We show that Kirchhoff ’s law of conservation holds for non-commutative graph flows if and only if the graph is planar. We generalize the theory of (Euclidean) lattices to infinite dimension and... Show moreWe show that Kirchhoff ’s law of conservation holds for non-commutative graph flows if and only if the graph is planar. We generalize the theory of (Euclidean) lattices to infinite dimension and consider the ring of algebraic integers as such a lattice. We compute some invariants using capacity theory and obtain a partial solution to the (algorithmic) closest vector problem. We generalize the results on (universally) graded rings by Lenstra and Silverberg. We study the special case of group rings, and show that under similar assumptions rings can be uniquely decomposed into a group ring in a maximal way. We give a functorial algorithm to compute roots of fractional ideals of orders in number rings. Show less
In analogy to mathematical proofs, the goal of a proof system is for a prover to convince a verifier of the correctness of a claim. However, by contrast, probabilistic proofs allow the verifier to... Show moreIn analogy to mathematical proofs, the goal of a proof system is for a prover to convince a verifier of the correctness of a claim. However, by contrast, probabilistic proofs allow the verifier to make mistakes, i.e., to accept false claims or reject true claims. Further, probabilistic proofs may have multiple rounds of interaction between the prover and the verifier, in which case they are also referred to as interactive proofs. These two relaxations revolutionized the theory of proofs. For instance, by trading absolute certainty for high probability and allowing interaction, it is possible to prove claims without revealing anything beyond their correctness, i.e., in zero-knowledge. Nowadays, zero-knowledge proofs are widely deployed; they are for instance essential in the public-key infrastructures (PKIs) that manage digital identities and secure communication channels on the internet. Especially the theory of Σ-protocols provides a well-understood basis for the modular design of zero-knowledge proof systems in a wide variety of application domains. However, recently a new folding mechanism was introduced as a drop-in replacement for Σ-protocols, significantly reducing the communication costs in many practical scenarios. In this dissertation, we show that the folding mechanism can be cast as a significant strengthening, rather than a replacement, of Σ-protocol theory, thereby reconciling it with the established theory. In addition, we close several gaps in the theory of probabilistic proofs that were exposed due to the introduction of these efficiency improvements. Show less
Combinatorial games are games for two competing players, moving in a turn-by-turn fashion, in which there is no chance nor hidden information. Chess, checkers and the simpler tic tac toe are well... Show moreCombinatorial games are games for two competing players, moving in a turn-by-turn fashion, in which there is no chance nor hidden information. Chess, checkers and the simpler tic tac toe are well-known examples of this class of games, as well as game of go. Though these games are by no means simple, there does exist a beautiful mathematical framework for their analysis. Using this theory, it is possible to efficiently determine the outcome of a given position of a game without having to explicitly compute the results for every possible move. Moreover, the theory provides a measure of how profitable a given position is to either player, often denoted by the `value’ of a position. An example application of the theory is research on endgames in go.However, not all games are combinatorial games. The game of poker, for example, introduces hidden information. In practice, impressive results have been obtained for these non-combinatorial games using artificial intelligence, but theory and understanding are perhaps lacking. In this thesis, the main question we address is whether the existing theory for combinatorial games can be adapted or extended to non-combinatorial ones. Show less
Organisms often need to adapt more efficiently and devise new strategies for surviving difficult ecological circumstances. Mammals indeed spend the winter in hibernation to conserve energy, food,... Show moreOrganisms often need to adapt more efficiently and devise new strategies for surviving difficult ecological circumstances. Mammals indeed spend the winter in hibernation to conserve energy, food, etc., for future purposes. Microbial populations also possess similar characteristics, where organisms enter into a state of low metabolic activity in response to adverse environmental conditions. In plant populations, the analogous strategy is the suspension of seed germination for an extended period of time. Several studies suggest that this bet-hedging strategy has important evolutionary consequences and plays a crucial role in maintaining genetic diversities in a population. In this thesis, we draw motivations from biological populations featuring this trait and investigate its effect in a probabilistic framework. In particular, we introduce a mathematical notion of dormancy in several well-known stochastic interacting systems and study how it changes the qualitative and quantitative properties of the systems by characterizing their behaviors in the long run. The construction of our model is built upon a well-known stochastic process in mathematical population genetics called the Moran model. The Moran model describes the genetic evolution of a single, reproductively active, finite population without seed-bank. We modify the model to include dormancy and extend it to the context of spatially structured populations with varying sizes. Show less
