The proposed project focuses on creating novel mathematical tools to analyse complex datasets in biology using topology, geometry, and machine learning. Building upon the success of the Centre for Topological Data Analysis (TDA), this new initiative aims to establish and strengthen collaborations with researchers in Saxony, Germany, specifically at the Max Planck Institute for Mathematics in the Sciences (MPI-MiS) and the Max Planck Institute of Molecular Cell Biology and Genetics (MPI-CBG). These institutes are closely associated with the Center for Scalable Data Analytics and Artificial Intelligence in Dresden/Leipzig (ScaDS.AI) and the Centre for Systems Biology Dresden (CSBD). These institutions are at the forefront of cutting-edge research in computational geometry, machine learning, and systems biology. The project's main objective is to advance topological data analysis (TDA) through the integration of data science techniques, algebraic and geometric methods, and topology. By working closely with experimentalists and modellers at MPI-CBG, the project aims to push the boundaries of TDA and apply it to biological systems, creating an iterative cycle between real-world applications and methodological advancements. This collaborative programme seeks to uncover shapes and structures within biological data, ultimately leading to groundbreaking insights in molecular biology. Biological datasets are often complex, noisy, and high-dimensional. Traditional methods, such as clustering or regression, have limitations when it comes to capturing the intricate shape of the data and cannot identify higher-order structures. TDA offers a unique approach to understanding multiscale systems by characterising and quantifying their inherent shape or structure. While TDA has already demonstrated its effectiveness in medicine, including applications in tumour-immune interactions and vascular networks--even led to the discovery of new subtypes of breast cancer-- the focus of this proposal is to expand the field of topological data analysis (TDA) to handle biological datasets encountered in (spatial) systems biology. Extending the mathematics in TDA will provide a versatile toolkit that can handle a wide range of data with multiple parameters. Through close collaboration with experimentalists and modellers at the Max Planck Institute of Molecular Cell Biology and Genetics, the project will have access to diverse biological datasets, enabling the team to push the theoretical, computational, and practical boundaries of TDA. The core focus of this programme is the expansion of TDA with other areas of mathematics and data science techniques. This multidisciplinary approach will create an iterative cycle between practical applications and methodological advancements. Through collaborations with leading researchers in applied algebraic geometry, differential geometry, and the AI/TDA interface, the project aims to develop new theoretical frameworks, case studies, and software. These resources will demonstrate the immediate applicability of topological and geometric tools for data analysis. In summary, the programme will contribute to the expansion of the UK topological data analysis (TDA) community and pave the way for future involvement in larger-scale projects. The proposed research project aims to develop innovative mathematical approaches for analysing spatial and temporal multi-parameter biological datasets. By harnessing the power of topology, geometry, and machine learning, the project seeks to unlock mechanistic insights and reveal structures within biological systems and revolutionise our understanding of biology. The collaboration with international research centres will maximise impact.

Context of the research: How does the brain work? This is one of the most fundamental questions a human being can ask. There are many sides to this question, philosophical as well as biological. My project focuses on particular proteins, the so-called Unc5 receptors, that are located on the surface of cells. Much like a navigation system, these proteins direct brain cells along the right paths and help them connect to each other. This gives rise to the neuronal networks that underlie the functions of our brain. These proteins also direct other cells in our body. For example, Unc5 receptors control the development of blood vessels is. They do this by regulating those cells that are responsible for growing new blood vessels. Some aspects of how Unc5 receptors guide and direct neurons and other cells are understood. Just like a navigator would do, they recognise specific signals (in this case other compatible molecules, referred to as ligands) in their environment. The recognition event involves physical interaction of the Unc5 receptor and ligands. Most ligands of Unc5 receptors also interact with other molecules. The result is a network of interactions, with Unc5 receptors constituting major nodal points. How Unc5 receptors guide cells depends on the surrounding molecules present in its environment. Aims and objectives: How do Unc5 receptors interact with specific ligands, and how do these ligands interact with further molecules? I will study these interactions with high resolution methods to understand the molecular details that define them. Following this, I will use this information to understand the specific biological functions of each of the interactions. For example, I will test what happens when Unc5 no longer recognises one of its ligands. WIll this affect specific cell properties, leaving others unaffected? Will it affect the way that Unc5 moves around on the surface of cells? My experiments will answer these questions by zooming in to the cell surface, using advanced microscopy methods, and by looking at the behaviour of multiple cells, using cell biology assays. Potential applications and benefits: Unc5 receptors are involved in fundamental processes including brain development and blood vessel formation. My results will provide an essential piece to the puzzle of how these processes are regulated. Given this information, could potentially be used to design new drugs related to diseases of the vascular and neuronal systems. Unc5 receptors also play important roles in human cancers. In most cases, cancer cells have a reduced ability to use Unc5 receptors for their guidance. By revealing how Unc5 receptors control cell behaviour, I will also generate knowledge for the specific treatment of cancers with Unc5 receptor-related defects. All information generated by this project will be published and made available for further exploitation. During the course of the project, I will engage in public outreach activities to communicate aspects of my work to a broad community.

