Doctoral Theses / Doktorsavhandlingar Institutionen för matematiska vetenskaper

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Hyperuniformity and Hyperfluctuations for Random Measures on Euclidean and Non-Euclidean spaces
(2025-05-05) Byléhn, Mattias
In this thesis we study fluctuations of generic random point configurations in Euclidean and symmetric curved geometries. Mathematically, such configurations are interpreted as isometrically invariant point processes, and fluctuations are recorded by the variance of the number of points in a centered ball, the number variance. Hyperuniformity and hyperfluctuation of such configurations in the sense of Stillinger-Torquato is characterized in terms of large-scale asymptotics of the number variance in relation to that of an ideal gas, and equivalently by small-scale asymptotics of the Bartlett spectral measure in the diffraction picture. Appended to the Thesis are three papers: In Paper I we provide lower asymptotic bounds for number variances of isometrically invariant random measures in Euclidean and hyperbolic spaces, generalizing a result by Beck. In particular, we find that geometric hyperuniformity fails for every isometrically invariant random measure on hyperbolic space. In contrast to this, we define a notion of spectral hyperuniformity which is satisfied by certain invariant random lattice configura- tions. In Paper II we establish similar lower asymptotic bounds for number variances of automorphism invariant point processes in regular trees. The main result is that these lower bounds are not uniform for the invariant random lattice configurations defined by the fundamental groups of complete regular graphs and the Petersen graph. We also provide a criterion for when these lower bounds are uniform in terms of certain rational peaks appearing in the diffraction picture. In Paper III we prove the existence and uniqueness of Bartlett spectral measures for invariant random measures on a large class of non-compact commutative spaces, which includes those in Papers I and II. For higher rank symmetric spaces governed by simple Lie groups, we prove that there is a power strictly less than 2 of the volume of balls that asymptotically bounds the number variance of any invariant random measure from above. Moreover, we derive Bartlett spectral measures for invariant determinantal point processes on commutative spaces and define a notion of heat kernel hyperuniformity on Euclidean and hyperbolic spaces that is equivalent to spectral hyperuniformity.
Enabling mechanistic understanding of cellular dynamics through mathematical modelling and development of efficient methods
(2024-11-13) Persson, Sebastian
Cell biology is complex, but unravelling this complexity is important. For example, the recent COVID-19 pandemic highlighted the need to understand how cells function in order to develop efficient vaccines and treatments. However, studying cellular systems is challenging because they are often highly interconnected, dynamic and contain many redundant components. Mathematical modelling provides a powerful framework to reason about such complexity. In the four papers underlying this thesis, our aim was twofold.The first was to unravel mechanisms that regulate cellular dynamic behaviour in the model organism Saccharomyces cerevisiae. In particular, by developing single-cell dynamic models, we investigated how cells respond to changes in nutrient levels. We identified mechanisms behind the reaction dynamics and uncovered sources of cell-to-cell variability. Additionally, by developing reaction-diffusion modelling, we studied the size regulation of self-assembled structures and demonstrated how the interplay of feedback mechanisms can regulate structure size. Our second aim was to develop methods and software to facilitate efficient modelling. Modelling often involves fitting models to data to verify specific hypotheses, and it is beneficial if models inconsistent with the data can be discarded rapidly. To this end, we developed software for working with single-cell dynamic models that, in contrast to previous methods, imposes fewer restrictions on how cell-to-cell variability is modelled. Moreover, we developed and evaluated software for fitting population-average dynamic models to data. This software outperforms the current state of the art, and to make it accessible, we released it as two well-documented open-source packages. Taken together, this thesis sheds light on fundamental regulatory mechanisms and introduces software for efficient modelling.
