Our papers are the official record of our discoveries. They allow others to build on and apply our work. Each paper is the result of many months of research, so we make a special effort to make them clear, beautiful and inspirational, and publish them in leading journals.
Mahler measuring amoebae
Genetic symbolic regression methods reveal the relationship between amoebae from tropical geometry and the Mahler measure from number theory.
By approximating the basis of eigenfunctions, we computationally determine the harmonic modes of bundle-valued Laplacians on Calabi-Yau manifolds.
A new connection between continued fractions and the Bourgain–Gamburd machine reveals a girth-free variant of this widely-celebrated theorem.
Complex digital cities
A complexity-science approach to digital twins of cities views them as interwoven self-organising phenomena, instead of machines or logistic systems.
Sum-product with few primes
For a finite set of integers with few prime factors, improving the lower bound on its sum and product sets affirms the Erdös-Szemerédi conjecture.
On John McKay
This obituary celebrates the life and work of John Keith Stuart McKay, highlighting the mathematical miracles for which he will be remembered.
BERT enhanced with recurrence
The quadratic complexity of attention in transformers is tackled by combining token-based memory and segment-level recurrence, using RMT.
Random Chowla conjecture
The distribution of partial sums of a Steinhaus random multiplicative function, of polynomials in a given form, converges to the standard complex Gaussian.
Landau meets Kauffman
A new, simple approach to the critical Kauffman model with connectivity one sharpens the bounds on the number and length of attractors.
Single-input Boolean networks
A new, simpler approach to the critical Kauffman model with connectivity one reveals that it has more attractors than previously believed.
Cell soup in screens
Bursting cells can introduce noise in transcription factor screens, but modelling this process allows us to discern true counts from false.
Kerr black holes symmetry
Effective field theories for Kerr black holes, showing the 3-point Kerr amplitudes are uniquely predicted using higher-spin gauge symmetry.
Multiplicativity of sets
Expanding the known multiplicative properties of large difference sets yields a new, quantitative proof on the structure of product sets.
Applying diffusion-based graph operators to complex networks identifies the proper spatiotemporal scales by overcoming small-world effects.
Clustered cluster algebras
Cluster variables in Grassmannian cluster algebras can be classified with HPC by applying the tableaux method up to a fixed number of columns.
Bounding Zaremba’s conjecture
Using methods related to the Bourgain–Gamburd machine refines the previous bound on Zaremba’s conjecture in the theory of continued fractions.
Optimal electronic reservoirs
Balancing memory from linear components with nonlinearities from memristors optimises the computational capacity of electronic reservoirs.
Gauge theory and integrability
The algebra of a toric quiver gauge theory recovers the Bethe ansatz, revealing the relation between gauge theories and integrable systems.
Flowers of immortality
The eigenvalues of the mortality equation fall into two classes—the flower and the stem—but only the stem eigenvalues control the dynamics.
Structure of genetic computation
The structural and functional building blocks of gene regulatory networks correspond, which tell us how genetic computation is organised.
AI classifies space-time
A neural network learns to classify different types of spacetime in general relativity according to their algebraic Petrov classification.
Algebra of melting crystals
Certain states in quantum field theories are described by the geometry and algebra of melting crystals via properties of partition functions.
Combinatorics, Number theory
Set additivity and growth
The additive dimension of a set, which is the size of a maximal dissociated subset, is closely connected to the rapid growth of higher sumsets.
Machine learning Hilbert series
Neural networks find efficient ways to compute the Hilbert series, an important counting function in algebraic geometry and gauge theory.
AI for cluster algebras
Investigating cluster algebras through the lens of modern data science reveals an elegant symmetry in the quiver exchange graph embedding.
Line bundle connections
Neural networks find numerical solutions to Hermitian Yang-Mills equations, a difficult system of PDEs crucial to mathematics and physics.
Unsupervised machine-learning of the Hodge numbers of Calabi-Yau hypersurfaces detects new patterns with an unexpected linear dependence.
Mahler measure for quivers
Mahler measure from number theory is used for the first time in physics, yielding “Mahler flow” which extrapolates different phases in QFT.
Recursively divisible numbers
Recursively divisible numbers are a new kind of number that are highly divisible, whose quotients are highly divisible, and so on, recursively.
Learning the Sato–Tate conjecture
Machine-learning methods can distinguish between Sato-Tate groups, promoting a data-driven approach for problems involving Euler factors.
Universes as big data
Machine-learning is a powerful tool for sifting through the landscape of possible Universes that could derive from Calabi-Yau manifolds.
True scale-free networks
The underlying scale invariance properties of naturally occurring networks are often clouded by finite-size effects due to the sample data.
