Developing intelligent inference by treating topological patterns in graphs and networks as constraints on a random graph ensemble.

Artificial intelligence of graphs

Capturing in simulations and mathematical form the surface structure of crystals and how they coalesce when heated but not melted.

At the surface of crystals

Applying spectral-like theories to the bipartite network of products and capabilities to find latent potential in countries and firms.

Building blocks of economic complexity

Demonstrating collective creativity: anonymous collaboration under constrained freedom that transcends the creativity of the individual.

Collective creativity

Reconstructing the 3d shape distribution of grains or other objects randomly packed together with access only to 2d slices  through them.

Deducing three dimensions from two

Developing new local and global measures for networks derived from social interactions to infer social structure, sentiment and behaviour.

Extracting meaning from social networks

Simulating the molecular structure of materials under pressures so extreme that we are not yet able to study them in the laboratory.

Extreme pressure surprises

Extending the theory of thermodynamics to account for quantum mechanics and the optimisation of energy transfer in quantum systems.

Foundations of quantum thermodynamics

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

Fundamental advances in machine learning

Understanding the dynamics of memristor networks, a new approach to low-power computation inspired by the structure of the brain.

How to remember currents and voltages

Taming limitations of general relativity, such as the big bang singularity, by formulating theories that admit bouncing or cyclic universes.

Implications of alternative universes

Creating discrete models of space that obey Euclid’s axioms over long distances and open up new possibilities for non-continuum physics. 

Is continuous space an illusion?

Developing a statistical physics model of recursive innovation in which technologies become the building blocks for new technologies.

Is technology a machine for creating itself?

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

Leaps in long-range connectivity

Developing a mathematical theory of how trust trees grow and how we can traverse them to exploit trust corridors in society for searching.

Making six degrees of separation a reality

Examining the effect of public opinion on stock market returns and harnessing social sentiment to make quantitative market predictions.

Markets and the mind

Background Innovation is to technology what evolution is to life: it is how technology improves and adapts to changing needs and environments. Yet despite advances in our understanding of evolution, what drives innovation has remained elusive. Is the innovation process essentially Darwinian—mutation, selection and inheritance—or are there fundamentally different rules at play?
Project We develop a simple mathematical model of innovation in which technologies are made up of components and new components become available over time. While the choices that organizations make play an important role in determining their success, this is countered by an intrinsic innovation rate specific to each domain. These opposing forces are reminiscent of nurture versus nature in human traits.
Consequences Mathematical models can help structure our empirical investigation of innovation in the real world, and better capture the intrinsic versus intervenable aspects of technological change. The ability to influence the rate of innovation in sectors such as energy production, software development and drug discovery will have enormous benefits for sustainability, technological progress and the health of the nation.

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

Mathematical structure of innovation

Background Modern cancer medicine can access vast amounts of patient-specific data, such as individual genomes, blood bio-markers and medical images. But it fails to use this information effectively. Analysing next-generation data with previous-generation tools causes avoidable patient harm and misses clues to new treatments. It also contributes to the high failure rates of clinical trials.
Project Most mathematical and machine learning methods for predicting cancer outcomes have a common Achilles heel. When the amount of information becomes too large, they confuse noise with real patterns and miss the wood for the trees. This breakdown is known as overfitting. We combat it by creating more powerful mathematical methods based on treating topological patterns in data as constraints on a random graph ensemble. These can be coded in algorithms that let multiple computers work in parallel.
Consequences More accurate clinical prediction reduces the likelihood of harmful, ineffective treatments and saves lives. Each doomed trial that gets stopped early by better analysis of preliminary data saves over £100 m for the public health services and the drug industry. And rescuing failing trials by identifying appropriate drug recipients broadens the range of available drugs.

Creating powerful mathematical methods for predicting cancer outcomes that can be coded in algorithms for fast parallel processing.

Mathematics for personalized medicine

Designing optimal self-similar structures for compact counter-current heat exchangers to reduce heating costs and greenhouse emissions. 

Moving mass while keeping heat stationary

Investigating the adverse effects of information asymmetry and deliberate errors in social media and the press and attempts to remedy them.

News and fake news in a connected world

Background Densely packed objects of a similar shape, from sedimentary rocks to foams to spheres, have intrigued mathematicians since Apollonius of Perga. The mechanical properties of these materials are determined by how big the gaps are between them—in particular, how close they are to their maximum random-packed density. Predicting this maximum, and how the material properties vary around this point, is a geometric and physical challenge.
Project We investigate the packing of spheres of the same size and mixtures of spheres with a range of sizes. While simulating packing is easy to program, the computational time scales poorly. To get around this obstacle, we develop new mathematical techniques for determining the properties of dense packings using a simplified 1d version. This powerful approach captures essential features of higher-dimensional systems, including an estimator of the maximum density.
Consequences Understanding how objects fit closely together under pressure or by design is at the foundation of materials science and branches of fluid dynamics. Predicting the behaviour of densely packed systems also sets the stage of reverse-engineering new composite materials to have desired properties.

