Quantifier Elimination (QE) is a very general challenge in mathematics: we consider the case of the real numbers. Saying "there exists a y such that x = y2 is equivalent to "x = 0", and saying "for all x, there do not exist y such that danger(x,y)" is a proof of correctness, in this respect, of a safety-critical system. Many safety-critical systems are currently proved safe by similar techniques over Booleans, but not reasoning over the reals as such. However , there are very hard instances of QE (DavenportHeintz1988 etc.), which has led to research into worst-case algorithms by the PI and many others. However, little research has been done on "good on average problems" algorithms and heuristics, which are what the industrial community actually need. We can draw a strong parallel with SAT-solving for Boolean expressions, which is in daily use in industry, even though this is NP-complete (only known algorithms are exponential) in the worst case. The initial area of research will be in Virtual Term Substitution (VTS) to pre-condition purely existential QE problems before solution, for example, by Cylindrical Algebraic Decomposition (CAD). This will require, amongst other things, significant programming in Maple using the package RegularChains. From there theory and experimentation into how much benefit is being gained from using such techniques can be looked into, in particular analysis on running times of the algorithms on a repository of real world example problems. In parallel, other recent developments such as NuCAD (Brown2017) should be investigated. As such early hypotheses and minor results should be expected before a visit to the CASE sponsor (Maplesoft) in 2018, equipped with early versions of useful code from such areas of research, written to appropriate specification via the standards of the symbolic computation community.

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Research into the feasibility of an additive-manufactured ultra high efficiency, high temperature micro gas turbine. The project aims to carry out fundamental research into a highly novel micro gas turbine by designing, manufacturing and testing a combustion system with industry support from HiETA Technologies utilising Additive Manufacturing to create high efficiency cooling systems. The objective is to prove the feasibility of running a system at very high gas temperatures to yield efficiency improvements. To start, research will be conducted on already existing combustor designs for similar micro-gas turbine applications, to gain an understanding of the already existing technology in the market and identify possible improvements that can be implemented with the use of additive manufacturing. This research will then feed into the initial proof of concept design that will then be analysed using CFD, manufactured by the project industrial partner HiETA and tested in the hot gas stand cell at Bath once it is fitted with a high temperature turbine. Further research on state of the art combustion cooling designs and CFD analysis on fuel delivery and combustion processes will follow, which will lead to multiple designs for a state of the art combustion system, which HiETA will assist in manufacturing. The designs will then be tested at high temperatures in the hot gas stand test cell at Bath again to validate the designs.

The project aims to translate existing academic research to enable the production of interpenetrating network (IPN) gels of starch and nanocellulose fibres using side-by-side enzymatic and/or bacterial synthesis routes. These IPN gels will be produced with the end aim of use in the food and home care industries. The project will develop tandem enzymatic processes for the production of small fragments of starch, xyloglucan and cellulose, assembly of components into gels in the presence of water, and characterisation of the mechanical and physical properties of the gels. We will valorise waste, making new products, reducing costs and CO2 emissions for the companies we will support through this technology. Industrial input will guide development of the gels, targetting key industrial applications. No research has yet developed gels based on industrial biotechnology approaches to produce small molecules of starch combining them with cellulose fibrils.

There are many instances in life when two numbers might be the same and yet there is no relation between the objects that are being counted. For example, the number of coins 1p, 2p, 5p,..., £2 in British currency, and the number of legs on a healthy spider. However, such coincidences in mathematics are sometimes merely the shadow of a much more interesting relation that exists on a deeper level between different structures. When this can be explained, we not only understand the original coincidence but, better yet, we can translate our understanding from one structure to the other. Put simply, we uncover the dictionary between two languages that we might only partially understand. One such example is provided by a positive number that appears in two apparently different mathematical contexts: one is equal to the number of certain types of symmetry that are defined in a rather abstract, algebraic way; and the other is obtained by counting certain geometrically-defined objects. In fact, this coincidence can be explained quite beautifully by a relation known as the "McKay correspondence". Roughly speaking, this correspondence describes in a very concrete way in which two rather abstract objects (called triangulated categories), one defined in terms of algebra and the other defined in terms of geometry, are in fact the same. Every such category encodes certain numbers, and the original coincidence boils down to the simple observation that two identical categories encode precisely the same numbers! This, then, is one of the primary goals of a pure mathematician: to investigate whether apparent coincidences can be explained in a natural way, and it is precisely this search for the "natural" notion that makes pure mathematics important to so many fields of science and engineering. The current proposal aims to do precisely this. As with the McKay correspondence described above, it has been known for some time that many such correspondences (called equivalences of categories) do exist even for rather different types of geometry which encode the same kind of numbers, and some of these have been described very elegantly by the work of several mathematicians over the last fifteen years or so. Even now, the general picture eludes us, but the following question has been posed by mathematicians Bondal and Orlov: "If we have two types of geometry that, while being different are nevertheless similar in a controlled way, does there exist a correspondence as above to explain the similarity?". Here we aim to lay the foundation for a new, geometric approach to this problem by introducing an abstract generalisation of a particular map - a kind of "machine" - that the PI has studied in depth. Crucially, we believe that we understand precisely the right level of abstraction to shed light on the correct path: too little abstraction may illuminate nothing at all; while too much abstraction may be so blinding as to provide no help whatsoever. While we do not have the full picture, we do believe that we have found the correct foundation for the problem, and to provide a "proof of concept" for our approach we will demonstrate that it works for an interesting class of examples. The results that will come from this proposal will, we believe, provide solutions to several interesting problems that, taken together, provide an important, geometric step towards our understanding of the celebrated conjecture of Bondal and Orlov.