Financial networks are like sprawling extended families. Their components are all interconnected, but they aren’t obviously dependent on each other until something goes wrong. Distress or failure in one area can propagate through the network via the very same avenues that made it seemingly strong and diverse.
This project captures the intricacies of financial systems using complex networks. By modelling the channels through which risk and stress in the network is communicated, it is possible to predict how different reactions to shocks will affect the network as a whole. Focusing on the microscopic theory of these networks, the DebtRank algorithm is studied as a tool for predicting the likelihood of systemic failure in a network, and appropriate immunisation strategies for limiting the spread of distress are proposed. 
The sentiment of borrowers and lenders in a financial network is what drives markets to success, but also to ruin. Zooming in on the non-linear links between these players quantifies the likely reactions to key events, and predicts how distress will spread. Modelling these weaknesses will enable strategies to limit system-wide catastrophic failure to be tuned, and ultimately prevent future financial crashes.

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

Systemic risk

A fix for failure that’s contagious

The secrets of untapped properties of materials are often hidden in their microstructures. Being able to push a material to its limit, be that in terms of temperature, pressure or loading, is vital for understanding and predicting its limiting behaviour. However, this can be difficult and expensive to do experimentally.
Instead, in this project, we use numerical techniques to study the microstructural properties of both known and novel materials. A Monte Carlo based model describes the growth of particles in metallic and ceramic powders; the semi-metallic properties of newly proposed two-dimensional borane are put to the test with simulations; and new low and high energy grain boundaries are found in graphene and strontium titanate. The results are a testament to taking an analytical approach to a problem that might otherwise have to be tackled by experimental trial-and-error.
Computational experiments are able to test and analyse the behaviour of materials under extreme conditions that may not be feasible or repeatable in the laboratory. Such fine-grained understanding of the micro-evolution of structures will enable the design and use of known materials to be optimised, and provide channels for the discovery of next-generation smart materials.

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

At the edge of crystals

At the surface of crystals

The process of economic development is difficult to quantify. Standard economic theories predict that a high level of specification in a country’s exports largely defines an economy’s growth. However, new network models of the links between countries and their exports suggest otherwise.
This project finds that more sophisticated measures of countries and their products point to a different defining factor of economic growth. Modelling a country’s competitiveness and fitness alongside the complexity of their products in a bipartite network structure shows that it is the diversity of a country’s exports which is the biggest indicator of economic development. Developed countries export a large variety of products, both simple and complex, whereas less developed countries export the modal exports only. In addition, the dynamics of the systems can exhibit behaviour which cannot be analysed using traditional regression methods.
Our improved non-linear modelling of bipartite networks has allowed us to form a deeper understanding of the links between countries and their products. This has in turn uncovered hidden potential for economic development and growth and expedited the need for more sophisticated metrics and statistical analysis tools.

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

Economic complexity

Building blocks of economic complexity

Homer’s Iliad and Wikipedia. One, the first great work in Western literature; the other, the world’s most-used work of reference. The link? Both are the products of collective creativity. The Iliad was created over generations by a series of oral poets, each of whom improved on the previous version, deleting some passages and adding others. Meanwhile, Wikipedia relies on contributions from the public, on the same basis of deleting and inserting. Shared knowledge curation efforts and open source software suggest that explicit cooperation may not be necessary for integrating multiple collaborators in their efforts to achieve a complex goal.
In this project we ask whether anonymous collaboration with constrained freedom can give rise to the spontaneous display of creativity within a non-cooperating group, greater than any one person could achieve alone. By studying simple creativity tasks and the dynamics of games, we devise a theoretical model for collective creativity and extrapolate key design principles for platforms capable of achieving it.
Collective creativity could profoundly extend the limits of human creativity in areas such as scientific collaboration, coding, creative writing and illustration.

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

Collective creativity

Dynamics of collective creativity

Pushing the boundaries of physics often requires exploring extreme conditions. Sometimes those conditions are too expensive or time-consuming to conjure up in the laboratory, and sometimes we simply haven’t worked out how to create the required environment in a controlled way yet. In these circumstances, theory and simulations come to the rescue.
This project uses theoretical and numerical techniques to model materials under extreme pressures. This approach enables us to study the most extreme processes in nature from the comfort of our own laptops. For example, simulating the structures and stability of calcium and magnesium carbonates at very high pressures gives insight as to what’s going on inside the Earth’s mantle and the deep carbon cycle. Furthermore, we investigate how a newly proposed theoretical material called two-dimensional borane could be synthesized if extremely high pressures are applied. 
Despite an apparent experimental ceiling as to the physical conditions that we are able to probe, it is in fact possible to make progress without entering the laboratory. This theoretical approach to materials physics answers questions about unreachable environments and provides clues to the existence of new and useful substances.

