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

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

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

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

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. We 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?

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.