What? The aim of the project is to develop an efficient organo-catalytic methodology for the enantioselective [1,2]-rearrangement of allylic ammonium ylides in the formation of amino acids. The project will involve the development of new catalysts to promote this transformation. Furthermore, once the method is optimized, the mechanism will be elucidated to give an even better insight into this important reaction. Why? Amino acids are one of the fundamental building blocks of life. For this reason, they play an essential role in both medicine and science. This method allows for a straightforward synthesis of amino acids from simple starting materials, generating chiral centers and C-C bonds in the process. Moreover, it explores a significant and recognized challenge in organic chemistry. Contrary to the [1,2], the [2,3]-sigmatropic rearrangement is a well-known reaction. A challenge of this project will be to favor the [1,2]-rearrangement over the [2,3]-rearrangement, allowing for a much wider scope for these transformations. How? The project will be carried out in collaboration with Prof. Andrew Smith from the University of St Andrews. Prof. Smith and his team are expert in organo-catalysis, using BTM as a catalyst. Building on previous results using BTM to promote the [2,3]-sigmatropic rearrangement of allylic ammonium ylides, BTM analogues will be synthesized and optimized for properties to promote the [1,2]-rearrangement. Furthermore, it will be investigated which structural motifs will bias towards the [1,2]-rearrangement. With this knowledge in hand, the reaction will be fine-tuned by applying meticulous optimization of temperature, reaction time, solvent, and additives. This in turn is expected to yield a well performing reaction of allylic ammonium ylides to amino acids by a [1,2]-rearrangement.

Semigroups are fundamental algebraic objects. They arise in many areas throughout mathematics and theoretical computer science. While semigroups have a rich theory, the less-rigid structure they have compared to groups makes them more difficult to work with. In fact, a common theme in Computational Semigroup Theory (CST) is considering semigroup problems as a problem in group theory together with a problem in combinatorics. However, the added combinatorial sub-problem can lead to problems for semigroups being much more computationally expensive than the corresponding problems for groups. Combinatorial optimisation is concerned with efficiently finding solutions to combinatorial problems which cannot realistically be found with an exhaustive search. A classic example is the "travelling salesman" problem, which asks for the shortest distance a salesman must travel to visit a number of cities exactly once. For 20 cities, there are over 2 quintillion different possible routes to check, which is unfeasible in any reasonable amount of time. However, far more efficient algorithms exist which can find optimal routes for even much larger numbers of cities. In addition, there are heuristic methods which can find approximate solutions very quickly. Heuristics, approximation, parallelization and reduction of search space through symmetries are examples of techniques from combinatorial optimisation which have applications throughout mathematics and computer science. As many areas can benefit from combinatorial optimisation techniques, general-purpose solvers for combinatorial problems have been developed. However, these solvers are not well-suited to problems in CST, as they cannot efficiently represent certain aspects of the problems. This project will bring the techniques of such solvers to bear on problems in CST. Specific problems in mind include: (Di)graph homomorphisms and endomorphisms Translations and the translational hull of semigroups Maximal subsemigroups Minimal generating sets of semigroups Normalisers and automorphism groups of semigroups Semigroup isomorphisms and homomorphisms Lattices of congruences Intersections of subsemigroups and inverse subsemigroups For all of the identified problems, theoretical research will lead to practical computer implementations. Completed work will be included in the Semigroups package (J. D. Mitchell and others, 2017) for the free and open source Computational Algebra software GAP (The GAP Group, 2017). Work in development will be available through a hosting service for open source software, GitHub. Algorithms that are performance-critical will be implemented in a standalone C/C++ library, accessible through Python. A recent trend in mathematics has been the use of computational tools to analyse examples in search of theoretical results. For example, researchers may wish to verify a new conjecture on all finite groups less than a certain order. This might not be realistic for a researcher in Semigroup Theory: the number of semigroups of size 10 is a 20-digit number. By providing new and improved tools in CST, and enabling currently-unfeasible computations, this project will open new avenues of mathematical research. J. D. Mitchell and others. (2017). Semigroups - GAP package, Version 3.0.7. doi:10.5281/zenodo.592893 The GAP Group. (2017). GAP - Groups, Algorithms, and Programming, Version 4.8.8 (http://www.gap-system.org).

Microcantilevers are used as sensors in atomic force microscopy (AFM) to determine the topography of surfaces as well as to measure properties such as adhesion, stiffness, or charge density, to name a few. Micro- and nanocantilevers are also increasingly employed as freestanding sensors to detect tiny amounts of masses or to determine the density and viscosity of small fluid volumes. If data based on such cantilevers is interpreted it is almost always assumed that the cantilevers show ideal behaviour, i.e., that they can be described as an object with perfect geometric shape and that their material properties are accurately known. Measurements with real cantilevers, however, do show some deviation from this ideal behaviour, for example due to non-perfect dimensions and a geometric shape that deviates from the ideal one. This can lead to errors when interpreting the data and has an influence on the results obtained. In this project we will use cantilevers for AFM measurements to explore material properties and to model what effect deviations from ideal cantilever behavior can have on the results. In addition, we plan to explore via simulations if cantilevers of geometric shapes that are different to those currently available might have advantages in being more sensitive to probing certain properties.