Coronary heart disease (CHD) is the UK's single biggest killer, leading to the death of nearly one in six men and one in ten women in the country. Approximately 2.3 million people in the UK are living with CHD. In 2009, the healthcare costs of this (medications, accident & emergency, primary/outpatient/inpatient care) amounted to nearly £2 billion, whilst informal care costs and productivity losses due to mortality and morbidity amounted to an additional £5 billion. The incidence of CHD correlates strongly with deprivation - mortality rates are almost twice as high in deprived areas compared with affluent areas. CHD occurs when the blood vessels of the heart (coronary arteries) become narrowed by fatty material (atheroma), reducing blood flow to the heart muscle. If the atheroma breaks off it can lead to the formation of a blood clot that could potentially block the coronary artery, cutting off the oxygen-rich blood supply to a part of the heart muscle and risking irreversible damage. The death of this part of the heart muscle is called a heart attack, also known as myocardial infarction (MI). Whilst the heart muscle that dies forms a scar, the surviving heart muscle around the scar undergoes numerous maladaptive changes that can dictate the outcome for the patient. Collectively, these changes are called pathological cardiac remodelling and can lead to a dilated heart that is unable to pump efficiently. A significant proportion of post-MI patients undergo progressive worsening of pathological cardiac remodelling and develop heart failure (HF), meaning that the heart can no longer pump enough blood to meet the needs of the body. Although improving, population morbidity and mortality remain high and new treatments are urgently required for patients with MI and HF. Runx1 is a protein that regulates the activity and expression of a number of other proteins important for the normal functioning of the body. Recently it has been discovered in patients with MI that increased levels of Runx1 are produced in the surviving heart muscle around the scar. Until now the function of Runx1 in the heart remained unknown. We have induced MI in a genetically modified mouse with selective reduction of Runx1 in the heart muscle cells (cardiomyocytes). Our experiments demonstrate compelling evidence that Runx1 is linked to how well the heart is able to pump post-MI. This proposal aims to uncover the mechanisms involved in this important link and in doing so drive forward future studies to determine the therapeutic potential of this novel target.

