Homological algebra and friezes

PhD ThesisOver the last decade frieze patterns, as introduced by Conway and Coxeter in the 1970's, have been generalised in many ways. One such exciting development is a homological interpretation of frieze patterns, which we call friezes. A frieze in the modern sense is a map from a triangulated category C to some ring. A frieze X is characterised by the propety that if x ! y ! x is an Auslander-Reiten triangle in C, then X( x)X(x)X(y) = 1. A canonical example of a frieze is the Caldero-Chapoton map. The more general notion of a generalised frieze was introduced by Holm and J rgensen in [25] and [26]. A generalised frieze X0 carries the more general property that X0( x)X0(x) X0(y) 2 f0; 1g. In [25] and [26] Holm and J rgensen also introduced a modi ed Caldero- Chapoton map, which satis es the properties of a generalised frieze. This thesis consists of six chapters. The rst chapter provides a detailed outline of the thesis, whilst setting some of the main results in context and explaining their signi cance. The second chapter provides a necessary background to the notions used throughout the remaining four chapters. We introduce triangulated categories, the derived category, quivers and path algebras, Auslander-Reiten theory and cluster categories, including the polygonal models associated to the cluster categories of Dynkin types An and Dn. The third chapter is based around the proof of a multiplication formula for the modi ed Caldero-Chapoton map, which signi cantly simpli es its computation in practice. We de ne Condition F for two maps and , and show that when our category is 2-Calabi- Yau, Condition F implies that the modi ed Caldero-Chapoton map is a generalised frieze. We then use this to prove our multiplication formula. The de nition of the modi ed Caldero-Chapoton map requires a rigid subcategory R that sits inside a cluster tilting subcategory T. Chapter 4 proves several results showing that in the case of the cluster category of Dynkin type An, the modi ed Caldero-Chapoton map depends only on the rigid subcategory R. These results then allow us to prove a general formula for the group Ksplit 0 (T)=N, which is used in the de nition of the modi ed Caldero-Chapoton map. Chapter 5 provides a comprehensive list of exchange triangles in the cluster category of Dynkin type Dn. Chapter 6 then proves several similar results to Chapter 4 in the case of the cluster category of Dynkin type Dn. We prove that the modi ed Caldero-Chapoton map depends only on the rigid subcategory R before again producing a general formula for Ksplit 0 (T)=

Cryptic population structure and insecticide resistance in Anopheles gambiae from the southern Democratic Republic of Congo

The Democratic Republic of Congo (DRC) suffers from one of the highest malaria burdens worldwide, but information on its Anopheles vector populations is relatively limited. Preventative malaria control in DRC is reliant on pyrethroid-treated nets, raising concerns over the potential impacts of insecticide resistance. We sampled Anopheles gambiae from three geographically distinct populations (Kimpese, Kapolowe and Mikalayi) in southern DRC, collecting from three sub-sites per population and characterising mosquito collections from each for resistance to pyrethroids using WHO tube bioassays. Resistance to each of three different pyrethroids was generally high in An. gambiae with < 92% mortality in all tests, but varied between collections, with mosquitoes from Kimpese being the most resistant. Whole genome sequencing of 165 An. gambiae revealed evidence for genetic differentiation between Kimpese and Kapolowe/Mikalayi, but not between the latter two sample sites despite separation of approximately 800 km. Surprisingly, there was evidence of population structure at a small spatial scale between collection subsites in Kimpese, despite separation of just tens of kilometres. Intra-population (H12) and inter-population (FST) genome scans identified multiple peaks corresponding to genes associated with insecticide resistance such as the voltage gated sodium channel (Vgsc) target site on chromosome 2L, a Cyp6 cytochrome P450 cluster on chromosome arm 2R, and the Cyp9k1 P450 gene on chromosome X. In addition, in the Kimpese subsites, the P450 redox partner gene Cpr showed evidence for contemporary selection (H12) and population differentiation (FST) meriting further exploration as a potential resistance associated marker

Phloem sap intricacy and interplay with aphid feeding

Aphididae feed upon the plant sieve elements (SE), where they ingest sugars, nitrogen compounds and other nutrients. For ingestion, aphid stylets penetrate SE, and because of the high hydrostatic pressure in SE, phloem sap exudes out into the stylets. Severing stylets to sample phloem exudates (i.e. stylectomy) has been used extensively for the study of phloem contents. Alternative sampling techniques are spontaneous exudation upon wounding that only works in a few plant species, and the popular EDTA-facilitated exudation technique. These approaches have allowed fundamental advances on the understanding of phloem sap composition and sieve tube physiology, which are surveyed in this review. A more complete picture of metabolites, ions, proteins and RNAs present in phloem sap is now available, which has provided large evidence for the phloem role as a signalling network in addition to its primary role in partitioning of photo-assimilates. Thus, phloem sap sampling methods can have remarkable applications to analyse plant nutrition, physiology and defence responses. Since aphid behaviour is suspected to be affected by phloem sap quality, attempts to manipulate phloem sap content were recently undertaken based on deregulation in mutant plants of genes controlling amino acid or sugar content of phloem sap. This opens up new strategies to control aphid settlement on a plant host