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

The design of efficient exchange devices is an important problem in biology and engineering. In nature, lungs, leaf venation and blood circulation networks have evolved to meet multiple physiological needs. In industrial settings a variety of heat exchangers are employed, such as plate, coil and counter-current devices. A distinctive feature of the biological examples is their fractal structure, with branching and usually anastomosing geometries. The fractal design provides a large surface for exchange but occupies a compact volume.
In this project we design optimal heat exchangers for man-made use that incorporate the design efficiencies of their biological counterparts. We develop a theory that links efficiency to the geometry of the exchange surface and supply network. By crumpling the exchange area into a fractal surface, our designs can be optimized for volume as well as performance. They are especially suited to being made by 3D printing because they are not load bearing.
Optimal heat exchangers increase the efficiency of industrial processes and reduce the cost and size of passive heat exchangers in houses. The underlying theory also provides insights into respiration, which may help develop diagnostic tests for breathing disorders.

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

Fractal heat exchange

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 hint at fractal design principles; but the small amount of hierarchy and lack of quantitative understanding suggest these are heuristic.
In this project we mathematically prove the existence of 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. We also quantify the role of imperfections in creating potential failure modes.
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.

Fractal structures

The quantity of data in our increasingly digitised world is now far too great to be processed by hand. This has led to the rebirth of artificial neural networks as a framework for automated data processing. Whilst these networks can deal with an unprecedented amount of information, their capabilities and limitations are not yet fully understood.
This project investigates the inner workings of both classical and quantum neural networks, in an attempt to unpick their “black box” natures. Analytic methods shed light on the functions computed deep within two different types of classical neural networks. We show that the simpler set-up computes the same range of functions as the more complex one, meaning that resources can be saved. Furthermore, we pose a way of generalising classical neural networks so that they can deal with quantum inputs, and also maintain the reversible nature of a quantum signal’s progression through a network. 
Developing our understanding of classical and quantum neural networks simultaneously allows for the two fields to grow together. They are having profound effects on the way society runs through their use in energy harvesting, autonomous vehicles and social media, but as such, need to be treated with care.

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

Fundamental advances in AI

Fundamental advances in machine learning

The devil is in the detail, when it comes to complex networks. Whilst holistic behaviour and evolution can be understood by studying the global properties of a network, many properties are hidden within the substructure. Solving graphs analytically cannot capture the relevant small-scale dynamics, whilst using brute-force numerics is extremely expensive for large networks, if feasible at all. 
In this project, novel methods and measures tease out exact solutions for the evolution of hierarchical structures within complex networks. Ideas from information theory, statistical models, and graph theory seek out links and groups quantitatively, enabling community detection and an understanding of how these substructures feed forward to the global behaviour of the network. 
Real-world networks, from biological systems to interconnected financial institutions, are systems that thrive or fail due to small-scale effects that propagate. Honing in on the communities and links that drive these networks in a way that doesn’t require unlimited computational resources, will help to optimise and safeguard their evolution.

Employing theoretical measures to detect communities and connections in complex networks.

Hidden communities

Hidden communities in networks

The coronavirus pandemic is an evolving testbed for improving our understanding of how invasions are fought off by both healthy and compromised immune systems.
In this project, we zoom in on the case of cancer patients who contract coronavirus. We find correlations between long-term immune system weaknesses and the type of cancer that the patient has. This could inform decisions on which patients are the most vulnerable, and who may need adjustments to treatment after their recovery from the virus. We also explore advances that may be possible in our understanding of the immune system with theoretical insights from graph theory. We model how a network of white blood cells defends itself against pathogens that enter the system, in varying levels of saturation, and find that immune networks possess multi-tasking capabilities to deal with such invasions. 
Our immune systems respond to threats in sophisticated and highly organised ways. A combination of both theoretical modelling and experimental advances, along with stringent analysis of patient data, has and will lead to successes in the management and treatment of both viruses and underlying conditions.

Using both theory and experimental analyses to understand how the immune system responds to an alien invasion such as coronavirus.

Organising a defence

How the immune system organises a defence

On Earth, solar, and galactic length scales, Einstein’s theory of General Relativity is unbeatable. However, the theory breaks down at the smallest length scales, due to the presence of singularities and the lack of a complete quantum description.
In order to rectify these shortcomings, modifications to General Relativity are necessary. Alternative theories predict signatures that may be observable, or at least falsifiable. For instance, some theories require that the Universe undergoes cyclic phases of contraction and expansion, as opposed to the usual Big Bang. It is possible that physical quantities preserve memory through the bounces, and could hence be measured. Furthermore, the scale at which modifications become important in classes of modified theories of gravity can be identified, signifying where and when deviations should show up.
A full theory of quantum gravity, and a unification of gravity with the other fundamental forces of nature is the holy grail of modern astrophysics and cosmology. The fact that such a successful theory as General Relativity needs modification demonstrates how precise our understanding of the Universe has become, and understanding these fine details may illuminate other mysteries, such as the nature of dark matter and dark energy.