Explaining treatment response variability between and within patients can support treatment and dosing optimization, to improve treatment of individual patients. This thesis discussed multiple... Show moreExplaining treatment response variability between and within patients can support treatment and dosing optimization, to improve treatment of individual patients. This thesis discussed multiple aspects of treatment variability and the associated statistical learning techniques which can be used to explain and/or predict part of that variability. Even though in recent times the availability of several high-throughput measurement technologies has created many new opportunities to develop improved treatment strategies, deriving actionable insights from such data remains a challenge. To this end, the use of longitudinal and high-dimensional data analysis techniques is needed to explore omics data for explaining treatment response and clinical course, and to answer clinical questions from routine healthcare data from hospitals and research institutes. Show less
The main theme of this thesis is the theoretical study of Gaussian processes as a tool in Bayesian nonparametric statistics. We are interested in the frequentist properties of Bayesian... Show moreThe main theme of this thesis is the theoretical study of Gaussian processes as a tool in Bayesian nonparametric statistics. We are interested in the frequentist properties of Bayesian nonparametric techniques in an asymptotic regime. We will be focusing specifically on consistency, convergence rates, uncertainty quantification and adaptation. These properties will be studied in the context of non-parametric problems, that is to say models with few modeling constraints. Moroever, the thesis will cover the topic of scalability of Bayesian techniques. Indeed, Bayesian methods are computation-hungry and rapidly become intractable as the number of observations grows. This issue led to the introduction of distributed Bayesian methods in order to decrease the computational complexity of the techniques. Show less
Inverse problems are problems where we want to estimate the values of certain parameters of a system given observations of the system. Such problems occur in several areas of science and... Show moreInverse problems are problems where we want to estimate the values of certain parameters of a system given observations of the system. Such problems occur in several areas of science and engineering. Inverse problems are often ill-posed, which means that the observations of the system do not uniquely define the parameters we seek to estimate, or that the solution is highly sensitive to small changes in the observation. In order to solve such problems, therefore, we need to make use of additional knowledge about the system at hand. One such prior information is given by the notion of sparsity. Sparsity refers to the knowledge that the solution to the inverse problem can be expressed as a combination of a few terms. The sparsity of a solution can be controlled explicitly or implicitly. An explicit way to induce sparsity is to minimize the number of non-zero terms in the solution. Implicit use of sparsity can be made, for e.g., by making adjustments to the algorithm used to arrive at the solution.In this thesis we studied various inverse problems that arise in different application areas, such as tomographic imaging and equation learning for biology, and showed how ideas of sparsity can be used in each case to design effective algorithms to solve such problems. Show less
Arithmetic geometry concerns the number-theoretic properties of geometric objects defined by polynomials. Mathematicians are interested in the rational solutions to these geometric objects. However... Show moreArithmetic geometry concerns the number-theoretic properties of geometric objects defined by polynomials. Mathematicians are interested in the rational solutions to these geometric objects. However, it is usually very difficult to answer questions like this.A. Beilinson and S. Bloch conjectured a very general height theory in 1980s, which was used by B. Gross and R. Schoen in their study of the Gross-Schoen cycles. The height of canonical Gross-Schoen cycles is conjectured to be non-negative. This was verified when the curve is an elliptic or hyperelliptic curve, while very few are known in the non-hyperelliptic case.During my PhD study, I study the Beilinson-Bloch height of canonical Gross-Schoen cycles on curves with an emphasis on the genus 3 case (almost all genus 3 curves are non-hyperelliptic). I studied its unboundedness and singular properties, and did explicit computation for the height of the canonical Gross-Schoen cycle of a specific plane quartic curve.The method used in my thesis should be helpful for verifications. Show less
