In this proposal we seek to establish a Centre for Doctoral Training in Mathematical Analysis and its Applications. The main purpose of the centre is to train upwards of 60 new PhD students in this area over several years, and in doing so address the proven skills need for highly-trained researchers in this area. The centre will be founded on rigorous mathematical analysis and its applications, with a strong focus on nonlinear partial differential equations, under three broad themes: theoretical, stochastic and numerical. Its scope includes harmonic analysis, mathematical analysis of large-scale discrete structures, applied analysis, dynamical systems, stochastic analysis, financial mathematics, applied probability and computational mathematics. There will be a special emphasis on the connections and interactions between these areas, and their applications, and active collaboration with industry -- in the formulation of student projects, in mentoring PhD students, in developing work placements for the students, and more broadly in two-way knowledge exchange -- will be a key feature of this CDT. The need for mathematicians trained in this centre is manifest in real-world phenomena where cutting-edge differential and/or stochastic models are needed, for example in oil extraction, in power grid renewable energy strategies, in finance processes, in ecological impacts of climate change, and in procceses inside the brain. We shall provide a flow of such PhDs with multiple skill sets and the ability to deal with the sophisticated challenges arising in mathematical modelling: they will be able to both analyse and implement and will be in a position to mount rapid and agile responses to current and future challenges. MIGSAA training will be constructed on two main pillars: outstanding academic provision and early-stage career development. These are underpinned by development of a strong sense of cohort. As a fully integrated joint 4 year PhD programme, it will offer much more than the standard UK Mathematics PhD model. Initial academic training will build upon the firm foundation provided by SMSTC, and will feature a strong taught and assessed component. Students will also complete two assessed projects during their first year. It is intended that the two projects will span the areas of MIGSAA, and will provide a firm basis for choosing the topic for the main PhD dissertation towards the end of Year 1. The main PhD project, which will be challenging and substantial, will lead to original research findings at the cutting edge of mathematical endeavour. A tranche of specially designed, more advanced courses will be available for students in Year 2 and beyond so that students will continue to consolidate the available knowledge and expertise as they continue on their main research project. Students will be further supported by a carefully-planned programme of complementary research activities. There will be a strong focus on early-stage career development in its broadest sense. This will include training in public engagement, effective collaboration, understanding the impact agenda and responsible innovation, leadership, outreach, media training, engagement with industry and networking. Central to our vision for MIGSAA is the sense of cohort which it will foster. Beginning with the annual induction event, the cohort environment already offered by participation in SMSTC will be significantly enhanced by provision of contiguous office accommodation and dedicated common spaces, with Year 1 students collocated at ICMS in central Edinburgh. For the later years cohort activities include: physical attendance at higher level courses, research seminars and generic skills courses; active working groups encouraging peer-to-peer learning; annual residential symposia.

Metals have a finite supply, thus metal scarcity and supply security have become worldwide issues. We have to ensure that we do not drain important resources by prioritizing the desires of the present over the needs of the future. To solve such a global challenge we need to move to a circular, more sustainable economy where we use the resources we have more wisely. One of the founding principles of a circular economy is that waste is an unused feedstock; that organic and inorganic components can be engineered to fit within a materials cycle, by the design, engineering and re-purposing of waste streams. In this fellowship I propose to design and engineer bacteria to repurpose our waste streams for us. I plan to use the new tools and techniques provided by advances in biology to engineer a microbe with the ability to upcycle critical metal ions from waste streams into high value nanoparticles. Certain bacteria have the ability to reduce metal cations and form precipitates of zero-valence, pure metals, as part of their survival mechanism to defend against toxic levels of metal cations. I will adopt the modular approach used in Synthetic Biology alongside iterative design, build and test cycles in order to enhance, manipulate and standardise the biomanufacture of these nanosize precipitates as high value products. With training in life cycle assessment, I will determine the financial benefits for business of adopting biological waste treatment methods with high value resource recovery and I will provide biogenic material to other researchers (academic and industrial) free of charge to encourage user pull for the technology.