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

Alternative universes

Implications of alternative universes

In modern statistical inference the number of inferred parameters often grows with the size of the data set. Inference in this high-dimensional regime is a serious challenge for classical statistics, which was developed for when the number of parameters is fixed. As a result, high-dimensional inference is often employed without understanding how it works. In the words of da Vinci, ‘he who loves practice without theory is like a sailor who boards a ship without rudder or compass and knows not where he is cast’.
In this project we use analytic tools from the statistical physics of disordered systems to develop a theory of high-dimensional statistical inference. The parameters of the statistical model play the role of the degrees of freedom and the likelihood of their values plays the role of energy. The inferred parameters correspond to the lowest energy state. Among other techniques, we harness the replica trick to average over the disorder embodied by the data.
A current problem with high-dimensional inference is the propensity to generate unsubstantiated and often unfalsifiable conclusions. A foundational theory will help discriminate between reliable and unreliable inference, as well as inform the design of efficient inference algorithms.

Developing a theory of high-dimensional statistical inference using analytic tools from the statistical physics of disordered systems.

Inference in many dimensions

Is space a continuum or is it composed of a discrete set of points? Once a purely philosophical question, there is now a pressing need to describe space and space-time as discretized in order to resolve quantum gravity. Yet despite sophisticated attempts at discretizing space, researchers have failed to construct even the simplest geometries.
In this project we investigate models of discrete space in which continuum-like behavior is recovered at large lengths. We show that if space is discrete, it must be disordered, and develop explicit recipes for growing disordered discrete space by sampling distributions of graphs at low temperature. We also pursue graph formulations of non-Euclidean metrics, especially the Minkowski metric from special relativity.
Models of discrete space and space-time could have profound implications for physics. They may tame the infinities that arise from quantizing gravity, and dispense with the machinery of the real numbers, which has no direct observational support. They also set the stage for digital physics, in which information takes centre stage in describing the universe.

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

Is continuous space illusory?

Is continuous space an illusion?

Bootstrap percolation originated as a simple cellular automata model of the spread of infection. Given a set of initially infected sites, in consecutive rounds sites become infected if they satisfy a certain infection rule, for example, more than a given number of their neighbours are infected. Percolation occurs if every node is eventually infected.
In this project we investigate the typical and extremal properties of bootstrap percolation for various infection rules and underlying networks. For sites that are initially randomly infected, we study the critical density of infection that leads to percolation. Alternatively, if the initial infected sites are set by hand, we study how many infected nodes are necessary to infect the whole network.
Bootstrap percolation can be used to model many real-life phenomena, such as pandemics, computer network failure, ferromagnetic behaviour, the spread of voting preferences and information processing in neural networks. A better mathematical theory will help us understand these processes and draw reliable conclusions well beyond what can be achieved through computer simulations.

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

Bootstrap percolation

Leaps in long-range connectivity

String theory is the most promising candidate for combining gravity with the Standard Model. It predicts 10 dimensions, four of which are space-time. The rest are folded up at a tiny scale, and their geometry determines the properties of our universe. However, there are a vast number of such manifolds, and finding the right one is hard. There is no theoretical selection principle, and exhaustive search is computationally infeasible.
In this project, we use techniques from machine learning to find a geometry which gives our observed universe. The search is informed by recently compiled data, such as the complete intersection Calabi-Yau manifolds, the Kreuzer-Skarke reflexive polytopes, as well as G2 manifolds. One advantage of these datasets is that they are expressed in the tensor language of algebraic and combinatorial geometry, which are well suited for deep neural networks, especially convolutional and graph networks.
This work will uncover new patterns in the space of manifolds and suggest principles for selecting the right one, ultimately bringing us closer to a Theory of Everything. It will also generate new benchmarks for machine learning, and could lead to new conjectures in pure mathematics by finding simpler or unexpected formulae.

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

Learning the universe

Machine learning the geometry of the universe

A complete understanding of gene regulatory networks would allow us to identify the subsets of genes that program specific cell types, saving resources and time. While a complete understanding is still far off, machine learning techniques are enabling us to build black box models that mimic the behaviour of cells. By simulating a cell’s response to different gene combinations, we can disentangle genes that cause programming from those that are a consequence of programming.
In this project we combine autoencoder techniques with inference methods to build low-dimensional models of cell dynamics. This allows us to fuse multiple sequencing measurements into a single low-dimensional cell state space. In this space, we infer a dynamical model which allows us to simulate how cells transition from one state to another. By examining the response of this model to different combinations of genes, we could deduce the combinations that program new types.
The structure of the low-dimensional cell state space can lead to new insights into cell differentiation and disease. By using this to improve the prediction of genes, we can accelerate the production of new programmed cell types, reducing the cost of drug discovery, cell therapies and basic research.

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

Learning the cell-state space

Machine learning the regulatory structure of cell states

Real-world networks are complicated beasts. In order to capture all of the detailed time evolution and correlations that drive the nuances of these networks, it is necessary to model them extremely accurately. However, their non-linear nature can make their analysis intractable, or very computationally expensive at best. Instead, measures must be sought that encapsulate the important and defining aspects of these networks, which can then be analysed in full.
This project defines new local and global measures of networks which are related to their geometries. The notion of graph temperature enables the prediction of distinct topological properties of real-world networks simultaneously, whilst the hyperbolicity of a network is able to measure how democratic or aristocratic it is. From these properties it is possible to deduce the sentiment and behaviour of social and economic networks over time.
The ability to predict how a social or economic network will behave or react in a given circumstance is a powerful tool for driving social and economic growth. Modelling and analysing these networks using ideas and measures from graph theory makes this possible, and paves the way for a fuller understanding of the interconnected nature of real-world networks.

