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Innate immune activation is thought to play a central role in IBD pathogenesis because genetic polymorphisms in NOD2 and NLRP3, cytosolic innate immune receptors belonging to the NLR family, have been associated with IBD susceptibility. However, the mechanisms through which NLR mutations predispose to IBD remain unclear. The aim of this project was to dissect the functional roles of different NLRs in intestinal inflammation. Using the well-established DSS-induced colitis model as well as experimental models of IBD based on infection with Helicobacter hepaticus, we found that Nod2 expression was significantly increased at the peak of intestinal inflammation, and remained elevated throughout the resolution process. This observation suggests a possible role for Nod2 in the resolution of inflammation. Conversely, upon infection with the acute intestinal pathogen Citrobacter rodentium, Nlrp3-/- mice suffered from increased bacterial colonization as early as 3 days post infection, resulting in exacerbated intestinal inflammation and severe weight loss. Analysis of irradiation bone marrow chimeras showed that the protection required Nlrp3 activation in the non-haematopoietic compartment. Furthermore, this protective mechanism was independent of the inflammasome-associated cytokines IL-1β or IL-18. Therefore, this study implicates Nlrp3 activation in intestinal tissue cells as having a crucial role in controlling pathogenic bacterial colonization, providing a potential mechanism by which NLRP3 polymorphisms could lead to increased susceptibility to IBD.

Due to the high toxicity and ubiquitous presence of inorganic arsenic (As) in groundwaters, there have been a number of studies investigating the removal of inorganic arsenic though most of the existing technologies are constrained by low energy efficiency and ineffectiveness in As(III) removal under circumneutral pH conditions. In this work we present proof-of-concept of a modified double potential step chronoamperometry (DPSC) method which achieves in-situ As(III) oxidation and concomitant electro-sorption of As(V) onto the electrode. The use of both carbon cloth and redox active polyvinylferrocene (PVF) functionalized carbon nanotube electrodes are investigated in this work. Our work with the carbon cloth electrode showed that in-situ anodic As(III) oxidation and concomitant sorption of As(V) formed was achieved at an applied voltage of 1.2 V. Our results further showed that the sorbed As(V) was effectively electro-desorbed by reversing the polarity thereby regenerating the electrode. The in-situ anodic As(III) oxidation, sorption of As(V) and desorption of As(V) are affected by aqueous pH with high oxidation and sorption/desorption rates observed at elevated pH. The increase in As(III) oxidation and concomitant As(V) adsorption at higher pHs is related to (i) the rapid oxidation of the deprotonated species compared to the protonated species and (ii) stronger electrochemical interaction between the multi-charged As(V) species and the electrodes. At 1.2 V and an electrical energy consumption of 0.06 kWh m−3, the total As concentration can be reduced from 150 to 15 µg L−1 using an electrochemical cell with electrode area of 10 cm × 8 cm and electro-sorption time of 120-min. Based on the experimental results, we have developed a mathematical model to describe the kinetics and mechanism of arsenic removal by the modified DPSC method using the carbon cloth electrode with this model of use in predicting, and potentially optimising, process performance under various conditions. While the carbon cloth electrode was effective in As removal in the absence of competing anions, in the presence of anions such as Cl−, SO42− and NO3−, the As(III) removal efficiency by the carbon cloth electrode was very low. In order to improve the selectivity of As over competing anions that are typically present in groundwaters, we investigated the sorption of arsenic from simulated groundwaters by a redox active PVF functionalized electrode. Our results show that effective and sustainable As(III) removal was achieved even at 0 V once the electrode is activated via anodic polarization. During activation, ferrocene (Fc) in PVF is oxidized to the ferrocenium ion (Fc+) with the latter facilitating As(III) sorption and subsequent oxidation as well as As(V) sorption. The high affinity of Fc+ to As and weak attraction to competing anions at 0 V ensure high selectivity of As over Cl−, SO42− and NO3− at concentrations typical of groundwaters. Following the removal process, efficient regeneration of the electrode is achieved at −1.2 V wherein Fc+ is reduced to Fc thereby facilitating As desorption from the electrode surface. Our