Predicting the geometry of densely packed objects in the natural and man-made worlds, from rocks to foams to polydisperse spheres.

Packing things into a crowded place

Background In information theory, messages are sent as strings of characters which are subject to errors in transmission or copying. If we take the distance between two strings to be the number of spelling mistakes between them, then the space defined by this distance function is a Hamming graph over all possible strings. To correct for mistakes, a string and its variants must all code for the same message, and this bundle of strings forms a subgraph of a Hamming graph.
Project Biological information is carried by strings of nucleotides or amino acids. Point mutations correspond to traversing an edge in the associated Hamming graph. But not every genetic change affects an organism, so genetic space (the Hamming graph of all genotypes) is split up into subgraphs called neutral networks, each representing a distinct phenotype, or message. We investigate bounds on the spectral radius of subgraphs of Hamming graphs, which directly relate to the robustness and number of allowable phenotypes.
Consequences The theory of neutral evolution is only one application of the properties of subgraphs of Hamming graphs. Understanding how to pack Hamming balls into a Hamming graph is at the heart of many error correcting codes, and Hamming graphs feature in association schemes in coding theory.

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

Point, line, square, cube, tesseract

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

Reconstructing a credit network

Developing a new approach to resilience in which mistakes and unexpected events are mitigated by easy repairs rather than redundancy.

Repairable as an alternative to robust

Background Most simple rules give rise to simple behaviours. But occasionally they generate surprisingly rich and varied dynamics. Some simple rules, such as Conway’s game of life, are even capable of universal computation. While applying simple rules is easy, deducing the rules from observed behaviour is both important and usually hard, as evidenced by the deduction of the laws of physics through science.
Project We investigate different classes of simple rules for discrete dynamics. We help classify elementary cellular automata by deducing general characteristics of their behaviour. We study the self-assembly of polyominoes and the reverse engineering of structures made from them. In simple models of competition, mathematical models help us track the long-term fitness of the population.
Consequences Reverse engineering simple rules that generate desired dynamics can help us design efficient algorithms and low-cost methods of assembly. Understanding of which rules give rise to behaviour that is not fully stable nor entirely chaotic may help design systems for artificial life and synthetic biology.

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

Surprising results of simple rules

Forecasting the rate of technological progress by harnessing empirical regularities captured by Moore’s law and Wright’s law.

The future of technological progress

Background Networks represent interactions between a number of similar objects. While the increasing availability of data makes it easy to build networks, extracting meaning from network structure remains a challenge. Part of the problem is that mathematical methods for characterizing networks are primitive and sundry. While the theory of graphs—idealized networks containing internal symmetry—is more sophisticated, the mathematical foundations for both networks and graphs, and their relation, are underdeveloped.
Project We investigate new mathematical methods to describe the geometry and topology of networks and graphs, without regard to the kind of data from which they are derived. The techniques under development are broad and include applications and extensions of spectral graph theory, tailored graph ensembles, replica methods, community detection and combinatorics.
Consequences A more complete understanding of the structure of networks and graphs underpins many other research endeavours. Examples range from biomedical data for personalized medicine to predicting sentiment and behaviour from social interaction networks to the stability of interconnected financial institutions.

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

The structure of how things relate

Applying ideas from diversification and cascading failures to mitigate the propagation of risk across inter-connected institutions.

What to do when failure is contagious

Background A fractal has the property that any part of it, when magnified, resembles the whole. Remarkably, biology employs fractal designs to achieve high efficiency: spongy bone for stiffness and self-similar spiral shells for pressure resistance. But in the man-made world, fractal designs are rarely used for mechanical efficiency. Transmission towers and the branching columns of Gaudi’s cathedral Sagrada Familia hint at fractal design principles; but the small amount of hierarchy and the lack of quantitative understanding suggest these are heuristic.
Project We discover and investigate a new class of light and strong structures based on fractal principles with new strength-to-mass scaling laws. Our fractal structures offer extraordinary mechanical efficiency because the advantage achieved at each generation can be passed on to the next generation, and so on down to the smallest scale.
Consequences Fractal structures can extend the limits of architecture and civil engineering. They can increase the range and payload capabilities of drones and planes. In space, where launching structures is weight-critial, they offer new opportunities for building space stations and exploring space.

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