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

Sense from social networks

Making sense of social networks

The rise and fall of the stock market is highly dependent on public opinion. As news travels among investors, they make decisions which directly affect stock values. It is now possible to track people’s engagement with the news through their digital footprint and interactions on social media. By analysing such data, we can uncover correlations between interactions with the news and the behaviour of the market.
In this project, we study web search, Twitter and news outlet data to discover hidden indicators of financial markets. We show that the volume of web search queries of individual stocks over short times is predictive of stock prices. In addition, we can identify the sentiment of tweets from textual analysis by using supervised learning methods. With this, correlations between social media and markets during key news events can be clearly picked out. Finally, while the effect of news stories on markets cannot be deduced from the stories alone, the extent of interactions with them can forecast price movements.
The collective behaviour of investors is responsible for financial swings and crises. By using sophisticated analyses of people’s online interactions with financial-related news, downturns can be predicted and crises mitigated.

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

Markets and the mind

Many geological processes can be inferred from the distribution of the size and shape of the grains in rocks. But figuring out this information involves costly and tedious imaging techniques. In one approach, a thin layer at the surface of the rock is repeatedly sliced off. In another, the structure is deduced by merging x-ray images taken at different angles.
In this project we develop a mathematical formalism for capturing the 3D structure of a rock’s grains from just one 2D slide through it. Our approach is based on the theory of packing spheres and other shapes. We derive exact relations between the distribution of the volume, surface area or shape characteristics of grains and simple measurements from a thin section through the rock. We also determine how many grains to measure to achieve a desired level of certainty. We test our method on real minerals and simulated materials made up of randomly oriented ellipsoids and blocks.
Our new approach will help geologists and material scientists determine the structure of a material’s grains at a fraction of the cost of current methods. It could shed light on fundamental processes of rock formation, help optimize oil extraction, and better connect geological percolation with percolation theory.

Reconstructing the 3D shape distribution of rock grains or other randomly packed objects with access to only a 2D slice through them.

Dimension extension

Materials dimension extension

Modern 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, misses clues to new treatments and contributes to the high failure rate of clinical trials.
In this project we create more powerful mathematical methods for predicting disease outcomes. Most present-day mathematical and machine learning methods have a common Achilles heel. When the amount of information becomes too large, they confuse real patterns with noise and miss the wood for the trees—a breakdown known as overfitting. Our approach to bypassing this is founded on treating topological patterns in medical data as constraints on a random graph ensemble.
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 £100m for 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 the outcomes of diseases that pinpoint the right treatments and speed up drug trials.

Mathematical medicine

Mathematics for personalized medicine

Social media is eradicating physical separation as a barrier to communication. The result is a world that is highly virtually connected. There are direct paths between producers and consumers of content, as well as ways to monitor an audience’s responses and reactions.
In this project, we investigate how the collective behaviour of social media users can be understood based on their data footprints. We study how users interact with news and fake news, and the effect of interventions to debunk the latter. Using data from a vast set of Facebook users, we identify distinct groups who follow either scientific or conspiracy-based content. Our analysis reveals how each polarised group responds to material aiming to discredit false rumours. We use Twitter data to attribute users’ online engagement with a political party to either support or opposition. Trends in long-term data can predict outcomes of political elections, whereas short-term data is unreliable.
The real-time flow of social media data offers a unique opportunity to understand a population’s sentiment towards ideas. Predicting their reaction to content can inform political and commercial decisions. It can also help curb false belief based on nonfactual sources and its propagation.

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

News and fake news in a connected world

Densely packed objects of a similar shape have intrigued mathematicians since Apollonius of Perga. The collective mechanical properties of these objects are determined by how big the gaps are between them—in particular by how close they are to their maximum random-packed density. Predicting this maximum, and properties of the system close to it, is a geometric and physics challenge.
In this project we investigate the packing of spheres of the same size and mixtures of spheres with a range of sizes. While simulations of packing are easy to program, their run times scale poorly with system size. To overcome this, we develop mathematical techniques for determining the properties of dense packings. We use a simplified 1d model to analytically capture essential features of higher-dimensional systems. A symmetric effective medium approach approximates the viscosity of a mixture of spheres. On the applied side, we analyse the linear elastic behaviour of dispersions of cells to better stabilise them.
Understanding how objects fit closely together is at the heart of materials science and branches of fluid dynamics. Predicting the behaviour of dense packings also sets the stage for reverse-engineering new composite materials to have desired properties.

