Heather Harrington Group

This project builds on the UK EPSRC 2018 grant "Application Driven Topological Data Analysis" (EP/R018472/1), which established the Centre for Topological Data Analysis (TDA). The centre fostered collaborations between mathematicians and scientists, leading to innovative tools and applications. Biology emerged as a key area, highlighting the need to integrate topology with machine learning and statistics for multiscale data analysis. The Oxford team, led by Professors Heather Harrington, Ulrike Tillmann, and Vidit Nanda, will partner with the Max Planck Institutes in Leipzig (MPI-MiS) and Dresden (MPI-CBG), renowned for their expertise in mathematics, data science, and systems biology. These German institutions, linked via ScaDS.AI, offer world-class facilities and expertise to drive the development of new theory, algorithms, and applications in TDA for biological systems.

This collaboration will focus on bridging the gap between Topological Data Analysis (TDA) and algebraic geometry, deep learning, biological data and software. TDA offers robust, shape-based insights into datasets and has already advanced areas like image processing, molecular stability, and network analysis. There are four main work packages proposed:

1. Algebraic Geometry of multiparameter persistent homology (MPH) modules (extending work on Harder-Narasimhan Filtrations and Groebner Bases computations of MPH)
2. Topological deep learning focusing on graph neural networks and differentiability in persistent homology (NA)
3. Applying mathematics created during the Center for TDA grant on biological datasets 
4. Creating code and software of mathematics created by the Centre for TDA.