Life, learning and emergence

Developing mathematical foundations for order in matter, organisation in living things, machine intelligence, and other emergent phenomena.

Many of nature's most striking phenomena are emergent: their behaviour belongs to a system's organisation rather than its substance. Statistical physics makes this precise, and its reach extends from electrons in a solid to the origin of living order and to the nature of learning.

Perhaps the most intriguing emergent phenomenon is life itself. Darwin explained how life evolves, but not how it began. We investigate the thermodynamic basis of self-replication and adaptation, and how flows of energy and matter keep organisms in non-equilibrium states. Which of these principles extend beyond biology, can they guide the creation of artificial life, and can evolution be made predictive?

Living systems also process information. The genome is less a blueprint than a program, run on gene regulatory networks. We study how these networks compute and what their architecture rules out, seeking a theory of cell programming precise enough to steer a cell to a chosen state.

Nature found its computers; we have to design ours. How do we make intelligent machines? We seek mathematical principles for how learning, memory and generalisation arise in biological and artificial networks, and for the part played by causal reasoning, modularity and representations of the environment. AI-assisted discovery treats machine intelligence as an instrument; here it is an object of theory.

Emergence is not confined to things that are alive or that learn. In condensed matter, interactions lead to emergence of new (quasi)particles that are responsible for the remarkable properties of materials. Among the best-known examples are superconductivity, underpinned by Cooper pairing of electrons, and the fractional quantum Hall effect, in which magnetic fields combine to produce excitations carrying a fraction of the electron's charge. But even spacetime — the stage on which these collective phenomena unfold — may itself be an emergent feature of quantum dynamics, rather than a fundamental ingredient of nature. We study phase transitions and the emergence of order in classical and quantum systems, at equilibrium and far from it. Quantum field theory is the common language here, as it is for the fundamental forces, and ideas have long travelled both ways.

We are drawn, too, to questions still taking shape — quantitative measures of selection and assembly, open-ended evolution, collective intelligence and the emergence of autonomy and agency. Our aim is a mathematics of emergence: laws that turn on organisation rather than substance, that hold for matter, organisms and machines alike, and that do not dissolve when a system is taken apart.

  • Machine learning the regulatory structure of cell states

    Learning the cell-state space

    Building machine learning models that mimic the behaviour of cells in silico to improve the prediction of genes for cell programming.

  • Mathematics of immortality

    Deriving the mortality equation, which governs the dynamics of an ageing population, and solving it to crack the evolutionary origin of ageing.

  • Informative experiment design

    Developing the mathematical structure of experiments using information theory and combinatorics to speed up the discovery of new cell types.

  • Theory of genetic computation

    Understanding genetic computation using regulatory motifs, a new kind of structural and functional building block of gene regulatory networks.

  • Learning the universe

    Using machine learning to search the vast space of 10-dimensional geometries for ones that predict the Standard Model from string theory.

  • Recursively divisible numbers

    Generalizing the divisor function to find a new kind of number that can be recursively divided into parts, for use in design and technology.

  • Fundamental advances in AI

    Developing radical new approaches to inference and automated decision making using advances in quantum information and statistical physics.

  • The structure of innovation

    Creating a mathematical model of combinatorial innovation to understand how innovation rates can be influenced as components are acquired.

  • Bootstrap percolation

    Advancing the mathematical theory of bootstrap percolation, where active cells on a lattice with few active neighbours cease to be active.

  • Surprises from simple rules

    Understanding complex dynamical behaviours generated by simple rules, such as cellular automata, polyominoes and models of competition.

  • Fractal structures

    Using fractal, or self-similar, patterns to design the lightest possible load-bearing structures with new strength-to-mass scaling laws.

  • Structure of how things relate

    Creating mathematical tools for characterizing the structure of ideal graphs and irregular networks, and the behaviour of processes on them.

  • Reconstructing credit networks

    Using ideas from statistical physics to reconstruct the average properties of financial networks from partial sets of information.

  • Information thermodynamics

    Understanding the physical nature of information and how it relates to energy transfer and new technologies that make use of these insights.

  • Spectre of hypercubes

    Exploring the spectral properties of subgraphs of the hypercube and Hamming graphs for insights into coding theory and models of evolution.

  • Puzzles in packing

    Predicting the geometry and behaviour of densely packed objects from first principles, from spheres to polydisperse spheres to cells.

  • Is continuous space illusory?

    Creating discrete models of space and spacetime that appear continuous over long lengths and set the stage for non-continuum physics.

  • Remembering to learn

    Understanding the dynamics of networks of memristors, a new paradigm for low-power computation inspired by the structure of the brain.

  • Intelligence of graphs

    Predicting the behaviour of graphs and processes on them by treating topological patterns as constraints on a random graph ensemble.