Number Theory is the study of the integers and their arithmetic applications. While problems in Number Theory can be easy to state, their solutions often become extremely intricate. For example, Fermat's Last Theorem - was formulated in the 17th century, yet only resolved in the 1990's. A fundamental approach in mathematics is to transform a seemingly difficult problem from one area to another, where it becomes tractable or even obvious. A famous example, is Wiles' proof of Fermat's Last Theorem; the key change in perspective transforming a problem about certain arithmetic objects (Galois representations of elliptic curves) into one about analytic objects (modular forms). This correspondence established by Wiles completing the proof of Fermat's Last Theorem is a very special case of a broad web of predicted correspondences and connections between analysis and arithmetic, collectively known as the Langlands Programme. The Local Langlands Programme is the specialization of the Langlands Programme at a prime number, and this is where the bulk of the research of our project takes place. The language of the Local Langlands Programme is in a branch of Algebra called Representation Theory, which deals with symmetries of spaces. The Local Langlands Programme is a deep statement that certain fundamental symmetries of finite dimensional spaces which arise in Number Theory can be understood in terms of completely different symmetries of infinite dimensional spaces, and conversely. The finite and infinite dimensional spaces considered are built on top of the complex numbers. A natural question now arises, why the complex numbers? Is there a more fundamental arithmetic connection hiding behind this? In this project, using explicit constructions of representations, we study integral structures in the Local Langlands Programme and their relation.

Hospital neurology and neurophysiology services are increasingly overwhelmed. With a growing and ageing population, the incidence of many brain conditions (such as dementia and epilepsy) are rapidly increasing. Compounded by the COVID-19 pandemic, there are now over 10,000 people in the UK waiting more than a year for an appointment with a neurologist. Things must change! The purpose of our Network is to address these challenges through the development of new technologies that enable diagnosis and management in the community. These services could be provided in a community diagnostic hub, by high-street healthcare professionals, in a GP surgery, in a mobile unit or even in the home environment. Our focus will be on new digital solutions built around neural interfacing, signal processing, machine learning and mathematical modelling. We will work closely with partners developing technologies for measuring brain, eye, spinal, and peripheral nerve activity using wearable technology and minimally invasive devices. Collectively, this will contribute to a significant increase in capacity that will augment the expertise provided in neurology services. To achieve this, we will build a network of partners with backgrounds spanning academia, industry, hospitals and GP surgeries, charities and policy makers. Crucially we will ensure that people with lived experience of neurological conditions are at the heart of our network. Their experience will inform debate and shape our research priorities, ensuring feasibility and acceptability of emerging technologies. We will empower people from different backgrounds and career stages to work together on challenging problems whose solutions will lead to societal benefit. To enable this we plan a suite of activities built around the principles of connect, communicate and collaborate. To connect people we will build a website and social media presence, create a public representation group and build new parnterships. We will establish a mentorship scheme and post opportunities for people at different career stages to undertake secondments with partner organisations. To facilitate communication, we will engage with stakeholders including the public, people with neurological conditions, healthcare providers and policy makers. We will host workshops on emerging areas of interest, as well as an annual conference to celebrate findings from across the network. To enable collaboration we will host events including stake-holder led study groups, sandpits and research incubators: where teams of partners will work collaboratively in a facilitated environment, conducting feasibility studies over 6-9 months.