Yeast at the Crossroads: Nutrient Signalling Paths and Stressful Turns
(2024-11-11) Braam, Svenja
Nutrients serve essential functions as building blocks for cellular components. Intriguingly, they serve as signalling elements that control basic cellular functions. In microorganisms such as Saccharomyces cerevisiae, the availability of nutrients also comes along with further challenges such as fluctuations in osmolytes, oxygen levels, or pH. Adaptation to these factors requires a coordinated response from various cellular pathways to ensure survival. This thesis explores the intersection of nutrient signalling and osmotic stress responses in S. cerevisiae. Emphasis is placed on (I) the Snf1/Mig1 network, which regulates the response to carbon source availability, (II) the High Osmolarity Glycerol (HOG) pathway, governing the osmotic stress response (III) the transcriptional regulators Nrg1 and Nrg2, whose roles in coordinating nutrientand osmotic stress responses are not fully understood. Of particular interest in this study is how cells adapt to lithium ions. Lithium salts have also gained attention in aging research, yet despite their long history in mood disorder treatments, their exact effects on cellular signaling remain unclear. We found that low concentrations of glucose mitigate survival of yeast cells exposed to lithium chloride. While Nrg1/2 play distinct roles in the response to lithium chloride exposure, deleting NRG1 markedly increases growth rate in lithium chloride and glucose. Deletion of both genes confers phenotypic enhancement, resulting in a distinctive growth pattern with an initial surge and subsequent drop in growth. Separately, we found fluctuations in shuttling kinetics of Snf1 are influenced by the presence of non-fermentable carbon sources. Additionally, we employed a genome-wide genetic screen to link mitochondrial gene expression with nuclear genome regulation, offering new insights into the crosstalk between cellular subsystems. These findings contribute to our understanding of the complex crosstalk between nutrient signalling and osmotic stress responses. By shedding light on the regulatory processes involved in cellular adaptation, this research adds to our knowledge of how cells respond to environmental stressors. The implications of these mechanisms extend to broader biomedical contexts, including aging and age-related diseases such as metabolic disorders and cancer, where similar signalling pathways play a critical role.
Hodge Theory in Combinatorics and Mirror Symmetry
(2024-10-22) Pochekai, Mykola
Hodge theory, in its broadest sense, encompasses the study of the decomposition of cohomology groups of complex manifolds, as well as related fields such as periods, motives, and algebraic cycles. In this thesis, ideas from Hodge theory have been incorporated into two seemingly unrelated projects, namely mathematical mirror symmetry and combinatorics. Papers I-II explore an instance of genus one mirror symmetry for the complete intersection of two cubics in five-dimensional projective space. The mirror family for this complete intersection is constructed, and it is demonstrated that the BCOV-invariant of the mirror family is related to the genus one Gromov-Witten invariants of the complete intersection of two cubic. This proves new cases of genus one mirror symmetry. Paper III defines Hodge-theoretic structures on triangulations of a special type. It is shown that if a polytope admits a regular, unimodular triangulation with a particular additional property, its $\delta$-vector from Ehrhart theory is unimodal.
Extreme rainfall modelling under climate change and proper scoring rules for extremes and inference
(2024-09-06) Ólafsdóttir, Helga Kristín
Model development, model inference and model evaluation are three important cornerstones of statistical analysis. This thesis touches on all these through modelling extremes under climate change and evaluating extreme models using scoring rules, and by using scoring rules for statistical inference on spatial models. The findings are presented in three papers. In Paper I, a new statistical model is developed, that uses the connections between the generalised extreme value distribution and the generalised Pareto distribution to capture frequency changes in annual maxima. This allows using high-quality annual maxima data instead of less-well checked daily data to separately estimate trends in frequency and intensity. The model was applied to annual maximum data of Volume 10 of NOAA Atlas 14, showing that in the Northeastern US there are evidence that extreme rainfall events are occurring more often with rising temperature, but that there is little evidence that there are trends in the distribution of sizes of individual extreme rainfall events. Paper II introduces the concept of local weight-scale invariance which is a relaxation of local scale invariance for proper scoring rules. This relaxation is suitable for weighted scores that are for example useful when comparing extreme models. A weight-scale invariant version of the tail-weighted continuous ranked probability score is introduced and the properties of the different weighted scores were investigated. Finally, Paper III continues on the path of scoring rules, but instead uses scoring rules for statistical inference of spatial models. The proposed approach estimates parameters of spatial models by maximising the average leave-one-out cross-validation score (LOOS). The method results in fast computations for Gaussian models with sparse precision matrices and allows tailoring estimator's robustness to outliers and their sensitivity to spatial variations of uncertainty through the choice of the scoring rule which is used in the maximisation.
