Mathematics that unifies

Understanding the relationships that bind different branches of mathematics and physics, and building the overarching theories they demand.

The unity of mathematics is a discovery rather than an assumption. Subjects that developed independently have often proved to share the same structures and phenomena, and when that happens each gains a vocabulary it lacked; what resisted proof in one can become almost easy in the other.

The deepest correspondences of this kind are still being worked out. The Langlands programme relates number theory to analysis and geometry; monstrous moonshine tied the largest sporadic simple group to modular functions; one short list of symmetries classifies objects in fields that otherwise have nothing in common. Physics supplies more, since a duality between two quantum field theories is a mathematical identity waiting to be proved. We pursue these correspondences and look for others.

Some of our own work runs in the other direction, taking classical questions and attacking them with imported tools. We study how the roots of equations permute, a question as old as Galois, using the topology of braids and the geometry of polytopes. We study symmetry in infinite dimensions, where the familiar theory breaks down, and find representation theory reaching into number theory. We work on the complex geometries that string theory needs and mathematics has yet to classify, on the number theory that appears unbidden inside quantum field theory, and on the boundary between the discrete and the continuous, where combinatorial graphs acquire the geometry of smooth spaces. We study not only the interplay between different branches of mathematics — algebra, geometry, and number theory — but also how the same structures appear in other fields. That is why we are always looking not only to strengthen but to broaden and complement our thematic coverage.

There is another reason to range widely. Machines are changing what a mathematician is for. As the technical work becomes cheaper, breadth and judgement become scarce. We use machines to find patterns in mathematical data, and then do the harder part: turning a pattern into a theorem.

Judgement of that kind is learned by proximity. At the London Institute, mathematics is not an adjunct to the science but part of it. Our mathematicians sit among physicists, which keeps both abreast of the other's tools and occasionally reveals that they were working on the same problem.

  • 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.