Reflexions on Mahler
Unifying Mahler measure, dessins and gauge theory, Newton polynomials reveal a 1-to-1 correspondence between Mahler measure and dessins.
Transitions in loopy graphs
The generation of large graphs with a controllable number of short loops paves the way for building more realistic random networks.
Coexistence in diverse ecosystems
Scale-invariant plant clusters explain the ability for a diverse range of plant species to coexist in ecosystems such as Barra Colorado.
Cancer screening with MRI
An ongoing study tests the feasibility of using MRI scans to screen men for prostate cancer in place of unreliable antigen blood tests.
Quick quantum neural nets
The notion of quantum superposition speeds up the training process for binary neural networks and ensures that their parameters are optimal.
Going, going, gone
A solution to the information paradox uses standard quantum field theory to show that black holes can evaporate in a predictable way.
A delicate balance between white blood cell protein expression and the molecules on the surface of tumour cells determines cancer prognoses.
Breaking classical barriers
Circuits of memristors, resistors with memory, can exhibit instabilities which allow classical tunnelling through potential energy barriers.
QFT and kids’ drawings
Groethendieck's “children’s drawings”, a type of bipartite graph, link number theory, geometry, and the physics of conformal field theory.
Neurons on amoebae
Machine-learning 2-dimensional amoeba in algebraic geometry and string theory is able to recover the complicated conditions from so-called lopsidedness.
New approaches to the Monster
Editorial of the last set of lectures given by the founder, McKay, of Moonshine Conjectures, the proof of which got Borcherds the Fields Medal.
Physics of financial networks
Statistical physics contributes to new models and metrics for the study of financial network structure, dynamics, stability and instability.
Channels of contagion
Fire sales of common asset holdings can whip through a channel of contagion between banks, insurance companies and investments funds.
Risky bank interactions
Networks where risky banks are mostly exposed to other risky banks have higher levels of systemic risk than those with stable bank interactions.
Cancer and coronavirus
Cancer patients who contract and recover from Coronavirus-2 exhibit long-term immune system weaknesses, depending on the type of cancer.
Scale of non-locality
The number of particles in a higher derivative theory of gravity relates to its effective mass scale, which signals the theory’s viability.
I want to be forever young
The mortality equation governs the dynamics of an evolving population with a given maximum age, offering a theory for programmed ageing.
Exact linear regression
Exact methods supersede approximations used in high-dimensional linear regression to find correlations in statistical physics problems.
Biological logics are restricted
The fraction of logics that are biologically permitted can be bounded and shown to be tiny, which makes inferring them from experiments easier.
In life, there are few rules
The bipartite nature of regulatory networks means gene-gene logics are composed, which severely restricts which ones can show up in life.
Energy bounds for roots
Bounds for additive energies of modular roots can be generalised and improved with tools from additive combinatorics and algebraic number theory.
Ample and pristine numbers
Parallels between the perfect and abundant numbers and their recursive analogs point to deeper structure in the recursive divisor function.
Network valuation in finance
Consistent valuation of interbank claims within an interconnected financial system can be found with a recursive update of banks' equities.
Deep layered machines
The ability of deep neural networks to generalize can be unraveled using path integral methods to compute their typical Boolean functions.
Replica analysis of overfitting
Statistical methods that normally fail for very high-dimensional data can be rescued via mathematical tools from statistical physics.
Theory of innovation
Insights from biology, physics and business shed light on the nature and costs of complexity and how to manage it in business organizations.
Inference, Statistical physics
We optimize Bayesian data clustering by mapping the problem to the statistical physics of a gas and calculating the lowest entropy state.
Theory of innovation
Recursive structure of innovation
A theoretical model of recursive innovation suggests that new technologies are recursively built up from new combinations of existing ones.
Bursting dynamic networks
A mathematical model captures the temporal and steady state behaviour of networks whose two sets of nodes either generate or destroy links.
Renewable resource management
Modern portfolio theory inspires a strategy for allocating renewable energy sources which minimises the impact of production fluctuations.
Energy harvesting with AI
Machine learning techniques enhance the efficiency of energy harvesters by implementing reversible energy-conserving operations.
Geometry of discrete space
A phase transition creates the geometry of the continuum from discrete space, but it needs disorder if it is to have the right metric.
A simple solvable model of memristive networks suggests a correspondence between the asymptotic states of memristors and the Ising model.
Physics of networks
Statistical physics harnesses links between maximum entropy and information theory to capture null model and real-world network features.
Theory of innovation
The rate of innovation
The distribution of product complexity helps explain why some technology sectors tend to exhibit faster innovation rates than other sectors.