This dissertation focuses on developing new mathematical and statistical methods to properly represent time-varying covariates and model them within the context of time-to-event analysis. This... Show moreThis dissertation focuses on developing new mathematical and statistical methods to properly represent time-varying covariates and model them within the context of time-to-event analysis. This research topic is motivated by specific clinical questions aimed at gaining insights into personalised treatments for cardiological and oncological patients. The main purpose is to enrich the knowledge available for modelling patients’ survival with relevant features related to the time-varying processes of interest.The efforts of this work address the complexities of both (i) developing adequate dynamic characterizations of the processes under study (i.e., representation problem) and (ii) identifying and quantifying the association between time-varying processes and patient survival (i.e., time-to-event modelling problem). In both cases, the main issue is dealing with complex data sources while taking into account the nature of the processes and managing the complex trade-off between clinical interpretability and mathematical formulation.By solving the aforementioned statistical complexities, this work is not only impacting the community of researchers in mathematics and statistics. The development of these novel methodologies may represent a significant step forward in the definition of customized and flexible monitoring tools to support doctors and clinicians in their work.*********This doctoral dissertation was part of a cotutelle agreement between the Politecnico di Milano and Leiden University Show less
In this thesis we study bistable reaction-diffusion equations on lattice domains. The power of reaction-diffusion equations is that they can successfully model various natural and social phenomena... Show moreIn this thesis we study bistable reaction-diffusion equations on lattice domains. The power of reaction-diffusion equations is that they can successfully model various natural and social phenomena with their intuitive and relatively simple (mathematical) representation. One of the main features of reaction-diffusion equations, both on discrete and continuous domains, is that they admit special solutions, so-called ‘travelling waves’, which we can describe as fixed profiles that move in a particular direction with some speed. Depending on their shape, we can roughly divide waves into three categories: pulses or solitons, periodic pulses (wave trains), and monotone wave fronts that connect two constant states. In this thesis we focus on the latter type of wave and we study their existence, propagation and long term behaviour on two type of discrete domains - infinite trees, and two-dimensional square lattices. Show less
This dissertation is a collection of four research articles devoted tothe study of Kummer theory for commutative algebraic groups. In numbertheory, Kummer theory refers to the study of field... Show moreThis dissertation is a collection of four research articles devoted tothe study of Kummer theory for commutative algebraic groups. In numbertheory, Kummer theory refers to the study of field extensions generatedby n-th roots of some base field. Its generalization to commutativealgebraic groups involves fields generated by the division points of afixed algebraic group, such as an elliptic curve or a higher dimensionalabelian variety. Of particular interest in this dissertation is the degreeof such field extensions. In the first two chapter, classical results forelliptic curves are improved by providing explicitly computable bounds anduniform and explicit bounds over the field of rational numbers. In thelast two chapters a general framework for the study of similar problemsis developed. Show less
In this thesis we study the moduli space of genus g curves, and the differential forms that occur naturally on this moduli space. We show that the rings of these tautological differential forms are... Show moreIn this thesis we study the moduli space of genus g curves, and the differential forms that occur naturally on this moduli space. We show that the rings of these tautological differential forms are finite-dimensional, and discuss algorithms that can be used to compute relations among tautological differential forms. Show less
Science is typically a patchwork of research contributions without much coordination. Especially in clinical trials, the follow-up studies that we do fail to be the most promising. They are also... Show moreScience is typically a patchwork of research contributions without much coordination. Especially in clinical trials, the follow-up studies that we do fail to be the most promising. They are also not always designed for the extra evidence that is needed. If they are, standard statistics makes it impossible to take such strategy into account.***This dissertation is about accumulating scientific knowledge, about the statistical problems with existing methods (accumulation bias) and about new statistical methods to do better. We can summarize scientific results efficiently by going ALL-IN. Science is a major gamble: there is little certainty when we embark on a new study. But gambling can be done strategically and clinical trials can use earlier results to decide whether a new study is necessary and optimally designed.***ALL-IN meta-analysis can help scientists to prioritize research, interpret findings in the context of existing results and gamble strategically with their next study. Hence reducing so-called Research Waste. But there is more to it. ALL-IN meta-analysis can be a bottom-up approach. Statistical results become much easier to communicate. Instead of progressing one publication at a time, with everyone focusing on their own paper, clinical science can be more of a continuous collaborative effort. Show less