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

Puzzles in packing

Packing things into a crowded place

The simplest model of dimension is the hypercube. When the coordinates are given by a binary string, the distance between two strings is the number of spelling mistakes between them. By considering strings made of up more than two symbols, and connecting strings that differ by just one spelling mistake, the hypercube can be generalized into Hamming graphs. In information theory, messages are sent as strings which are subject to errors in transmission. To correct these mistakes, variants of a string must code for the same message as the string itself. In this way error correction is deeply connected to partitioning the Hamming graph into compact clusters.
In this project, we investigate the spectral properties of subgraphs of the hypercube and Hamming graphs. By studying a diffusing population on these graphs, we develop an analogy between mutational robustness and the eigenvalues of subgraphs that code for the same phenotype.
The spectral theory of Hamming graphs has direct applications to the theory of neutral evolution, in which biological information is carried by strings of nucleotides or amino acids. It is also at the heart of Hamming and other error correcting codes, and Hamming graphs feature in association schemes in coding theory.

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

Spectre of hypercubes

Point, line, square, cube, tesseract

When piecing together the fractures and weaknesses that might have led to a system’s failure, the first obstacle is in obtaining all of the data. In the case of financial networks, a full set of data is rarely available due to privacy and non-disclosure issues. This means that statistical methods must be used in order to reconstruct the networks from partial data.
This project seeks new and innovative solutions to this problem, by focusing on the topological structure of financial networks. Whilst it may not be possible to fully reconstruct a financial network and all of its connections, reconstructing the topological features that describe how distress propagates through the network provides the key information in the most efficient way. It becomes possible to understand the higher order behaviour of the network without mapping out each node and link one by one.
Novel approaches lead to surprising results, and indeed structures that were thought to strengthen financial networks are found to be some of the most vulnerable to failure. Identifying these characteristics will help to assess the structure of current financial networks and mitigate the likelihood of the cascading failures that caused recent financial disasters.

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

Reconstructing credit networks

Divisible numbers are useful whenever a whole needs to be divided into equal parts. But sometimes it is necessary to divide the parts into subparts, and subparts into sub-subparts, and so on, in a recursive way. For example, websites are divided into columns for different types of content, each of which is divided into sub-columns of elements or text.
In this project we introduce and study recursively divisible numbers, starting with the recursive divisor function: a recursive analog of the usual divisor function. We give a geometric interpretation of recursively divisible numbers and study the number and sum of the recursive divisors. Just as the divisor function motivates the abundant and perfect numbers, the recursive divisor function motivates their recursive analogs, which we investigate. The striking parallels between these two families of numbers hint at deep links between the usual and recursive divisor function.
By computing the most recursively divisible numbers, we recover many of the grid sizes commonly used in graphic design and digital technologies, and suggest new grid sizes which have yet to be adopted. These are especially relevant to recursively modular systems which operate across multiple organizational length scales.

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

Recursively divisible numbers

Memristors are resistors that preserve memory. They can be used to regulate and limit the amount of current that flows through a circuit, as they are able to change the value of their resistance based on their memory of the current and the voltage in the network. 
This project investigates how an analytical description of simple memristive circuits can be used to understand the properties of more complicated networks of memristors. Our overall approach uses idealised models where the dynamics can be solved exactly to uncover emergent behaviour. For example, we explore the relationship between the topology of the circuit and the dynamics of memristors using techniques from graph theory. Using mean field theory, we identify the asymptotic behaviour of memristor dynamics beyond the linear regime. We verify our analytic results with numerical solutions, demonstrating the ability of simple models to explain the collective behaviour of complex networks. 
The idea that memristors have internal memory means that it is possible for a network of them to learn. This ability can be harnessed to implement models of neural systems, and hence study and mimic the brain. It also makes memristors a potential alternative paradigm for solid-state memory.

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

Remembering to learn

To be sure of achieving a goal in the face of an uncertain future, individuals and organizations pursue plans that can cope with mistakes and unexpected events. One way of doing so is to choose a robust plan: one that is able to absorb any one setback and thereby achieve the goal with certainty. A different strategy is to choose a repairable plan: one that can be easily modified in the face of setbacks such that the goal is achieved, even though the original plan might have failed without modification.
In this project we develop a theory of systems that achieve resilience by being repairable instead of robust. Our approach is to minimize the typical cost of a workaround, averaged over all possible failure modes. We deduce design principles for easily repairable systems and show that repairability is a superior strategy whenever the number of failure modes is high.
Adopting a paradigm of repairability will increase the resilience of technologies and infrastructure to natural and man-made disasters when the next type of failure mode is hard to predict. This includes communication grids, transport networks and technologies for exploring space. It also suggests alternative strategies for sustainability in the face of environmental change.

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

Repairable instead of robust

Repairable as an alternative to robust

The cell coding company Bit Bio is hunting for the recipe to produce every cell in the human body. They are reprogramming induced pluripotent stem cells by switching on the correct genetic characteristics for each type of target cell. They’ve already had some exciting experimental success, but there are too many combinations of switches to test them all. That’s where we come in.
Our physicists are harnessing methods from network science to model the interactions between genes and proteins, which are fundamental to understanding how a cell functions. We are developing analytical tools which strike a balance between having to focus on only small-scale processes to make an experimental test tractable, and having to make too many assumptions for a large-scale statistical study. We want to understand these fundamental feedback loops and interactions in order to find groupings and patterns that will circumvent the need for an expensive brute-force approach.
Introducing mathematical methods to a traditionally experimental field of biology is challenging, but the rewards will be significant. The ability to reprogram cells will provide unprecedented advances in new drug research, cell therapy and even a glimpse into the process of ageing.