Unsupervised methods for integrative data analysis
(2024-05-17) Held, Felix
Unsupervised data analysis methods are important for data exploration to introduce structure, reduce data dimensions, or extract interpretable knowledge. Integrative analysis of two or more data sets is crucial to gain understanding of local and global effects within and across data sources. Recent technological advancements in large scale collection of single cell data require efficient and scalable methods to process the increasing size of available data. Integration of data sources with secondary data, also known as side information, can improve prediction of missing data and is important for recommender systems. However, many currently existing methods cannot accommodate the scale or complexity of available data. Therefore, there is a need for new methods for unsupervised integrative data analysis that scale well with input data size, can be applied efficiently, and provide flexible support for complex data input. In Paper I, a novel scalable method is proposed which integrates gene clustering of single cell data with selection of cluster-specific gene regulators having sign-consistent correlation and therefore well-defined effect within each cluster. An efficient alternating two-step algorithm for parameter estimation is developed, along with criteria for optimal hyperparameter and cluster count selection. Applications to single cell data demonstrate the methods capability to identify regulators of intratumoral heterogeneity, primarily in neural cancers. In Paper II, a low-rank matrix factorization model is proposed which allows flexible integration of input data sources and produces interpretable estimates of orthogonal latent factors. Parameter estimation is performed efficiently within an ADMM framework and its convergence theory is extended to support embedded manifold constraints such as orthogonality. Simulation studies show that the method performs well in comparison to established methods and the importance of support for flexible data input layouts is demonstrated. The lack of scalable flexible matrix integration methods is addressed in Paper III by reformulating the data integration problem as a graph estimation problem. A novel algorithm is proposed, using matrix denoising and the asymptotic geometry of singular vectors in noise-perturbed low-rank matrices, to perform estimation within the graphical framework. Simulation studies demonstrate the method's high scalability in comparison to established methods. Software packages with easy-to-use interfaces for each paper are publicly available. The methods presented in this thesis contribute to the development of efficient, flexible, and scalable unsupervised methods for integrative data analysis.
New AI-based methods for studying antibiotic-resistant bacteria
(2023-11-03) Inda Díaz, Juan Salvador
Antibiotic resistance is a growing challenge for human health, causing millions of deaths worldwide annually. Antibiotic resistance genes (ARGs), acquired through mutations in existing genes or horizontal gene transfer, are the primary cause of bacterial resistance. In clinical settings, the increased prevalence of multidrug-resistant bacteria has severely compromised the effectiveness of antibiotic treatments. The rapid development of artificial intelligence (AI) has facilitated the analysis and interpretation of complex data and provided new possibilities to face this problem. This is demonstrated in this thesis, where new AI methods for the surveillance and diagnostics of antibiotic-resistant bacteria are presented in the form of three scientific papers. Paper I presents a comprehensive characterization of the resistome in various microbial communities, covering both well-studied established ARGs and latent ARGs not currently found in existing repositories. A widespread presence of latent ARGs was observed in all examined environments, signifying a potential reservoir for recruitment to pathogens. Moreover, some latent ARGs exhibited high mobile potential and were located in human pathogens. Hence, they could constitute emerging threats to human health. Paper II introduces a new AI-based method for identifying novel ARGs from metagenomic data. This method demonstrated high performance in identifying short fragments associated with 20 distinct ARG classes with an average accuracy of 96. The method, based on transformers, significantly surpassed established alignment-based techniques. Paper III presents a novel AI-based method to predict complete antibiotic susceptibility profiles using patient data and incomplete diagnostic information. The method incorporates conformal prediction and accurately predicts, while controlling the error rates, susceptibility profiles for the 16 included antibiotics even when diagnostic information was limited. The results presented in this thesis conclude that recent AI methodologies have the potential to improve the diagnostics and surveillance of antibiotic-resistant bacteria.
Efficient training of interpretable, non-linear regression models
(2023-06-30) Allerbo, Oskar
Regression, the process of estimating functions from data, comes in many flavors. One of the most commonly used regression models is linear regression, which is computationally efficient and easy to interpret, but lacks in flexibility. Non-linear regression methods, such as kernel regression and artificial neural networks, tend to be much more flexible, but also harder to interpret and more difficult, and computationally heavy, to train. In the five papers of this thesis, different techniques for constructing regression models that combine flexibility with interpretability and computational efficiency, are investigated. In Papers I and II, sparsely regularized neural networks are used to obtain flexible, yet interpretable, models for additive modeling (Paper I) and dimensionality reduction (Paper II). Sparse regression, in the form of the elastic net, is also covered in Paper III, where the focus is on increased computational efficiency by replacing explicit regularization with iterative optimization and early stopping. In Paper IV, inspired by Jacobian regularization, we propose a computationally efficient method for bandwidth selection for kernel regression with the Gaussian kernel. Kernel regression is also the topic of Paper V, where we revisit efficient regularization through early stopping, by solving kernel regression iteratively. Using an iterative algorithm for kernel regression also enables changing the kernel during training, which we use to obtain a more flexible method, resembling the behavior of neural networks. In all five papers, the results are obtained by carefully selecting either the regularization strength or the bandwidth. Thus, in summary, this work contributes with new statistical methods for combining flexibility with interpretability and computational efficiency based on intelligent hyperparameter selection.
