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Computing the dominant Fourier coefficients of a vector is a common task in many fields, such as signal processing, learning theory, and computational complexity. In the Sparse Fast Fourier Transform (Sparse FFT) problem, one is given oracle access to a $d$-dimensional vector $x$ of size $N$, and is asked to compute the best $k$-term approximation of its Discrete Fourier Transform, quickly and using few samples of the input vector $x$. While the sample complexity of this problem is quite well understood, all previous approaches either suffer from an exponential dependence of runtime on the dimension $d$ or can only tolerate a trivial amount of noise. This is in sharp contrast with the classical FFT algorithm of Cooley and Tukey, which is stable and completely insensitive to the dimension of the input vector: its runtime is $O(N\log N)$ in any dimension $d$. In this work, we introduce a new high-dimensional Sparse FFT toolkit and use it to obtain new algorithms, both on the exact, as well as in the case of bounded $\ell_2$ noise. This toolkit includes i) a new strategy for exploring a pruned FFT computation tree that reduces the cost of filtering, ii) new structural properties of adaptive aliasing filters recently introduced by Kapralov, Velingker and Zandieh'SODA'19, and iii) a novel lazy estimation argument, suited to reducing the cost of estimation in FFT tree-traversal approaches. Our robust algorithm can be viewed as a highly optimized sparse, stable extension of the Cooley-Tukey FFT algorithm. Finally, we explain the barriers we have faced by proving a conditional quadratic lower bound on the running time of the well-studied non-equispaced Fourier transform problem. This resolves a natural and frequently asked question in computational Fourier transforms. Lastly, we provide a preliminary experimental evaluation comparing the runtime of our algorithm to FFTW and SFFT 2.0.

Conophytum N.E.Br. (Aizoaceae) is a genus of dwarf succulent plants endemic to western South Africa and southern Namibia. It is highly diverse, with 106 constituent species and is considered a model genus for the Succulent Karoo Biome, an arid region with uniquely high desiccation-tolerant (succulent) plant diversity, based on its variety of distribution types and growth forms. The aim of this project was twofold: to provide a clearer understanding of the patterns and drivers of speciation in the genus, and to identify conservation strategies that maximise the likely survival of these species, based on spatial ecological analyses of their current distributions. The biodiversity of the Succulent Karoo biome is increasingly vulnerable to a multitude of threats, mostly as a consequence of growing levels of human activity. The current and future conservation of Conophytum is dependent on evidence-based interventions that, in particular, help mitigate the effects of a changing climate and illegal collecting. First, an updated phylogeny was produced providing the foundation to many of the analyses undertaken in this thesis, beginning with an assessment of the drivers of diversification. For the first time, nuclear gene regions and sequences from almost all species in the genus (102 of 106) were included in the development of phylogenetic hypotheses. From there, a selection of 12 pertinent characters was mapped over the phylogeny and phylogenetic signal was calculated in a univariate and multivariate manner. This, along with correlations in character evolution, was used to infer potential drivers of diversification. Nocturnal flowering was found to characterise certain clades and was correlated with pollen type D and flower structure A2, suggesting an influence of pollinator specialisation on evolution, while windowed leaves characterised one particularly large, strongly supported clade in the phylogeny, and, through correlated evolution with other traits such as the presence of bladder cells, suggested a key influence of the abiotic environment in this clade���s formation. This abiotic influence on speciation was further investigated using spatial environmental data to determine differences in the abiotic preferences of individual Conophytum species and five large, strongly supported clades in the phylogeny. Geological, topographical and climatic variables were mapped onto the phylogeny, while differences in clade and species��� environmental preferences were assessed. Associations between environmental variables and the evolution of certain traits were also analysed. Variation in precipitation/moisture, elevation and slope were identified as key drivers of diversification in the genus, with niche divergence prevalent in driving speciation. Additionally, some clades were characterised by both nocturnal flowering and certain leaf traits (e.g. very long trichomes, sunken stomata), suggesting a possible interaction between biotic and abiotic diversification drivers. These interactions were analysed mainly by calculating correlations between nocturnal flowering and abiotic variables. Nocturnal flowering was strongly correlated with a lower mean annual temperature, suggesting a possible climatic influence on pollinator assemblages potentially driving a greater likelihood of species in cooler locations to be nocturnally moth pollinated. After gaining a greater understanding of what drove diversity in Conophytum, attention switched to the vulnerability of this diversity to future changes in climate by projecting range change responses using species distribution models (SDMs). These analyses focused on the more broadly distributed species (a minimum of 20 occurrence points) to facilitate the building of statistically robust models, comparing climate change vulnerability at both the species- and clade-levels. For four of the five clades, and for the genus as a whole, range size reduction was prevalent, with a range contraction from the north occurring as conditions in the north of the genus��� range become unsuitable through lower winter rainfall and increased temperatures, while the more mesic southern reaches of South Africa remain unsuitable for Conophytum inhabitation. One clade, however, characterised by bladder cells and windowed leaves which give constituent species better adaptation to hotter conditions, may include some species that benefit from predicted future climate conditions. Two new regions were prioritised based on these projections for future protection, as these successfully cover the areas of high current and future diversity in the genus, along with protecting separate clades and thereby maintaining phylogenetic diversity. As SDMs excluded narrow endemics in the genus because of the low numbers of occurrence points available, a final assessment on the abiotic environmental preferences and trait associations of narrow endemic Conophytum species was performed to better understand their ecology and to aid their conservation. These taxa make up a sizable proportion of species in the genus (~33%) and are some of the most vulnerable Conophytum species through their small range and population sizes. This assessment demonstrated a preference of these species for more high-stress environments (steeper slopes, higher elevation and low moisture) compared to broadly distributed species, and identified that these taxa are predominantly diurnal flowering, although this was likely to be because narrow endemic distribution overlaps strongly with diurnal wasp pollinators. This study also identified an important region for the conservation of narrow endemics and these ecological findings will assist prioritising regions for protection or translocation. The findings of this project shed light on the evolution of Conophytum and its vulnerability to anthropogenic impacts and provide an improved basis for understanding of the genus��� ecology and biogeography. Based on these findings, Conophytum can now act as an ideal model genus for the Succulent Karoo, a uniquely biodiverse arid hotspot. An improved understanding of the evolution and vulnerability of Conophytum provides important evidence for drivers of rapid speciation in a region known for its diversifications and, perhaps even more importantly, highlights threats and important conservation strategies for the Succulent Karoo in an uncertain future.

Formation and structural modification of oxygenated polycyclic aromatic hydrocarbons (oxyPAHs) by UV irradiation on minerals have recently been proposed as a possible channel of PAH transformation in astrochemical and prebiotic scenarios of possible relevance for the origin of life. Herein, it is demonstrated that high-energy proton-beam irradiation in the presence of various meteorites, including stony iron, achondrite, and chondrite types, promotes the conversion of two representative oxyPAH compounds, 1-naphthol and 1,8-dihydroxynaphthalene, to complex mixtures of oxygenated and oligomeric derivatives. The main identified products include polyhydroxy derivatives, isomeric dimers encompassing benzofuran and benzopyran scaffolds, and, notably, a range of quinones and perylene derivatives. Addition of urea, a prebiotically relevant chemical precursor, expanded the range of identified species to include, among others, quinone diimines. Proton-beam irradiation of oxyPAH modulated by nitrogen-containing compounds such as urea is proposed as a possible contributory mechanism for the formation and processing of insoluble organic matter in meteorites and in prebiotic processes.

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