Applying mathematical tools to the holy grail of cellular biology: can we produce every type of human cell from within the laboratory?

Reprogramming the cell

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 equivalent to a universal Turing machine and can compute any computable function. But while applying simple rules is easy, deducing the rules from observed behaviour is both important and usually hard, as evidenced by our attempts to deduce the laws of physics through science.
In this project we investigate different families of simple rules for discrete dynamics. We help classify elementary cellular automata by deducing the apparent randomness of their behaviour. We investigate the self-assembly of polyominoes and the reverse engineering of structures made from them. Simple mathematical models of competition help us track the long-term behaviour of competitors and deduce which strategies maximize success.
Reverse engineering simple rules that generate desired kinds of behaviour can help us design efficient algorithms and low-cost methods for assembling complex structures. Understanding which rules give rise to dynamics that are at the border between order and chaos may offer fundamental insights into artificial life and synthetic biology, which have proved elusive.

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

Surprises from simple rules

Surprising results from simple rules

Most real-world networks are either a snapshot of a graph that evolves over time, or one instance of a collection of related graphs. Some aspects of the graph are random while others—such as the connectivity or density of cycles—are fixed. The trick is to figure out which is which. By treating a graph as a random variable, it is possible to deduce the constraints by comparing the behaviour they give rise to with real data. This is hard, and nearly all random graphs that can be solved analytically are locally tree-like. But empirically we know that realistic graphs have short loops.
In this project we develop sophisticated mathematical methods for deducing the behaviour of graphs and processes on them. By treating topological patterns as constraints on a random graph ensemble, we go beyond the replica trick and cavity method to handle graphs with a finite fraction of short loops. We start with triangles, the “harmonic oscillator” of loopy random graphs, and proceed to larger loops, along the way introducing the imaginary replica technique.
The theory of tailored graph ensembles offers a deeper understanding of graph structure. It enables us to predict the null behaviour of neural and genetic regulatory networks and to detect network communities. It also offers an indirect approach to longstanding problems on lattices, like the 3D Ising model.

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

Intelligence of graphs

The artificial intelligence of graphs

To understand a theoretical complex network, with simplified connections and structure, is no mean feat. It is another challenge entirely to shift that network into the real world, with all of its complications, and repeat the trick. However, it is vital to find a way to use what we have learnt from the solutions of idealised networks and extend them to practical scenarios.
In this project, we investigate how networks grow, synchronise and become unstable or disordered under real-world conditions. We tackle complications such as finite carrying capacity, and small-scale perturbations which lead to transient dynamics. These technicalities require fine-grained insight into the time evolution of the networks and sophisticated methods such as transfer operator analysis to get a handle on all of the moving parts.
The ability to solve realistic complex networks will make it possible to predict the behaviour of real-world systems under the imperfect stresses and changes that they are subjected to. These results will be of importance to studies of populations, ecosystems and financial systems, in order to inform societal and political decisions.

Improving our understanding of real-world networks with sophisticated analyses of realistic complications.

The fate of real systems

Technological progress is widely acknowledged as the main driver of economic growth, so any method for forecasting technological change is potentially very useful. But given that technological progress depends on innovation—generally thought of as something new and unanticipated—forecasting it might seem to be an oxymoron. In fact there are several postulated laws for technological improvement, such as Moore’s law and Wright’s law, but these are technology-dependent and we do not know how confident to be in their predictions.
In this project we investigate how to combine the forecasts of many technologies to make reliable distributional forecasts for a given technology. We use network theory and time series analysis to analyse performance curves and the large historical records of patenting activity to map the progress of technological ecosystems into the future.
The ability to forecast technological change with specified uncertainties can provide a foundation for a theory of economic growth. It can also offer clear recommendations for sustainability investments, where long-term planning is key. From a policy perspective, our insights provide an objective point of comparison to expert forecasts, which can be biased by vested interests.

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

Technological progress

The future of technological progress

Programming cells to adopt a particular cell type requires activating a small subset of genes. For most desirable cell types, we don’t know what these subsets are. Searching the space of gene combinations is a difficult experimental problem, because the space of combinations is large and because experiments are subject to noise, which complicates their interpretation.
In this project we reduce the experimental process to its mathematical core, mapping the problem of finding subsets of genes to a communication problem. Each cell in an experiment can be considered as a noisy message, communicating information about the subset of genes for the desired cell type. By combining many of these messages, we can infer the subset that generated them. This problem has its roots in combinatorial design in WWII, but the arena of cell programming inspires generalisations that require new techniques to analyse.
By establishing bounds from the noisy channel coding theorem, we can estimate the number of cells that experimenters must collect in order to find the desired subsets of genes. By maximising the number of bits communicated per cell, biologists can design more informative experiments and find new cell types that would otherwise be infeasible.