Statistical physics, Thermodynamics
One-shot analogs of fluctuation-theorem results help unify these two approaches for small-scale, nonequilibrium statistical physics.
Network users who have access to the network’s most informative node, as quantified by a novel index, the InfoRank, have a competitive edge.
A Hamiltonian recipe
An explicit recipe for defining the Hamiltonian in general probabilistic theories, which have the potential to generalise quantum theory.
Solvable memristive circuits
Exact solutions for the dynamics of interacting memristors predict whether they relax to higher or lower resistance states given random initialisations.
Grain shape inference
The distributions of size and shape of a material’s grains can be constructed from a 2D slice of the material and electron diffraction data.
A novel approach to volunteer clouds outperforms traditional distributed task scheduling algorithms in the presence of intensive workloads.
From ecology to finance
Bipartite networks model the structures of ecological and economic real-world systems, enabling hypothesis testing and crisis forecasting.
Hamming balls, subgraphs of the hypercube, maximise the graph’s largest eigenvalue exactly when the dimension of the cube is large enough.
Theory of innovation
Forecasting technology deployment
Forecast errors for simple experience curve models facilitate more reliable estimates for the costs of technology deployment.
Hierarchies in directed networks
An iterative version of a method to identify hierarchies and rankings of nodes in directed networks can partly overcome its resolution limit.
Exactly solvable random graphs
An explicit analytical solution reproduces the main features of random graph ensembles with many short cycles under strict degree constraints.
The interbank network
The large-scale structure of the interbank network changes drastically in times of crisis due to the effect of measures from central banks.
Theory of innovation
The science of strategy
The usefulness of components and the complexity of products inform the best strategy for innovation at different stages of the process.
Theory of materials
Dirac cones in 2D borane
The structure of two-dimensional borane, a new semi-metallic single-layered material, has two Dirac cones that meet right at the Fermi energy.
Modelling financial systemic risk
Complex networks model the links between financial institutions and how these channels can transition from diversifying to propagating risk.
Bayesian analysis of medical data
Bayesian networks describe the evolution of orthodontic features on patients receiving treatment versus no treatment for malocclusion.
Theory of innovation
The secret structure of innovation
Firms can harness the shifting importance of component building blocks to build better products and services and hence increase their chances of sustained success.
Quantum neural networks
We generalise neural networks into a quantum framework, demonstrating the possibility of quantum auto-encoders and teleportation.
Debunking in a world of tribes
When people operate in echo chambers, they focus on information adhering to their system of beliefs. Debunking them is harder than it seems
Memristive networks and learning
Memristive networks preserve memory and have the ability to learn according to analysis of the network’s internal memory dynamics.
Dynamics of memristors
Exact equations of motion provide an analytical description of the evolution and relaxation properties of complex memristive circuits.
Bipartite trade network
A new algorithm unveils complicated structures in the bipartite mapping between countries and products of the international trade network.
3d grains from 2d slices
Moment-based methods provide a simple way to describe a population of spherical particles and extract 3d information from 2d measurements.
Disentangling links in networks
Inference from single snapshots of temporal networks can misleadingly group communities if the links between snapshots are correlated.
Spectroscopy experiments show that energy shifts due to photon emission from individual molecules satisfy a fundamental quantum relation.
Financial network reconstruction
Statistical mechanics concepts reconstruct connections between financial institutions and the stock market, despite limited data disclosure.
Pathways towards instability
Processes believed to stabilize financial markets can drive them towards instability by creating cyclical structures that amplify distress.
Worst-case work entropic equality
A new equality which depends on the maximum entropy describes the worst-case amount of work done by finite-dimensional quantum systems.
Theory of innovation
Serendipity and strategy
In systems of innovation, the relative usefulness of different components changes as the number of components we possess increases.
The spectral density of graph ensembles provides an exact solution to the graph partitioning problem and helps detect community structure.
Complex networks, Financial risk
Non-linear distress propagation
Non-linear models of distress propagation in financial networks characterise key regimes where shocks are either amplified or suppressed.
Immunisation of systemic risk
Targeted immunisation policies limit distress propagation and prevent system-wide crises in financial networks according to sandpile models.
Optimal growth rates
An extension of the Kelly criterion maximises the growth rate of multiplicative stochastic processes when limited resources are available.
Quantum tunnelling only occurs if either the Wigner function is negative, or the tunnelling rate operator has a negative Wigner function.
The price of complexity
Increasing the complexity of the network of contracts between financial institutions decreases the accuracy of estimating systemic risk.
