Persistent Homology for Inferencing on Cancer
Anthea Monod (Imperial College of London)
In this talk, I will introduce persistent homology as a tool for data analysis, built from underlying principles of algebraic topology. I will then show how it can be applied to cancer images as a powerful tool to inference on and uncover insights on two vastly different cancer types: Glioblastoma multiforme (brain cancer) and acute myeloid leukaemia (blood cancer). The talk is based on joint works with my students and collaborators, Antoniana Batsivari, Dominique Bonnet, Andrew X. Chen, Lorin Crawford, Sayan Mukherjee, Raul Rabadan, Anna Song, and Qiquan Wang.
Topological Data Analysis and the Nature of Nuclear Matter
Jeffrey Giansiracusa (Durham University)
Modern physics describes the world in terms of quantum field theories, and the inside of protons and neutrons is described by one particular theory called quantum chromodynamics (QCD), which says that nuclear matter is made out of particles called quarks and gluons. It's a fabulously successful theory, except for one annoying detail: nobody has ever seen a quark or gluon! This paradox is explained away by a hypothetical mechanism called 'confinement' that nobody really understands. We see clear evidence of confinement in experiments, and we can watch it happen in Monte Carlo simulations on supercomputers. These simulations produce vast amounts of data, and so understanding confinement should be a data science problem. I'll explain how persistent homology might just be the tool we need for this problem.
Equitability and Cluster Synchronisation in Multiplex and Higher-order Networks
Ruben Sanchez Garcia (University of Southampton)
Cluster synchronisation is a key phenomenon observed in networks of coupled dynamical units. Its presence has been linked first to symmetry then more generally to equability of the underlying pattern of interactions between the dynamical units. In this talk, I will describe recent results that explain the relation between equitability and cluster synchronisation on a very general dynamical system which allows multi-layer and higher-order interactions. This is joint work with Kirill Kovalenko, Gonzalo Contreras-Aso, Charo del Genio, and Stefano Boccaletti.
From Collapses to Discrete Morse Theory: Algorithms for Shape Simplification
Ximena Fernandez (City St George’s University of London)
Classical collapse-based methods for simplifying simplicial complexes are powerful but limited, as many spaces (like triangulated manifolds) do not admit collapses. Discrete Morse theory, on the other hand, is a more flexible tool but often obscures the combinatorial structure of the simplified complex.
In this talk, I will reframe discrete Morse theory as a direct generalization of collapses and present an explicit, iterative construction of the reduced CW-complex. I'll also explore special cases where this process produces a reduced complex with a clear combinatorial description. This perspective offers new algorithmic tools for long-standing problems in computational topology and combinatorial group theory. This includes recent progress on the Andrews–Curtis conjecture and algorithmic approaches to simplicial-complex simplification beyond collapse-based methods.