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

Informative experiment design

The mathematical design of experiments for cell programming

The development and maintenance of living organisms requires a considerable amount of computation. This is performed at the molecular level through gene regulatory networks. But we have little understanding of how this computation is organised: the existence of computational modules and how they combine to perform more advanced computation.
In this project, we introduce and study regulatory motifs, a new kind of structural and functional building block of gene regulatory networks. They are a consequence of the bipartite nature of genes and transcription factors. Just as the global network structure is a combination of these regulatory motifs, the global network function is a combination of the biological logics that these motifs can carry out. We study the mathematical properties of these regulatory motifs, such as their restricted range of logics and their tendency towards simple logics.
Genetic computation is responsible for the creation of body structures, the determination of cell identity and resilience to environmental fluctuations. Regulatory motifs help us understand how this computation is built up. This opens the door to the creation of interventions for existing genetic circuits, and synthetic genetic circuits with novel function.

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

Theory of genetic computation

The mathematical structure of genetic computation

Innovation is to technology what evolution is to life: it is how technology improves and adapts to a changing environment. Yet despite advances in our understanding of evolution, what drives innovation has remained elusive. Is the process of innovation essentially Darwinian—variation, selection and inheritance—or are there fundamentally different forces at work? 
In this project we develop a simple mathematical model of innovation in which technologies are made up of components and new components become available over time. In this expanding space of building blocks, we find that innovation has a structure all its own. While the choice of components plays an important role in determining the innovation rate, this is countered by an intrinsic rate specific to each domain. 
Our work helps structure the empirical investigation of innovation in the real world, and highlights the intrinsic versus intervenable aspects of technological change. Speeding up innovation in sectors such as energy production, software development and drug discovery will have far-reaching effects on sustainability, technological progress and the health of society.

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

The structure of innovation

The mathematical structure of innovation

Ageing is generally thought to be a consequence of an accumulation of errors in the storage of genetic information. But mounting experimental evidence suggests that ageing can be slowed. Under what conditions, if any, is ageing a necessary consequence of mutation, inheritance and selection?
In this project, we introduce a mathematical framework for understanding the evolutionary value of ageing. We derive a simple mortality equation which governs the transition matrix of an evolving population with a given maximum age. Remarkably, this equation is independent of the choice of fitness function. We show how to solve it in terms of the spectrum of eigenvalues which governs the dynamics, and consider the value of ageing in both a constant and changing environment. We also seek complete analytic solutions to the mortality equation for specific fitness functions.
If programmed death is favoured by natural selection rather than a consequence of genetic breakdowns, then it may be possible to devise pharmacological and genetic interventions to combat disease and prolong life. A mathematical formulation of ageing puts to rest some of the controversy surrounding the subject, which tends to rely on qualitative rather than quantitative reasoning.

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

Mathematics of immortality

The mathematics of ageing and immortality

Networks represent interactions between multiple elements. Widely accessible and large data sets makes it easy to build networks, but extracting meaning from them 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 advanced, the mathematical foundations for both networks and graphs, and their relation, are underdeveloped.
In this project we investigate new mathematical methods to describe the topology and geometry of networks and graphs, without regard to the kinds of data from which they are derived. The broad range of techniques under development include applications and extensions of spectral graph theory, tailored graph ensembles, replica methods, community detection and enumerative combinatorics.
A deeper understanding of network and graph structure underpins many other research endeavours. Examples range from biomedical data for personalized medicine to predicting 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.

Structure of how things relate

The structure of how things relate

Ever since Maxwell’s demon helped establish the link between thermodynamics and information, scientists have been formulating the thermodynamics of memories, information flows and feedback processes, especially in the 21st century. Quantum information thermodynamics extends these ideas to quantum measurement and feedback.
In this project we investigate the relation between information and energy in the classical and quantum regimes. Using analytical techniques and simulations, we pursue a collection of hypotheses about the physical nature of information, how to extract energy from fluctuations in the environment, and new technologies that take advantage of these and further insights.
A better understanding of information thermodynamics will help advance the nascent fields of quantum computing, nano-scale devices and biomolecular engineering. Sustained energy extraction could reduce the need for batteries in small autonomous devices. Quantifying the energy required by organisms and synthetic devices to gather and act on information could help form a foundational theory of life and artificial life.

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

Information thermodynamics

Thermodynamics of information

Understanding the relations between different branches of pure mathematics, and creating overarching theories that bind them together.

Mathematics that unifies

Mathematics is central to theoretical research, and advancing pure mathematics advances science itself. This is especially true for mathematics that unifies seemingly unconnected fields.

Dualities play a key role in how we form insights. Examples include the Langlands programme, monstrous moonshine, the ADE classification and dualities across quantum field and string theories. We investigate extensions of these, and develop their consequences. Can we develop other dualities, especially ones that exploit our intuition for geometry and arithmetic?