Photonic Maxwell's demon
With inspiration from Maxwell’s classic thought experiment, it is possible to extract macroscopic work from microscopic measurements of photons.
Eigenvalues of neutral networks
The principal eigenvalue of small neutral networks determines their robustness, and is bounded by the logarithm of the number of vertices.
Cascades in flow networks
Coupled distribution grids are more vulnerable to a cascading systemic failure but they have larger safe regions within their networks.
Self-organising adaptive networks
An adaptive network of oscillators in fragmented and incoherent states can re-organise itself into connected and synchronized states.
Optimal heat exchange networks
Compact heat exchangers can be designed to run at low power if the exchange is concentrated in a crumpled surface fed by a fractal network.
Instability in complex ecosystems
The community matrix of a complex ecosystem captures the population dynamics of interacting species and transitions to unstable abundances.
Form and function in gene networks
The structural properties of a network motif predict its functional versatility and relate to gene regulatory networks.
Financial risk, Percolation theory
Clusters of neurons
Percolation theory shows that the formation of giant clusters of neurons relies on a few parameters that could be measured experimentally.
Cyclic isotropic cosmologies
In an infinitely bouncing Universe, the scalar field driving the cosmological expansion and contraction carries information between phases.
Theory of innovation
Predicting technological progress
A formulation of Moore’s law estimates the probability that a given technology will outperform another at a certain point in the future.
Bootstrap percolation models
A subset of bootstrap percolation models, which stabilise systems of cells on infinite lattices, exhibit non-trivial phase transitions.
News sentiment and price dynamics
News sentiment analysis and web browsing data are unilluminating alone, but inspected together, predict fluctuations in stock prices.
Communities in networks
A new tool derived from information theory quantitatively identifies trees, hierarchies and community structures within complex networks.
Effect of Twitter on stock prices
When the number of tweets about an event peaks, the sentiment of those tweets correlates strongly with abnormal stock market returns.
Democracy in networks
Analysis of the hyperbolicity of real-world networks distinguishes between those which are aristocratic and those which are democratic.
Protein interaction experiments
Properties of protein interaction networks test the reliability of data and hint at the underlying mechanism with which proteins recruit each other.
Combinatorics, Graph theory
Erdős-Ko-Rado theorem analogue
A random analogue of the Erdős-Ko-Rado theorem sheds light on its stability in an area of parameter space which has not yet been explored.
Collective attention to politics
Tweet volume is a good indicator of political parties' success in elections when considered over an optimal time window so as to minimise noise.
A measure of majorization
Single-shot information theory inspires a new formulation of statistical mechanics which measures the optimal guaranteed work of a system.
DebtRank and shock propagation
A dynamical microscopic theory of instability for financial networks reformulates the DebtRank algorithm in terms of basic accounting principles.
Spin systems on Bethe lattices
Exact equations for the thermodynamic quantities of lattices made of d-dimensional hypercubes are obtainable with the Bethe-Peierls approach.
Financial risk, Network theory
Fragility of the interbank network
The speed of a financial crisis outbreak sets the maximum delay before intervention by central authorities is no longer effective.
Theory of materials
Structure and stability of salts
The stable structures of calcium and magnesium carbonate at high pressures are crucial for understanding the Earth's deep carbon cycle.
Dynamics of economic complexity
Dynamical systems theory predicts the growth potential of countries with heterogeneous patterns of evolution where regression methods fail.
From memory to scale-free
A local model of preferential attachment with short-term memory generates scale-free networks, which can be readily computed by memristors.
Maximum percolation time
A simple formula gives the maximum time for an n x n grid to become entirely infected having undergone a bootstrap percolation process.
Random graphs with short loops
The analysis of real networks which contain many short loops requires novel methods, because they break the assumptions of tree-like models.
Taxonomy and economic growth
Less developed countries have to learn simple capabilities in order to start a stable industrialization and development process.
Viscosity of polydisperse spheres
A fast and simple way to measure how polydisperse spheres crowd around each other, termed the packing fraction, agrees well with rheological data.
Networks of credit default swaps
Time series data from networks of credit default swaps display no early warnings of financial crises without additional macroeconomic indicators.
Entropies of graph ensembles
Explicit formulae for the Shannon entropies of random graph ensembles provide measures to compare and reproduce their topological features.
Easily repairable networks
When networks come under attack, a repairable architecture is superior to, and globally distinct from, an architecture that is robust.
A review of the achievements concerning typical bipartite entanglement for random quantum states involving a large number of particles.
Theory of materials
Predicting interface structures
Generating random structures in the vicinity of a material’s defect predicts the low and high energy atomic structure at the grain boundary.