The soil that makes up the earth's substrate is an ever-changing assemblage of organic matter, minerals, organisms, gases, and liquids. Plants, animals, and humans in a host of ways mine this substrate to make life possible and use it as a container for things both precious and toxic. In this central function of earthly survival, soil's composition is both an archive of past life as well as the planet's largest carbon sink, capturing within it emissions from centuries ago. It is, in its very essence, a historical record that shapes our collective future. "Stories from the Substrate" reflects on the historical composition of soil by using it as a medium for engaging with and narrating East African history and as a point of view for considering the epoch of the Anthropocene. This project begins with both archival and fieldwork in East Africa (Tanzania, Uganda, and Kenya) that will form the basis of a collaborative, interactive soil map website and scholarly monograph that highlights how East Africans have relied on and shaped soil with the help of plants and animals. Previous scholarship on soil in East Africa has focused almost exclusively on colonial development projects and campaigns against erosion, this project takes a multispecies approach and centres East African epistemologies of soil to view it as not just a medium of agriculture, but as a building material, a source for mining mineral salts, the haven of "good" and "bad" microbial life, a sink for pollutants, a metaphor of home and belonging, and a valuable commodity of both local and global trade. Chronicling a history of the region from the soil seeks to set aside the prevailing themes of extant regional histories that narrate from political events and the ethnic and colonial logics of the archive. By considering the region through its substrate, this project hews close to the everyday and the often unseen or overlooked ways that people, animals, and their environments make one another and shape history. While each case study takes a particular place as its starting point, several themes will run throughout, assembling a picture of landscape, ecology, human health and labor in the region. In bringing together a variety of attachments to the soil in East Africa, this project aims to contribute to Black Ecologies, an interdisciplinary field seeking to examine the relationship between Black people and their environments. This work highlights the harm and toxicity that comprise many of these environments due to centuries of oppression and dispossession, but it also examines modes of knowing and being with nature that have gone under-explored. From this initial research, the project then shifts to collaborate with a wider research community to "Think with the Substrate". Bringing together scholars who work on histories of substrate materials, we will consider how humans have variably remade and relied on this middle layer between the subterranean and terrestrial. This inter-disciplinary project, in the spirit of other work on the Anthropocene such as the recent Plantationocene project, aims to engage with an unwieldy chunk of human history through a unifying object of inquiry. While the Anthropocene has been defined as the epoch when human impact has been the most consequential force shaping the earth, many scholars have critiqued the totalizing concept for eliding the fact that all societies have not had equal impact. To think about the Anthropocene "with the substrate" is not to assemble a list of different uses and interventions into the ground. It is a project of using historical engagements with substrate matter to reveal variable social relations, conceptions of nature, and engagements with nonhuman animals. These stories will no doubt chronicle the rise of global capitalism, but it will also capture the many alternative ontological and epistemological worlds that are sometimes forgotten and overshadowed in blunt narratives of capitalism and the Anthropocene.

A huge number of important and challenging applications in operational research are governed by optimization problems. One crucial class of these problems, which has significant applicability to real-world processes, is that of partial differential equation (PDE)-constrained optimization, where an optimization problem is solved with PDEs acting as constraints. To provide one illustration, such formulations arise widely in image processing applications: this produces a crucial link to scientific and technological challenges from far-and-wide, for example determining the health of complex human organs such as the brain, exploring underground geological structures, and enabling Google cars to function without a human driver by assessing traffic situations. The possibilities offered by PDE-constrained optimization problems are immense, and consequently they have recently attracted tremendous interest from researchers in mathematics, as well as applied scientists more widely. These formulations may also be used to describe processes in fields as wide-ranging as fluid dynamics, chemical and biological mechanisms, other image processing problems such as medical imaging, weather forecasting, problems in financial markets and option pricing, electromagnetic inverse problems, and many other applications of importance. The study of these problems is therefore a cutting-edge research area, and one which can forge a huge advance in the fields of operational research and optimization. There has been much theoretical work undertaken on these problems, however the construction of strategies for solving these optimization problems numerically is a relatively recent development. In this project I wish to build fast and effective solvers for the matrix systems involved (these systems contain all of the equations which arise from the problem). The solvers are coupled with the development of a powerful 'preconditioner' (the idea of which is to approximate the corresponding matrix accurately in some sense, but in a way that is cheap to apply on a computer). Carrying this out is a highly non-trivial challenge for many reasons, specifically that it is often infeasible to store the matrix in its entirety at any one time, it is very difficult to build an approximation that captures the properties of the matrix in an effective way and is also cheap to apply, it is frequently necessary to build solvers which are parallelizable (meaning that computations may be carried out on many different computers at one time), and one is often required to carry out the expensive process of re-computing many different matrices. The aim of this project is to build powerful solvers, which counteract the above issues, for PDE-constrained optimization problems of significant real-world and industrial value. I will consider four specific applications: optimal control problems arising from medical imaging applications, PDE-constrained optimization formulations of image processing problems, models for the optimal control of fluid flow, and control problems arising in chemical and biological processes. I will consider problem statements that have the maximum practical potential, and generate viable, fast and effective solution strategies for these problems.