Over the last three decades, our lives have been revolutionized by the availability of inexpensive CMOS-based CCD cameras whose ubiquitous nature has changed key aspects of security, communications, data handling, healthcare, commerce and leisure for almost all sections of society, regardless of wealth or geographical location. For example, it is estimated that over one half of all adults in the UK own a smartphone with imaging/video capability - a statistic considered unthinkable less than 10 years ago. The next revolution in imaging will almost certainly be spearheaded by sparse photon and three dimensional imaging, ultimately using the effects of quantum entanglement. Such a revolution will necessarily require fast timing of the single-photon detection, in the form of arrayed detectors or single-pixel cameras. The use of fast timing will permit effective time-of-flight based depth profiling at remote distances, and the effects of quantum entanglement could be utilised effectively in critical niche examples, such as imaging below the diffraction limit, wavelength transmutation or quantum secure imaging. These revolutionary changes represent a paradigm shift in terms of functionality, but present significant challenges in algorithm development and data processing, as well as data fusion with other imaging platforms, for example multispectral and regular video. This Fellowship will allow me to bridge the gap between the enabling quantum technology and the image processing community in order to improve the scope and overall performance of next generation imaging systems based on quantum technology.

This project brings atomic physics and cryogenic research together to establish the Geonium Chip as a pioneering, practical quantum technology. The chip's core element is the Coplanar Waveguide Penning trap, conceived and developed by the PI at the University of Sussex. It has a broad range of applications, including quantum computation and metrology, mass spectrometry and the physics of strongly correlated electrons. The project will focus on one concrete goal: the implementation of a broadband, tuneable, quantum non-demolition detector of single microwave photons. An efficient detector of single microwave (MW) photons is a fundamental tool still missing in quantum technology. Such detectors are essential for determining the quantum state of GHz radiation fields and thus vital for quantum communication/information applications with microwaves. While several alternatives based upon super- and semiconductor technologies are being developed, the first observations of individual microwave photons employed a trapped electron as transducer. We will develop the electrons as functional sensors, with unique capabilities for the observation and coherent manipulation of quantum MW fields, initially within the frequency range 3-60 GHz. Cryogenic Penning traps permit an accurate control of the dynamics of a trapped electron, at the level of inducing and observing quantum jumps between its Fock-states. The rest gas pressure in cryogenic vacuum chambers amounts to 10^(-16) mbar, allowing for a very prolonged capture (months) of the particles. The continuous Stern-Gerlach effect permits the detection and manipulation of the electron's spin, while the Purcell effect enhances the coherence time of its quantum state. Hence, cryogenic Penning traps are excellent quantum laboratories and trapped electrons have been proposed for implementing a quantum processor. A single electron in a Penning trap is also known as a geonium atom, as coined by the 1989 Nobel laureate Hans Dehmelt. It is outstanding for ultra-high precision metrology. Examples are the free electron's g-factor, measured with 10^(-13) relative uncertainty and the proton-to-electron mass ratio with 10^(-10). These, and other advanced Penning trap experiments, invariably employ a big, "room-size", superconducting solenoid. We propose to radically change that concept: integrating the trap and the magnetic field source in a single, scalable (2nd generation) Geonium Chip. Within this project we will develop the 2nd generation Geonium Chip into a practical quantum technology. A functional microwave photon detector must provide the following critical features: a) A tuneable, broadband detection range b) Quantum Non Demolition detection c) High quantum efficiency d) Coherent connectivity to other systems e) Scalability and a cost as low as possible. The currently most advanced Penning traps use superconducting solenoids, requiring highly specialised engineers to tune the trapping magnetic field -and hence the detection range-. Moreover, cooling to 100 mK or lower is done with extremely expensive (> £ 350 000) dilution refrigerators, difficult to install and operate. This contrasts radically with our novel Geonium platform, which will eliminate solenoid and dilution refrigerator altogether. With this pioneering approach, we will reduce the cost and complexity, enabling our chip Penning trap as a useful quantum 2.0 technology, particularly as a single microwave photon detector.

The vision for Edinburgh's Centre for Mammalian Synthetic Biology (SynthSys-Mammalian) is to pioneer the development of the underpinning tools and technologies needed to implement engineering principles and realise the full potential of synthetic biology in mammalian systems. We have an ambitious plan to build in-house expertise in cell engineering tool generation, whole-cell modelling, computer-assisted design and construction of DNA and high-throughput phenotyping to enable synthetic biology in mammalian systems for multiple applications. In this way we will not only advance basic understanding of mammalian biology but also generate tools and technologies for near-term commercial exploitation in areas such as the pharmaceutical and drug testing industries, biosensing cell lines sensing disease biomarkers for diagnositics, novel therapeutics, production of protein based drugs e.g. antibodies and also programming stem cell development and differentiation for regenerative medicine applications. In parallel we will develop and implement new understanding of the social and economic impact of this far-reaching technology to ensure its benefits to society.