución de Especies a nivel continental para los distintos taxa de megafauna extinta sudamericana que existieron durante la transición Pleistoceno-Holoceno. Estos modelos se generarán utilizando la locación geográfica e información cronológica del rico registro fósil de megafauna existente para Sudamérica, junto con información de las condiciones ambientales extraída de modelos climáticos generados para distintos momentos de la transición Pleistoceno-Holoceno. A partir de los Modelos de Distribución de Especies se identificarán las áreas de mayor biodiversidad (aquellas con la mayor cantidad de taxa) las cuales se comprarán con la distribución geográfica de sitios arqueológicos tempranos. Todo esto con la finalidad de identificar posibles rutas de migración humana hacia el continente. Para el caso del Desierto de Atacama (y de los Andes centrales) se aplicará la misma metodología descrita anteriormente pero haciendo uso del rico registro paleclimático existente para la región. Además, utilizaré los modelos para estimar la distribución pasada de especies actuales que habitan el sector desde el Pleistoceno tardío. Los Modelos de Distribución de Especies resultantes se compraran con la distribución geográfica de sitios arqueológicos para intentar reconocer las rutas migratorias usadas por los primeros humanos que colonizaban este ambiente extremo. Con la idea de obtener mayor información acerca de la megafauna extinta que habitó el Desierto de Atacama, me uniré a un grupo de investigadores en labores de rescate y análisis de sitios fósiles recientemente encontrados. Mi propósito es aportar con mi conocimiento de datación de megafauna extinta por radiocarbono y obtener más datos que mejoren la calidad de los modelos planeados. Resultados esperados: Se espera que la superposición de los modelos de distribución de megafauna extinta para la transición Pleistoceno-Holoceno muestre las áreas que probablemente exhibían la mayor biodiversidad para este periodo. Al comprar dichas áreas con la locación geográfica de los sitios tempranos de colonización humana en Sudamérica, se espera reconocer si las rutas de migración escogidas por los primeros cazadores-recolectores habrían sido determinadas por la presencia de mayor o menor biodiversidad de megafauna. Para el caso del Desierto de Atacama y los Andes Centrales, se espera obtener la probable distribución de las especies de megafauna que habitaban el área de manera más detallada, dado el uso de registros paleoclimáticos específicos para la zona como variables predictivas. Se espera que dichos Modelos de Distribución de Especies de megafauna permitan comprender e identificar la elección de rutas migratorias por los primeros cazadores-recolectores que habitaron estas regiones en particular.

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.

- Investigador asociado. Centro de Modelamiento Matemático (CMM). Centro Basal Nº AFB-170001, Director: Alejandro Maass. Marzo 2016-actual.