A concept of central importance in mathematics is that of symmetry. One used to think of symmetry as a property of geometric shapes, but in the 19th century Evariste Galois extended the concept of symmetry to algebraic objects, and today his insights are completely fundamental to pure mathematics. The underlying goal of this proposal, which is situated between Algebra, Number Theory, and Topology, relying also on techniques from Probability Theory and Additive Combinatorics, is to study symmetries of arithmetic and geometric objects. Number Theory is an ancient mathematical discipline with a rich history of over 2000 years, but also with spectacular developments in recent years. Some of the most impressive recent advances have happened in the area of Number Theory called Arithmetic Statistics: the groundbreaking contributions of Manjul Bhargava have been rewarded with a Fields Medal in 2014. The aim of Arithmetic Statistics is to understand the behaviour of arithmetic objects, such as (ray) class groups, in families. The birth of this area goes back to Gauss, who formulated some concrete conjectures concerning the behaviour of class groups of quadratic fields. It was given a huge boost in the 1980s, when Cohen and Lenstra proposed a general model that implied all the conjectures of Gauss, and more. Roughly speaking, they postulated that class groups of imaginary quadratic fields obey a probability distribution that assigns to a finite abelian group X a probability that is inverse proportional to the number of symmetries of X. This is, in fact, a very natural model for random algebraic objects. This was later generalised to other number fields by Cohen and Martinet, but in more general cases the probability distributions looked more mysterious. The Cohen-Lenstra-Martinet Heuristics have been used as a guiding principle in Arithmetic Statistics since then, and have found applications in many other areas, such as the theory of Elliptic Curves, in Combinatorics, and in Differential Geometry. This project will consist of a blend of theorems, conjectures, and computations. I will: - show that the original conjectures are false, as stated, - find the correct formulations, - put them on a more conceptual footing, by explaining the mysterious looking probability weights of Cohen-Martinet using a theory of commensurability of algebraic objects that I have been developing together with Hendrik Lenstra, - extend the scope of the heuristics, e.g. to ray class groups. Two other kinds of very basic objects whose symmetries one studies are finite sets and finite dimensional vector spaces. An old problem in Representation Theory, with applications to Number Theory and Differential Geometry, is to compare symmetries of sets with symmetries of vector spaces, and in particular to determine which symmetries of sets become isomorphic (essentially the same) when the sets are turned into vector spaces. There are two incarnations of this problem: one where the vector spaces are over a field of characteristic 0, e.g. over the real numbers, and one where they are vector spaces over a field of positive characteristic. In previous joint work with Tim Dokchitser we have solved the case of characteristic 0, thereby settling an over 60 year old problem. Using the techniques that we developed, and new ones, this project will settle the case of positive characteristic. Finally, I will also investigate symmetries of low-dimensional manifolds. These are the basic objects studied by modern geometry and topology, and it is an old and fruitful line of investigation to determine what one can say about the topology of the manifold from knowing its symmetries. In recent joint work with Aurel Page, I have introduced a new representation theoretic tool into the area, which I had worked on in number theoretic contexts. Using these new techniques, I am planning to shed more light on the connection between symmetries and the topology of the manifold.

Nanoparticles are key to a wide range of day-to-day technological applications, including petrochemical catalysis, biomedical imaging, optoelectronics, paints, inks, coatings, and nanomedicine. To develop and optimise such applications, one needs a good understanding of the atomic structures of the involved nanoparticles that is difficult to obtain from experiment alone. Similar structural models are relevant to understanding the toxicology and environmental effects of nanoparticles and their role in astrochemistry. We will develop a linked nanoscale structure prediction code (WASP@N) and web-interfaced database aimed at generating and archiving such structural models of nanostructures. This combination will provide a fast and efficient way of predicting the atomic structure of both fundamentally new systems that have never been studied before and known systems embedded in a realistic environment; e.g. in solution, in the pores or on the surface of a material, and/or with an organic capping agent, rather than isolated in a vacuum. The latter is crucial for understanding nanoparticles in many industrial processes (e.g. liquid phase nanoparticle catalysis, inks, coatings) and, for instance, nanotoxicology, but has not been done routinely before due to additional computational cost of including the environment. We strongly believe that the combination of new algorithmic approaches to be included in WASP@N and access to all low energy structures for the particles in vacuum in the cluster structure database will make predicting the atomic structure of nanoparticles insolution, for example, much more efficient and a standard technique in the repertoire of the applied computational chemist. The cluster structure database, finally, will also be useful as a stand-alone resource for experimental and computational chemists, chemical engineers, physicists, electronic engineers and toxicologists looking for information on the structure of materials on the nanoscale.

Space-borne synthetic aperture radar (SAR) observations can be used to measure structure and velocity within the Antarctic ice sheet. Most SAR missions to date have used L-band frequencies (1-2 GHz) but interest is now turning to lower-frequency P-band signals (around 430 MHz) because they have greater penetration of the ice. Both the University of Bath and BAS are currently involved in feasibility studies relating to P-band SAR design for future ESA satellites. P-band SARs in polar orbits such as the ESA BIOMASS satellite due to launch in 2014, have the potential to map out the three-dimensional structure of ice sheets. However, their signals will suffer from significant ionospheric effects including Faraday rotation, range defocusing, range delay and interferometric phase bias. The ionosphere must be taken into account in the system design but the necessary ionospheric measurements to do this do not currently exist. This project will deliver the measurements for the Antarctic region and lay the foundation for successful P-band SAR missions. This project involves equipment development, fieldwork and analysis. The objective of the fieldwork is to deploy modified GPS receiving equipment that will for the first time take measurements of total electron content (TEC), plasma velocity and ionospheric scintillation at remote locations across the Antarctic. To achieve this, eight new GPS receivers will be deployed to undertake long-term measurements in the auroral and polar-cap regions over a two year period. Additional data from lower Antarctic latitudes will be provided by international partners. The measurements will be used to develop a multi-scale model of the Antarctic ionosphere. This model will be a critical input to SAR design that will minimize the impact of the ionosphere on ice measurements for future satellite missions.