Limit Theorems for Lattices and L-functions
Holm, Kristian
This PhD thesis investigates distributional questions related to three types of objects: Unimodular lattices, symplectic lattices, and Hecke L-functions of imaginary quadratic number fields of class number 1. In Paper I, we follow Södergren and examine the asymptotic joint distribution of a collection of random variables arising as geometric attributes of the N = N(n) shortest non-zero lattice vectors (up to sign) in a random unimodular lattice in n-dimensional Euclidean space, as the dimension n tends to infinity: Normalizations of the lengths of these vectors, and normalizations of the angles between them. We prove that under suitable conditions on N, this collection of random variables is asymptotically distributed like the first N arrival times of a Poisson process of intensity 1/2 and a collection of positive standard Gaußians. This generalizes previous work of Södergren. In Paper II, we use methods developed by Björklund and Gorodnik to study the error term in a classical lattice point counting asymptotic due to Schmidt in the context of symplectic lattices and a concrete increasing family of sets in 2n-dimensional Euclidean space. In particular, we show that this error term satisfies a central limit theorem as the volumes of the sets tend to infinity. Moreover, we obtain new Lp bounds on a height function on the space of symplectic lattices originally introduced by Schmidt. In Paper III, we follow Waxman and study a family of L-functions associated to angular Hecke characters on imaginary quadratic number fields of class number 1. We obtain asymptotic expressions for the 1-level density of the low-lying zeros in the family, both unconditionally and conditionally (under the assumption of the Grand Riemann Hypothesis and the Ratios Conjecture). Our results verify the Katz--Sarnak Density Conjecture in a special case for our family of L-functions.
Sketches of Noncommutative Topology
(2022-11-07) Kuzmin, Alexey
This thesis thematically divided into two parts. In the first part we are mastering C*-isomorphism problem by using various techniques applied to different examples of noncommutative algebraic varieties. In the second part we apply noncommutative homotopy theory to C*-algebraic objects related to manifold theory, in such a way deriving results and formulas for such an object as differential operators. In the first article we consider C*-algebra Isom_{q_{ij}} generated by n isometries a_1, \ldots, a_n satisfying the relations a_i^* a_j = q_{ij} a_j a_i^* with \max |q_{ij}| < 1. This C*-algebra is shown to be nuclear. We prove that the Fock representation of Isom_{q_{ij}} is faithful. Further we describe an ideal in Isom_{q_{ij}} which is isomorphic to the algebra of compact operators. In the second article we consider the C*-algebra \mathcal{E}^q_{n,m}, which is a q-twist of two Cuntz-Toeplitz algebras. For the case |q| < 1, we give an explicit formula which untwists the q-deformation showing that the isomorphism class of \mathcal{E}^q_{n,m} does not depend on q. For the case |q| = 1, we give an explicit description of all ideals in \mathcal{E}^q_{n,m}. In particular, we show that \mathcal{E}^q_{n,m} contains a unique largest ideal \mathcal{M}_q. We identify \mathcal{E}^q_{n,m}/\mathcal{M}_q with the Rieffel deformation of \mathcal{O}_n \otimes \mathcal{O}_m and use a K-theoretical argument to show that the isomorphism class does not depend on q. The latter result holds true in a more general setting of multiparameter deformations. In the third article we consider the universal enveloping C*-algebra \mathsf{CAR}_\Theta of the *-algebra generated by a_1, \ldots, a_n subject to the relations a_i^* a_i + a_i a_i^* = 1, a_i^* a_j =e^{2\pi i \Theta_{ij}}a_j a_i^*, a_i a_j = e^{-2\pi i \Theta_{ij}} a_j a_i for a skew-symmetric real n x n matrix \Theta. We prove that \mathsf{CAR}_\Theta has a C(K_n)-structure, where K_n = [0, \frac{1}{2}]^n is the hypercube and describe the fibers. We classify irreducible representations of \mathsf{CAR}_\Theta in terms of irreducible representations of a higher-dimensional noncommutative torus. We prove that for a given irrational skew-symmetric \Theta_1 there are only finitely many \Theta_2 such that \mathsf{CAR}_{\Theta_1} \simeq \mathsf{CAR}_{\Theta_2}. Namely, \mathsf{CAR}_{\Theta_1} \simeq \mathsf{CAR}_{\Theta_2} implies (\Theta_1)_{ij} = \pm (\Theta_2)_{\sigma(i,j)} for a bijection \sigma of the set \{(i,j):i
Global residue currents and the Ext functors
(2022-09-09) Johansson, Jimmy
This thesis concerns developments in multivariable residue theory. In particular we consider global constructions of residue currents related to work by Andersson and Wulcan. In the first paper of this thesis, we consider global residue currents defined on projective space, and we show that these currents provide a tool for studying polynomial interpolation. Polynomial interpolation is related to local cohomology, and by a result known as local duality, there is a close connection with certain Ext groups. The second paper of this thesis is devoted to further study of connections between residue currents and the Ext functors. The main result is that we construct a global residue current on a complex manifold, and using this we give an explicit formula for an isomorphism of two different representations of the global Ext groups on complex manifolds.