To bring rigour to network science, we investigate graphical notions of geometry and topology compatible with their continuum analogues. We develop a theory of statistical inference suitable for high dimensions, which is the basis for much of artificial intelligence. There are compelling arguments for a computational understanding of the universe, and we seek mathematical insights into the scope and limits of computation, particularly the output of simple rules.

Embedding pure mathematics within theoretical science keeps theorists abreast of the latest tools, and inspires mathematicians to take up new directions. Can we re-conceive mathematics as a core part of science, rather than an adjunct, thereby ensuring it is favoured and funded in proportion to its value?

Tackling big questions about the fundamental forces, symmetry and information, and the intimate interplay between physics and mathematics.

The elegant universe

We lack a single theory that describes the universe. Gravity, described by general relativity, is not consistent with quantum field theory describing the other three forces. Will this be resolved by string theory, loop quantum gravity, or something new? What are the testable consequences of such a theory, which is currently beyond the limits of human experimentation? We seek the structure of the extra dimensions of space-time whose geometry determines our universe.

We study the large-scale structure of space-time,  especially in relation to singularities. Can black hole thermodynamics help prescribe a theory of everything? What are the implications of the holographic principle? What is behind dark energy and matter, which are posited to make sense of the cosmos?

We study the physical nature of information, and the extraction of energy from fluctuations in the environment. We seek a foundational understanding of the relation between information and energy in the classical and quantum regimes.

Wigner noted the unreasonable effectiveness of mathematics in physics. Today, we see the reverse: attempts to advance physics, such as string theory, are driving mathematics. We investigate the convergence between pure mathematics and fundamental physics, and how this vantage could help redress physics’ current impasse.

Developing mathematical foundations for life and artificial life, machine intelligence, and other emergent phenomena that defy reductionism.

Life, learning and emergence

What is life? Darwin’s theory provides a qualitative understanding of evolution. But from a physics perspective, we don’t know how life got started in the first place. What investigate the thermodynamic basis for emergent self-replication and adaptation, of which biology is just one instance. Can this be used to engineer artificial digital life? Can evolution itself be made a predictive science?

How do we make intelligent machines? Far from approaching artificial general intelligence, AI is stuck in high-dimensional curve-fitting. We seek mathematical insights that could lead to more intelligent AI, such as causal reasoning, reusable functional modules, and a representation of the environment. We investigate ways to use computation and AI to automate the search for new mathematical insights. Are there fundamental limits to AI, and what might this tell us about human intelligence?

What are the emergent properties of digital and neural computation, and might this shed light on autonomy and free will? We study information processing at the genetic level and the functional architecture of gene regulatory networks. We seek a theoretical understanding of cell programming and how to infer programming sets. Is causality itself an emergent phenomenon, as we traverse across different organisational length scales?

Establishing the scientific principles behind the technologies of the future, in order to transform work, health, defence and creativity.

Tomorrow’s technology

How do we reduce toil, improve health, boost security and enhance creativity? The technologies that address these fundamental human needs always rely on scientific principles, and we develop the theories behind the technologies of tomorrow.

The greatest labour-saving innovation today is efficient computation. We work on theoretical aspects of alternatives to electronic computing, such as memristors and photonic and quantum computing. We seek a deeper foundation for artificial neural computation, which currently resembles engineering more than science.

Can a deeper understanding of how life processes information set the stage for a biological analogue of the silicon revolution? We study whether aging is the result of a programme, rather than an entropic necessity. If so, how can we slow it? We develop new mathematics for high dimensional inference and apply it to precision medicine.

Working with the US Department of Defense and the Ministry of Defence, we lay the theoretical groundwork for moonshot technologies of national interest. These range from self-similar materials, to sensor dust, to repairable instead of robust, to harvesting energy from fluctuations in the environment.

In our theme on the theory of human enterprise, we investigate how we can surpass the limitations of the individual through collective creativity.

Developing mathematical models of markets, innovation and organisations, so that we can predict them and enhance them through interventions.

Theory of human enterprise

Science has traditionally been concerned with the natural world. But as society gets more interconnected and organisations get bigger, the man-made world needs a science of its own.

The sentiment of borrowers and lenders in a financial network is what drives markets to success, but also to ruin. We develop mathematical methods to predict how distress spreads, and determine strategies to limit system-wide catastrophic failure. We determine the latent potential in countries and firms by applying spectral-like theories to their networks of products and capabilities.

Despite advances in our understanding of evolution, what drives innovation remains elusive. Technological innovation operates in an expanding space of building blocks, in which combinations of technologies become new technologies. We characterise innovation in a mathematical way, extracting concepts and conservation laws, so that we can predict and influence it.

Organisations have emergent properties and capabilities that we are just coming to terms with. The success of some wikis suggests that many non-interacting agents can produce creative works superior to what any one person could do alone. What is the mathematical basis for collective creativity, and what sectors can we apply it to? Can it be used to speed up discovery in physics and mathematics? 

We do curiosity-driven theoretical research. We build theories—concise mathematical descriptions of patterns—and identify their testable consequences. We pursue the most intriguing patterns, without concern for their usefulness, which leads to the most transformative breakthroughs.