Microwave Imaging (MI) has gained a great deal of attention among researchers over the past decade, mainly due to its potential use in breast cancer imaging. MI is seen as a safe, portable and low-cost alternative to existing imaging modalities. Due to the breast tissue properties at microwave frequencies, MI benefits from significantly higher contrast than other techniques. The great excitement about MI radar system is that, using a multi-static real aperture technique and sophisticated signal processing, it has sufficient resolution to be clinically useful and is far better than simple wavelength assumptions would estimate. Whilst to date MI has been mainly proposed for breast cancer detection, some recent reports have also speculated a use of MI in extremities imaging, diagnostics of lung cancer, brain imaging and cardiac imaging. Despite the interest in Microwave Imaging among researchers, it has not moved far beyond numerical simulations and very simple experimental works without clinical realisation. Bristol is among two research groups in the world who have clinical experience with Microwave Imaging.Compared with other medical imaging techniques, microwave imaging is still in its infancy. One historical reason for this might due to the fact that most microwave systems-devices originated in military applications, radar being an obvious example. In recent years however, due to the mobile/wireless revolution, we have witnessed unprecedented progress in high performance microwave hardware as well as computing power. This opens up a unique opportunity for development of microwave imaging systems. The goal of this Career Acceleration Fellowship project is to explore a novel direction in MI, Differential Microwave Imaging (DMI), in clinical applications reaching far beyond breast cancer detection. In Differential Microwave Imaging, the goal is to image temporal changes in tissue, and not the tissue itself. This somewhat limits usability of DMI as an imaging technique on one hand, but at the same time it opens up totally new applications where standard Microwave Imaging could not be applied. The idea of DMI came from the discovery during world's first clinical trial of microwave radar imaging system in Bristol in 2009. During the clinical trials it was realised that the Microwave Imaging system was extremely sensitive to any changes occurring during the scan. Following this up it was then discovered that the local change in tissue properties can easily be detected and precisely located. Moreover, it was shown that this change in local properties of tissues can even be detected in very dense and heterogeneous breast tissues. The project will focus on two applications, serving as Proof of Principle:1. Nanoparticle contrast-enhanced DMI for cancer detection The proposed work on 3D detection of nanoparticles is of great interest to researchers working in the cancer imaging field. DMI could find applications not only in cancer detection, but it could also be used to find and evaluate the effectiveness of new cancer biomarkers, track nanoparticle-labelled cells or monitor delivery of nanoparticles for hyperthermia treatment. 2. Functional brain imaging using DMI radar systemDMI, as a general method, is also a promising concept for functional brain imaging. Development of the DMI system for functional brain imaging is timely related to current research activities in neuroscience. Functional imaging is used to diagnose metabolic diseases and lesions (such as Alzheimer's disease or epilepsy) and also for neurological and cognitive psychology research. This novel interdisciplinary project connects the fields of electronic engineering, nanotechnology and medical physics. The proposed research project addresses one of the EPSRC strategic priorities: Towards next generation healthcare. High calibre of clinical collaborators will ensure that research outcomes are relevant to end users.

Over 80 million patients worldwide suffer from hip osteoarthritis, and increasing numbers of patients are requiring total hip replacement surgery. This is considered to be a successful intervention, however, an ageing population with increasing orthopaedic treatment needs, greater levels of obesity and patient expectations, and reducing healthcare budgets and surgical training are conspiring to challenge this success. There is also increasing demand for surgical treatments in younger patients that will delay the need for hip replacement surgery, these interventions reshape bone and repair soft tissue. One of the major causes of failure in the natural hip and in hip replacements is impingement, where there is a mechanical abutment between bone on the femoral side and hip socket or hip replacement components. In the natural hip, surgery reshaping the bone can reduce this impingement and soft tissue damage can be repaired; however, the effects of the amount of bone that is removed is not well understood nor is the best way to repair soft tissue. The number of hip replacements needing to be removed from patients and replaced with a new one in revision surgery is increasing; damage to the cup rim because of impingement is often implicated. It is known that this is more likely if the components are not well aligned relative to each another, or relative to the load direction experienced in the body. In this proposal, I seek to ensure long term outcomes of early intervention and hip replacement surgery are always optimum by negating concerns about impingement. To do this, I will develop an experimental anatomical hip simulator. The simulator will apply loads and motions to the hip similar to those observed clinically, and include high fidelity phantoms that mimic the natural hip, into which hip replacement components can also be implanted. This anatomical simulator will be used to assess how variables such as those associated with the patient (e.g. their bony geometry), the extent of early intervention surgery (e.g. the amount of bone removed) or the design of the prosthesis and how the hip is aligned in the body will affect the likelihood of impingement. This improved understanding of factors affecting the likelihood and severity of impingement will enable better guidance on how the surgery should be performed to optimise outcomes to be provided. I will work with orthopaedic surgeons to integrate this improved understanding into their clinical practice and with an orthopaedic company to integrate the findings into new product development processes; so that future interventions and devices can be designed to provide better outcomes for all patients.