Mathematical Modelling of Cellular Ageing: a Multi-Scale Perspective
(2022-04-19) Schnitzer, Barbara
In a growing and increasingly older population, we are progressively challenged by the impact of ageing on individuals and society. The UN declared the years 2021-2030 as the Decade of Healthy Ageing, highlighting the efforts to minimise the burden of ageing and age-related diseases. A crucial step towards this goal is to elucidate basic underlying mechanisms on a molecular and cellular level. While much is known about individual hallmarks of cellular ageing, their interactive and multi-scale nature hinders the progress in gaining deeper insights into the emergent effects on an organism. In the five papers underlying this thesis, we aimed to study protein damage accumulation over successive cell divisions (replicative ageing), as one emergent factor defining ageing. We combined experimental data in the unicellular model organism yeast Saccharomyces cerevisiae with mathematical modelling, which offers systematic and formal ways of analysing the complexity that arises from the interplay between processes on different time and length scales. In that way, we showed how interconnections in the cellular signalling network are essential to ensure a robust adaption to stress on a short time scale, being crucial for preventing and handling protein damage. By linking different models for cellular signalling, metabolism and protein damage accumulation, we provided one of the most comprehensive mathematical models of replicative ageing to date. The model allowed us to map metabolic changes during ageing to a dynamic trade-off between protein availability and energy demand, and to investigate global metabolic strategies underlying cellular ageing. Going beyond single-cell models, we examined the synergy between processes that create, retain and repair protein damage, balancing the health of individual cells with the viability of the cell population. Taken together, by constructing, validating and using mathematical models, we unified different scales of protein damage accumulation and explored its causes and consequences. Thus, this thesis contributes to a more comprehensive understanding of cellular ageing, taking a step further towards healthy ageing.
On the mathematics of the one-dimensional Hegselmann-Krause model
(2022-01-19) Wedin, Edvin
This thesis contains three papers, which all in different ways concern the asymptotics of the Hegselmann-Krause model in opinion dynamics. In this model a set of conformist agents repeatedly and synchronously replace their real-valued opinions by the average of those opinions that are within unit distance of their own. The resulting dynamics become exceedingly subtle, and are sensitive to the precise initial configuration. In Paper I, my supervisor and I investigate the case when the system is initialised with a chain of equidistant opinions. This includes determining the precise evolution for every (possibly infinite) chain with inter-agent distance 1 or approximately 0.81. In Paper II, I expand the current theory for the evolution of systems where a very large number of opinions are independently drawn at random from the uniform distribution on an interval. I develop a method for approximating the updates of configurations with arbitrarily many agents by updates of smaller ones which can be explicitly computed. I use this method to show rigorously for the first time that for some interval lengths, one asymptotically almost surely reaches a consensus. This gives theoretical support for an observation by Lorenz, that sometimes a group that would not otherwise reach a consensus could do so by individually becoming more closed-minded. In Paper III, we show that if opinions are taken to be points on a circle with perimeter larger than 2 instead of being real numbers, the resulting sequence of updates must converge pointwise from any possible initial configuration.
