A Golod-Shafarevich Equality and p-Tower Groups

All current techniques for showing that a number field has an infinite p-class field tower depend on one of various forms of the Golod-Shafarevich inequality. Such techniques can also be used to restrict the types of p-groups which can occur as Galois groups of finite p-class field towers. In the case that the base field is a quadratic imaginary number field, the theory culminates in showing that a finite such group must be of one of three possible presentation types. By keeping track of the error terms arising in standard proofs of Golod-Shafarevich type inequalities, we prove a Golod-Shafarevich equality for analytic pro-p-groups. As an application, we further work of Skopin, showing that groups of the third of the three types mentioned above are necessarily tremendously large.Comment: 12 pages, pre-reviewer versio

Self-Automaton Semigroups

After reviewing automaton semigroups, we introduce Cayley Automata and the corresponding Cayley Automaton semigroups. We investigate which semigroups are isomorphic to their Cayley Automaton semigroup and give some results for special classes of semigroups. We answer a question posed by Cain relating to the dual construction.Comment: 18 page

Class number formulas via 2-isogenies of elliptic curves

A classical result of Dirichlet shows that certain elementary character sums compute class numbers of quadratic imaginary number fields. We obtain analogous relations between class numbers and a weighted character sum associated to a 2-isogeny of elliptic curves.Comment: 19 pages; To appear in the Bulletin of the London Mathematical Societ

Developing screening tools to identify novel, resistance breaking pesticides.

Pesticide resistance is estimated to cost the USA $1.4 billion annually. Not only is there a huge economic cost, but the loss of crop yield and higher doses of pesticides needed to control pests damages the ecosystem 1,2. The development of resistance to chemicals is a universal phenomenon and within insect pests more than 440 species are now resistant to one or more pesticidal compound 3,4. As increasing levels of resistance arise and new molecular tools become available the understanding of resistance mechanisms grows and the limitations of pesticides are clarified 5,6. Understanding resistance is vital to counter it 5,7,8. Still facing high levels of pesticide resistance and the damaging effects of the remaining effective compounds, I here look to identify a novel pesticidal compound to overcome current resistance mechanisms 9. Synthetic compounds made by industrial partner Darr House M.I. were tested for activity against Drosophila melanogaster and Myzus persicae. The first 18 compounds were expected to act on the nicotinic acetylcholine receptor using imidacloprid as a positive control. Four competitively active compounds were found but, following a ban on neonicotinoids in the EU in 2018 and a knock-on lack of interest on the part of major agrochemical companies in novel nAChR compounds, this part of the project was pursued no further 10,11. The next 30 compounds were then tested for activity against the neurotransmitter gamma-aminobutyric acid (GABA) receptor. Here activity was only found against Drosophila not Myzus. Five compounds showed activity against D. melanogaster susceptible strain Canton-S, four then showed activity against metabolic resistant strain Hikone-R with compound 47 being close to resistance breaking. While synthetic compounds are popular, natural sources are not only a source of inspiration for synthetic products but natural products used for pest control have advantages of being environmentally friendly and constantly evolving with their pests. I tested 9 botanical sources for insecticidal and repellent activity against D. melanogaster and the Peach potato aphid M. persicae. Extracts from samples were taken using a methanol extraction technique. Rosemary extract results suggest potential lethal effects on Drosophila but development of this product would be required to concentrate the lethal effects above 40%. All extracts: basil, chilli, garlic, lemongrass, nasturtium leaves, flowers and seeds and rosemary showed repellent activity against Myzus except dill extract which had no effect. An increase in nymph droppings was seen for Myzus treated with basil suggesting possible problems for use of this compound as aphid control. To address the problem of cost and identification of novel active pesticides a Fly-Tox panel was developed using D. melanogaster as a model screening tool containing metabolic P450 resistance genes from multiple economically important pests and pollinator species. In this thesis four lines were developed containing Cyp6cm1, Cyp6bq23, Cyp6bq9 and Cyp337b3 but conferral of resistance was unsuccessful. Alternative lines from the published Fly-Tox panel were used to test the use of the screening tool with novel insecticides from chapter 2 and 3; one nAChR and one GABA targeting compound. These novel compounds were compared against positive controls; imidacloprid and fipronil, and showed a successful test run of a section of the screening tool. No resistance breaking bee-safe compounds were identified in this thesis but there was a successful trial of the Fly-Tox screening tool of transgenic Drosophila showing the value of this new resource in pesticidal discovery science. There were also findings of broad metabolic capabilities of the gene Cyp6er1, known to metabolise neonicotinoids, but also found to be active against suspected GABA targeting novel compound 47.European Commission BBSR