results further show that O2 and associated generation of hydrogen peroxide (H2O2) during the electrode regeneration step drives the oxidation of Fc to Fc+, thereby maintaining the constant generation of Fc+ required to achieve As(III) removal in subsequent cycles. Our results show that 91.8%  0.6% of As(III) could be selectively removed from a simulated groundwater over 10 cycles at an ultralow energy consumption of 0.12 kWh m−3. We also investigated the influence of divalent cations (i.e. Ca2+and Mg2+) on the removal of As(III) by a PVF functionalized electrode. Our results show that in the absence of divalent cations, nearly 90% ± 0.9% of As(III) removal is achieved over ten continuous cycles by single-pass DPSC, even in the presence of competing anions (5 HCO3−, 3 mM Cl−, 0.5 mM NO3−, 2 mM SO42−) however the presence of divalent cations (Ca2+ and Mg2+) significantly inhibits electrode regeneration. While the presence of Ca2+ and Mg2+ facilitates As(III) removal in the 1st step of the DPSC, the regeneration of the electrode in the 2nd step of DPSC is significantly inhibited, thereby decreasing the As(III) removal in the subsequent cycles. Based on the results from various control experiments and surface characterization, it appears that Ca2+/Mg2+ either acts as a bridge between the electrode surface and As anions or the sorption of Ca2+/Mg2+ increases the positive charge on the electrode surface thereby facilitating the sorption of As. The Ca2+/Mg2+ assisted sorption occurs even during the electrode regeneration process, thereby decreasing the electrode regeneration efficiency and As(III) removal in subsequent cycles. Overall, the results show that the DPSC method with either carbon cloth or PVF functionalized electrodes achieves in-situ As(III) oxidation, concomitant sorption of As(V) and desorption of As(V). However, the adverse impact of the presence of divalent cations (Ca2+ and Mg2+) on electrode regeneration and thus the efficacy of the As removal process must be taken into account. Future work should focus on i) optimization of the configuration of the system in order to minimize the influence of divalent cations on regeneration of the electrodes and ii) scale up of the unit to a size that is meaningful for practical application. Additionally, development of more stable PVF electrodes and/or alternate electrode materials for better removal efficiency is also required.

The matrix plays an essential role in many theoretical computer science and machine learning problems. In this thesis, we develop a better understanding of matrices with a view towards these applications. Our insights yield improvements for a number of old, well-studied algorithmic problems. In this thesis, we study matrices from three perspectives. We first consider their role in optimization. We study a number of matrix optimization problems, and propose new solvers and results for linear programs, empirical risk minimization, ordinary differential equations, deep neural networks. We next consider how random matrices concentrate. Specifically, we generalize a number of scalar Chernoff-type concentration inequalities and the Spencer-type discrepancy theorems to matrices. Finally, we develop new algorithms for problems on matrices. These fall roughly into two sub-categories, namely matrix factorization problems and structured recovery problems. In the first category, we propose a number of new algorithms for a variety of low-rank matrix factorization problems. In the second category, we give new algorithms for some recovery tasks with structured matrices. We design matrices and corresponding algorithms for compressed sensing tasks, and we give fast algorithms for the sparse Fourier transform problem, which can be thought of as a sparse recovery problem where one cannot freely choose the matrix. We now describe our contributions in more detail. Linear programming is one of the fundamental problems in computer science. In both theory and practice, many problems can be solved via linear programming, and doing so often yields the best-known runtime. We present an algorithm that runs in the current matrix multiplication time, which breaks a thirty-year-old barrier. Furthermore, our technique can be generalized to speed up a large family of convex optimization problems, i.e., empirical risk minimization. Sampling logconcave functions which arise in statistics and machine learning has been the subject of intensive study. This problem can be thought of as a special case of solving multivariate ordinary differential equations (ODEs). Under sufficiently strong smoothness conditions, discrete algorithms of Hamiltonian Monte Carlo (HMC) have runtime and number of function evaluations growing with the dimension. We give new