Combinatorics of solvable lattice models with a reflecting end
(2021-04-21) Hietala, Linnea
I den här avhandlingen studerar vi några exakt lösbara, kvantintegrerbara gittermodeller. Izergin bevisade en determinantformel för partitionsfunktionen till sexvertexmodellen på ett gitter av storlek n × n med Korepins domänväggrandvillkor (domain wall boundary conditions – DWBC). Metoden har blivit ett användbart verktyg för att studera partitionsfunktionen för liknande modeller. Determinantformeln har också visat sig vara användbar för att lösa andra typer av problem. Genom att specialisera parametrarna i Izergins determinantformel kunde Kuperberg hitta en formel för antalet alternerande tecken-matriser (alternating sign matrices – ASMs). Bazhanov och Mangazeev introducerade speciella polynom, bland annat pn och qn, som kan användas för att uttrycka speciella komponenter av egenvektorerna för grundtillstånden till den supersymmetriska XYZ-spinnkedjan av udda längd. I artikel I hittar vi explicita kombinatoriska uttryck för polynomen qn i termer av trefärgsmodellen med DWBC och en (diagonal) reflekterande rand. Sambandet uppstår genom att specialisera parametrarna i partitionsfunktionen för den elliptiska sexvertexmodellen med DWBC och en (diagonal) reflekterande rand på Kuperbergs sätt. Som en följd av detta kan vi hitta resultat för trefärgsmodellen, exempelvis antalet tillstånd med ett givet antal rutor av varje färg. I artikel II studerar vi polynomen pn på ett liknande sätt. Kopplingen till den elliptiska sexvertexmodellen fås genom att specialisera alla parametrar utom en på Kuperbergs sätt. Genom att använda Izergin–Korepin-metoden i Artikel III hittar vi en determinantformel för partitionsfunktionen för den trigonometriska sexvertexmodellen med DWBC och en partiellt (triangulär) reflekterande rand på ett gitter av storlek 2n×m, där m ≤ n. Sedan använder vi Kuperbergs specialisering av parametrarna för att hitta ett explicit uttryck för antalet tillstånd av modellen som en determinant av Wilson-polynom. Vi kopplar också detta till en sorts ASM-liknande matriser. // In this thesis, we study some exactly solvable, quantum integrable lattice models. Izergin proved a determinant formula for the partition function of the six-vertex (6V) model on an n×n lattice with the domain wall boundary conditions (DWBC) of Korepin. The method has become a useful tool to study the partition functions of similar models. The determinant formula has also proved useful for seemingly unrelated questions. In particular, by specializing the parameters in Izergin’s determinant formula, Kuperberg was able to give a formula for the number of alternating sign matrices (ASMs). Bazhanov and Mangazeev introduced special polynomials, including pn and qn, that can be used to express certain ground state eigenvector components for the supersymmetric XYZ spin chain of odd length. In Paper I, we find explicit combinatorial expressions for the polynomials qn in terms of the three-color model with DWBC and a (diagonal) reflecting end. The connection emerges by specializing the parameters in the partition function of the eight-vertex solid-on-solid (8VSOS) model with DWBC and a (diagonal) reflecting end in Kuperberg’s way. As a consequence, we find results for the three-color model, including the number of states with a given number of faces of each color. In Paper II, we perform a similar study of the polynomials pn. To get the connection to the 8VSOS model, we specialize all parameters except one in Kuperberg’s way. By using the Izergin–Korepin method in Paper III, we find a determinant formula for the partition function of the trigonometric 6V model with DWBC and a partially (triangular) reflecting end on a 2n × m lattice, m ≤ n. Thereafter we use Kuperberg’s specialization of the parameters to find an explicit expression for the number of states of the model as a determinant of Wilson polynomials. We relate this to a type of ASM-like matrices.
Differential forms and currents on non-reduced complex spaces with applications to divergent integrals and the dbar-equation
(2020-12-16) Lennartsson, Mattias
Network modeling and integrative analysis of high-dimensional genomic data
(2020-05-07) Kallus, Jonatan
Genomic data describe biological systems on the molecular level and are, due to the immense diversity of life, high-dimensional. Network modeling and integrative analysis are powerful methods to interpret genomic data. However, network modeling is limited by the requirement to select model complexity and due to a bias towards biologically unrealistic network structures. Furthermore, there is a need to be able to integratively analyze data sets describing a wider range of different biological aspects, studies and groups of subjects. This thesis aims to address these challenges by using resampling to control the false discovery rate (FDR) of edges, by combining resampling-based network modeling with a biologically realistic assumption on the structure and by increasing the richness of data sets that can be accommodated in integrative analysis, while facilitating the interpretation of results. In paper I, a statistical model for the number of times each edge is included in network estimates across resamples is proposed, to allow for estimation of how the FDR is affected by sparsity. Accuracy is improved compared to state-of-the-art methods, and in a network estimated for cancer data all hub genes have documented cancer-related functions. In paper II, a new method for integrative analysis is proposed. The method, based on matrix factorization, introduces a versatile objective function that allows for the study of more complex data sets and easier interpretation of results. The power of the method as an explorative tool is demonstrated on a set of genomic data. In paper III, network estimation across resamples is combined with repeated community detection to compensate for the structural bias inherent in common network estimation methods. For estimation of the regulatory network in human cancer, this compensation leads to an increased overlap with a database of gene interactions. Software implementations of the presented methods have been published. The contributed methods further the understanding that can be gained from high-dimensional genomic data, and may thus help to devise new treatments and diagnostics for cancer and other diseases.
