Computational topology and geometry for systems biology

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.