A tidal basin is an inland sea which is almost entirely enclosed by land and connected to the open sea by a tidal inlet. Through the tidal inlets interaction takes place between the tidal basin and... Show moreA tidal basin is an inland sea which is almost entirely enclosed by land and connected to the open sea by a tidal inlet. Through the tidal inlets interaction takes place between the tidal basin and the open sea. Due to the tide in the open water, the water level of the tidal basin is driven by the water level on the sea. The water flows from the open sea through the tidal inlet to the tidal basin and back. Sand particles are being transported with the water which affects the bottom of the tidal basin and the tidal inlet. The position of the bottom itself also influences the movement of the sand particles. Due to this interaction, interesting bottom patterns with a fractal structure occur. It is these kind of patterns which we analyse in this dissertation. Therefore, we construct an idealised morphodynamic model. With this model, we can find an expression for four important variables: the water level, the water velocity, the sand transport and the height of the bottom. We investigate the existence of these patterns. Also, their sensitivity to variations of geometry and various physical parameters is analysed. We consider tidal inlet system with a various number of tidal inlets. Show less
Tomography is a powerful technique to non-destructively determine the interior structure of an object.Usually, a series of projection images (e.g.\ X-ray images) is acquired from a range of... Show moreTomography is a powerful technique to non-destructively determine the interior structure of an object.Usually, a series of projection images (e.g.\ X-ray images) is acquired from a range of different positions.from these projection images, a reconstruction of the object's interior is computed. Many advanced applications require fast acquisition, effectively limiting the number of projection images and imposing a level of noise on these images. These limitations result in artifacts (deficiencies) in the reconstructed images. Recently, deep neural networks have emerged as a powerful technique to remove these limited-data artifacts from reconstructed images, often outperformingconventional state-of-the-art techniques. To perform this task, the networks are typically trained on a dataset of paired low-quality and high-quality images of similar objects. This is a major obstacle to their use in many practical applications. In this thesis, we explore techniques to employ deep learning in advanced experiments where measuring additional objects is not possible. Show less
Central Limit Theorem associated to a particular case of ergodic total automorphisms. Finally, Chapter 6 proves that the isoperiodic foliation is connected for a meromorphic differential with three... Show moreCentral Limit Theorem associated to a particular case of ergodic total automorphisms. Finally, Chapter 6 proves that the isoperiodic foliation is connected for a meromorphic differential with three poles on a torus. Show less
One of the effects of climate change is the phenomenon of desertification, a process that occurs in semi-arid and arid areas and causes land degradation as well as vegetation loss. Due to the lack... Show moreOne of the effects of climate change is the phenomenon of desertification, a process that occurs in semi-arid and arid areas and causes land degradation as well as vegetation loss. Due to the lack of resources, vegetation self-organizes to sustain itself by forming large-scale spatial patterns. In this thesis, the underlying mathematical structure of these observed vegetation patterns is studied using partial differential equations models. The vegetation patterns are analyzed using techniques from geometrical singular perturbation theory and numerical simulations. Additionally, novel multi-front patterns are constructed that arise within one of the models studied. This interdisciplinary research allows for cross-fertilization of both mathematics and ecology. Show less
The thesis introduces three methods for high-dimensional prediction problems in the biomedical field. The methods make use of empirical and variational Bayes in the estimation. Several applications... Show moreThe thesis introduces three methods for high-dimensional prediction problems in the biomedical field. The methods make use of empirical and variational Bayes in the estimation. Several applications show that the proposed methods are competitive to existing methods. Show less