Percolation on Galton-Watson trees
The critical probability for bootstrap percolation, a process which mimics the spread of an infection in a graph, is bounded for Galton-Watson trees.
Memory effects in stock dynamics
The likelihood of stock prices bouncing on specific values increases due to memory effects in the time series data of the price dynamics.
Self-healing complex networks
The interplay between redundancies and smart reconfiguration protocols can improve the resilience of networked infrastructures to failures.
Fractal structures need very little mass to support a load; but for current designs, this makes them vulnerable to manufacturing errors.
Default cascades in networks
The optimal architecture of a financial system is only dependent on its topology when the market is illiquid, and no topology is always superior.
Random close packing fractions
Lognormal distributions (and mixtures of same) are a useful model for the size distribution in emulsions and sediments.
Multitasking immune networks
The immune system must simultaneously recall multiple defense strategies because many antigens can attack the host at the same time.
Metrics for global competitiveness
A new non-monetary metric captures diversification, a dominant effect on the globalised market, and the effective complexity of products.
Measuring the intangibles
Coupled non-linear maps extract information about the competitiveness of countries to the complexity of their products from trade data.
The temperature of networks
A new concept, graph temperature, enables the prediction of distinct topological properties of real-world networks simultaneously.
Scales in weighted networks
Information theory fixes weighted networks’ degeneracy issues with a generalisation of binary graphs and an optimal scale of link intensities.
Multi-tasking in immune networks
Associative networks with different loads model the ability of the immune system to respond simultaneously to multiple distinct antigen invasions.
The most efficient load-bearing fractals are designed as big structures under gentle loads, a common situation in aerospace applications.
Complex networks detect the driver institutions of an interbank market and ascertain that intervention policies should be time-scale dependent.
New mathematical tools can help infer financial networks from partial data to understand the propagation of distress through the network.
Network-based metrics to assess systemic risk and the importance of financial institutions can help tame the financial derivatives market.
Organized knowledge economies
The Yule-Simon distribution describes the diffusion of knowledge and ideas in a social network which in turn influences economic growth.
Bootstrapping topology and risk
Information about 10% of the links in a complex network is sufficient to reconstruct its main features and resilience with the fitness model.
Weighted network evolution
A statistical procedure identifies dominant edges within weighted networks to determine whether a network has reached its steady state.
Ultralight fractal structures
The transition from solid to hollow beams changes the scaling of stability versus loading analogously to increasing the hierarchical order by one.
Hierarchical space frames
A systematic way to vary the power-law scaling relations between loading parameters and volume of material aids the hierarchical design process.
Network analysis of export flows
Network theory finds unexpected interactions between the number of products a country produces and the number of countries producing each product.
Metric for fitness and complexity
A quantitative assessment of the non-monetary advantage of diversification represents a country’s hidden potential for development and growth.
Networks for medical data
Network analysis of diagnostic data identifies combinations of the key factors which cause Class III malocclusion and how they evolve over time.
Search queries predict stocks
Analysis of web search queries about a given stock, from the seemingly uncoordinated activity of many users, can anticipate the trading peak.
Unbiased randomisation processes generate sophisticated synthetic networks for modelling and testing the properties of real-world networks.
Robust and assortative
Spectral analysis shows that disassortative networks exhibit a higher epidemiological threshold and are therefore easier to immunize.
Edge multiplicity—the number of triangles attached to edges—is a powerful analytic tool to understand and generalize network properties.
What you see is not what you get
Methods from tailored random graph theory reveal the relation between true biological networks and the often-biased samples taken from them.
Shear elastic deformation in cells
Analysis of the linear elastic behaviour of plant cell dispersions improves our understanding of how to stabilise and texturise food products.
Dynamics of Ising chains
A transfer operator formalism solves the macroscopic dynamics of disordered Ising chain systems which are relevant for ageing phenomena.
Theory of materials
Diffusional liquid-phase sintering
A Monte Carlo model simulates the microstructural evolution of metallic and ceramic powders during the consolidation process liquid-phase sintering.
Tailored random graph ensembles
New mathematical tools quantify the topological structure of large directed networks which describe how genes interact within a cell.
The information needed to self-assemble a structure quantifies its modularity and explains the prevalence of certain structures over others.
Techniques from random sphere packing predict the dimension of the Apollonian gasket, a fractal made up of non-overlapping hyperspheres.
Random cellular automata
Of the 256 elementary cellular automata, 28 of them exhibit random behavior over time, but spatio-temporal currents still lurk underneath.
Single elimination competition
In single elimination competition the best indicator of success is a player's wealth: the accumulated wealth of all defeated players.