The construction, analysis and validation of mechanistic mathematical models of protein kinetics in the context of replicative ageing in budding yeast
(2020-05-04) Johannes, Borgqvist
Mathematical modelling constitutes a forceful tool for elucidating properties of biological systems. Using theoretical approaches in combination with experimental techniques it is possible to study specific molecular aspects of phenomena such as the ageing of human beings. In fact, as many processes are similar in simpler organisms such as the budding yeast \textit{Saccharomyces cerevisiae} it is possible to experimentally investigate for instance the accumulation of damaged proteins due to ageing in these biological systems. The aim of this thesis is to construct, analyse and validate mathematical mechanistic models of protein kinetics consisting of both ordinary and partial differential equations in the context of ageing. This is done both on a large time scale corresponding to the entire life span of cells and a short time scale corresponding to an isolated part of the cell division. The focus of the work on the large time scale is twofold, firstly the life span of individual yeast cells is modelled (Paper II) and secondly the life spans of vast numbers of cells in numerous populations are simulated (Paper III). Using a model of the accumulation of damage involving the forces cell growth, formation and repair of damage as well as the cell division, the impact of these individual parts on the overall fitness of individual cells and entire populations is investigated. On the short time scale, a more detailed model of a single protein called Cdc42 involved in the cell division is presented (Paper IV) and this theoretical framework has a high level of detail as it describes the spatial movement of the protein of interest within the cell over time. Given this precise description of the geometry of an individual cell, the mathematical properties of the model is analysed and these theoretical results are used to conduct numerical simulations of the activity of this protein. Lastly, an overall theme of the thesis is the difficulty of validating mechanistic models even in the presence of data. More precisely, as numerous and sometimes mutually exclusive models can describe a system equally well it is currently very hard, even by calibrating the models to experimental data using statistical methods, to differentiate between various models. To this end, a mathematical tool called symmetry methods is introduced as a potential remedy to this problem, and using this methodology it is possible to extract information in the data as well as in the model that is not available using standard approaches. To showcase the power of symmetries, a minimal example of the usage of these methods in the context of enzyme kinetics is presented (Paper V). In conclusion, this work suggests that novel analytical tools such as symmetry methods could complement and assist the current standard approaches for modelling protein kinetics where the purpose is to deduce the underlying mechanisms of biological systems.
Geometrical and percolative properties of spatially correlated models
(2020-03-10) Hallqvist Elias, Karl Olof
This thesis consists of four papers dealing with phase transitions in various models of continuum percolation. These models exhibit complicated dependencies and are generated by different Poisson processes. For each such process there is a parameter, known as the intensity, governing its behavior. By varying the value of this parameter, the geometrical and topological properties of these models may undergo dramatic and rapid changes. This phenomenon is called a phase transition and the value at which the change occur is called a critical value. In Paper I, we study the topic of visibility in the vacant set of the Brownian interlacements in Euclidean space and the Brownian excursions process in the unit disc. For the vacant set of the Brownian interlacements we obtain upper and lower bounds of the probability of having visibility in some direction to a distance r in terms of the probability of having visibility in a fixed direction of distance r. For the vacant set of the Brownian excursions we prove a phase transition in terms of visibility to infinity (with respect to the hyperbolic metric). We also determine the critical value and show that at the critical value there is no visibility to infinity. In Paper II we compute the critical value for percolation in the vacant set of the Brownian excursions process. We also show that the Brownian excursions process is a hyperbolic analogue of the Brownian interlacements. In Paper III, we study the vacant set of a semi scale invariant version of the Poisson cylinder model. In this model it turns out that the vacant set is a fractal. We determine the critical value for the so-called existence phase transition and what happens at the critical value. We also compute the Hausdorff dimension of the fractal whenever it exists. Furthermore, we prove that the fractal exhibits a nontrivial connectivity phase transition for dimensions four and greater and that the fractal is totally disconnected for dimension two. In the three dimensional case we prove a partial result showing that the fractal restricted to a plane is totally disconnected with probability one. In Paper IV we study a continuum percolation process, the random ellipsoid model, generated by taking the intersection of a Poisson cylinder model in d dimensions and a subspace of dimension k. For k between 2 and d-2, we show that there is a non-trivial phase transition concerning the expected number of ellipsoids in the cluster of the origin. When k=d-1 this critical value is zero. We compare these results with results for the classical Poisson Boolean model.