algorithms that obtain a nearly linear implementation of HMC for a broad class of smooth, strongly logconcave densities, with the number of iterations (parallel depth) and gradient evaluations being polylogarithmic in the dimension (rather than polynomial, as in previous work). Deep neural networks (DNNs) have demonstrated dominating performance in many fields; since AlexNet, networks used in practice are going wider and deeper. We prove why stochastic gradient descent (SGD) can find global minima on the training objective of DNNs in polynomial time. We only make two assumptions: the inputs are non-degenerate and the network is over-parameterized. Furthermore, our results also hold for Recurrent Neural Networks (RNNs) which are multi-layer networks widely used in natural language processing. They are harder to analyze than feedforward neural networks, because the same recurrent unit is repeatedly applied across the entire time horizon. The Chernoff bound for the concentration of scalar random variables is a fundamental tool in the analysis of randomized algorithms. Over the past decade, a matrix generalization of the Chernoff bound has also found widespread application, but this generalization is fairly restrictive. For a number of decades, it has been open whether one can remove some of these restrictions. We answer this question in the affirmative by giving a number of new matrix Chernoff bounds under more relaxed independence assumptions than before. Spencer's theorem is a famous result in discrepancy theory, and it is an important open question how to generalize this result to the matrix setting. We make progress on this, and prove a matrix generalization of this result, in some restricted settings. Our result also generalizes the famous Kadison-Singer conjecture. The classical low rank approximation problem is : given a matrix A, find a rank-k matrix B such that the Frobenius norm of A - B is minimized. It can be solved in polynomial-time using, for instance, singular value decomposition. If one allows randomization and approximation, it can be solved in time proportional to the number of non-zero entries of A with high probability. We consider a number of natural generalizations of this important problem. We generalize the problem to consider a number of different norms, including weighted norms, entry-wise L1, and more general loss functions including Huber and Tukey loss. We also consider the natural tensor version of the problem. For all these settings, we give new state-of-the-art algorithms, including a number of new fixed parameter tractable algorithms. Compressed Sensing, or sparse recovery, is a powerful mathematical framework whose goal is to reconstruct an approximately sparse signal from linear measurements. We consider the extensively studied problem of L2/L2 compressed sensing. Our algorithm has faster decoding time and significantly smaller column sparsity, answering two open questions of prior work. Previous work on sublinear-time compressed sensing employed an iterative procedure, recovering the heavy coordinates in phases. We completely depart from that framework, and give the first sublinear-time algorithm which achieves the optimal number of measurements without iterating. The Fourier transform is ubiquitous in computer science, mathematics, and beyond. In recent years, a number of works have studied methods for computing the Fourier transform in sublinear time if the output is sparse. We consider two important settings in which this occurs: namely, the sparse high-dimensional setting, and the sparse continuous setting. In the high dimensional setting, we consider the extensively studied problem of computing a k-sparse approximation to the d-dimensional Fourier transform of a length n signal. Our algorithm achieves a nearly optimal sample complexity and runs in time comparable to the Fast Fourier Transform. All previous algorithms proceed either via the Restricted Isometry Property or via filter functions. Our approach totally departs from the aforementioned techniques, and we believe is a new approach to the sparse Fourier transform problem. While many of the works on sparse Fourier transforms have focused on the discrete setting, in many applications the input signal is continuous and naive discretization significantly worsens the sparsity level. Assuming the frequency gap is known, we present an algorithm for robustly computing sparse Fourier transforms in the continuous setting. Knowing the frequency gap is necessary for robustly identifying individual frequencies. However, was unknown whether interpolating the signal also requires a frequency gap. We resolve this problem by giving an algorithm which shows that such a gap not necessary for estimating the signal.