Learning to solve problems that you have not learned to solve: Strategies in mathematical problem solving
(2019-08-16) Fülöp, Éva
This thesis aims to contribute to a deeper understanding of the relationship between problem-solving strategies and success in mathematical problem solving. In its introductory part, it pursues and describes the term strategy in mathematics and discusses its relationship to the method and algorithm concepts. Through these concepts, we identify three decision-making levels in the problem-solving process. The first two parts of this thesis are two different studies analysing how students’ problem-solving ability is affected by learning of problem-solving strategies in mathematics. We investigated the effects of variation theory-based instructional design in teaching problem-solving strategies within a regular classroom. This was done by analysing a pre- and a post-test to compare the development of an experimental group’s and a control group’s knowledge of mathematics in general and problem-solving ability in particular. The analysis of the test results show that these designed activities improve students’ problem-solving ability without compromising their progress in mathematics in general. The third study in this thesis aims to give a better understanding of the role and use of strategies in the mathematical problem-solving processes. By analysing 79 upper secondary school students’ written solutions, we were able to identify decisions made at all three levels and how knowledge in these levels affected students’ problem-solving successes. The results show that students who could view the problem as a whole while keeping the sub-problems in mind simultaneously had the best chances of succeeding. In summary, we have in the appended papers shown that teaching problem-solving strategies could be integrated in the mathematics teaching practice to improve students mathematical problem-solving abilities.
Spectral properties of elliptic operators in singular settings and applications
(2019-05-22) Nursultanov, Medet
The present thesis is focused on the investigation of the spectral properties of the linear elliptic operators in the presence of singularities. It is divided into three chapters. In the first chapter, we consider geometric singularities. We construct the heat kernel on surfaces with corners for Dirichlet, Neumann, and Robin boundary conditions as well as mixed problems. We compute the short time asymptotic expansion of the heat trace and apply this expansion to demonstrate a collection of results showing that corners are spectral invariants. The second chapter deals with linear elliptic second-order partial differential operators with bounded real-valued measurable coefficients. We emphasize that no smoothness assumptions are made on the coefficients. In the first half of this chapter, we study a time-harmonic electromagnetic and acoustic waveguide, modeled by an infinite cylinder with a non-smooth cross section. We introduce an infinitesimal generator for the wave evolution along the cylinder and prove estimates of the functional calculi of these first order non-self adjoint differential operators with non-smooth coefficients. Applying our new functional calculus, we obtain a one-to-one correspondence between polynomially bounded time-harmonic waves and functions in appropriate spectral subspaces. In the second half, we derive Weyl's law for the weighted Laplace equation on Riemannian manifolds with rough metric. Key ingredients in the proofs were demonstrated by Birman and Solomyak nearly fifty years ago in their seminal work on eigenvalue asymptotics. In the last chapter, we investigate spectral properties of Sturm-Liouville operators with singular potentials. We consider different types of singularities. We find asymptotic formulas for the eigenvalues of the Sturm-Liouville operator on the finite interval, with potentials having a strong negative singularity at one endpoint. We establish that, unlike the case of an infinite interval, the asymptotics for positive eigenvalues does not depend on the potential, and it is the same as in the regular case. The asymptotics of the negative eigenvalues may depend on the potential quite strongly. Next, we study the perturbation of the generalized anharmonic oscillator. We consider a piecewise Hölder continuous perturbation and investigate how the Hölder constant can affect the eigenvalues. Finally, for the the Sturm-Liouville operator with $\delta$-interactions, two-sided estimates of the distribution function of the eigenvalues and a criterion for the discreteness of the spectrum in terms of the Otelbaev function are obtained.

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