1

#### Apollonian packings

Leibniz considered covering the plane, starting with touching circles and repeatedly adding the largest circle which touches three others. Related constructions in 3d can be used as models for broken rock at fault zones and the vortex structure of turbulence; while their graphs afford models for social networks. The residual set of this Apollonian packing is fractal and its Hausdorff dimension captures important information. We have found a method to estimate this dimension in arbitrary dimension.

2

#### Complex Networks

Complex networks describe systems as diverse as the internet, financial agents and species predation in food webs. We use graph theoric techniques to provide a network description of these systems and establish rules that couple the evolution of network shape with function. Current projects are implementing these findings to understand how we can make networks such as financial systems and suuply networks resilient to attacks and failures.

3

#### Computational Topology

Computational topology, in which the theory of persistent homology, created by Herbert Edelsbrunner, plays a key role, develops efficient algorithms for solving topological problems. We develop new applications of results from topology. Recent work includes persistent local homology, persistent intersection homology and persistent characteristic classes.

4

#### Diffusion in Hamming space

Point mutations are single “spelling errors” in an organisms genome. The set of all possible point mutations form a Hamming graph. A sequence of point mutations means genomes perform a random walk in Hamming space. We have shown that because Hamming space is both high dimensional curved, diffusion in it is radically different to conventional, euclidean space diffusion, with novel scaling laws.

5

#### Fractal Materials

Compression members (such as columns) require more material than tension members (such as ropes) to support a given load, because of their vulnerability to buckling. Quantifying this in terms of non-dimensional loading & efficiency numbers, we have shown that fractal designs can be used to make very lightweight compression structures, by using one hierarchical level to protect against buckling of the next. More complex designs are needed for large structures or those required to withstand gentle forces.

6

#### Fracture Mechanics

The failure of brittle, polycrystals is governed by the nucleation and growth of cracks, which operate by focussing strain energy down from a macroscopic region onto the crack tip, which may be of molecular scale. By considering the geometric properties of a polycrystal containing a non-solidified matrix (for example sea ice, or a ceramic at high temperature), we find universal shapes for the voids, and by treating them as incipient fractures, deduce a general theory for the failure stress of brittle solids of this kind.

7

#### Jamming in Rheology

Concentrated suspensions of hard particles (e.g. corn flour) can jam when the strain rate is too high. The related phenomenon of dilatancy is seen when walking on wet sand: the material expands, drawing water from the surface & leaving a dry halo around the footprint. We build an analytic theory of this, based on particle clusters forming under flow & growing by cluster-cluster aggregation. The theory is solved by a continued fraction expansion in complex strain, showing the occurrence of a dynamic phase transition.

8

#### Materials at High Pressures

We have developed methods, dubbed â€œab initio random “structure searching” (AIRSS) which combines quantum mechanics random searching to predict material structures at high pressures. Far from being boring, these materials adopt intriguing complex structures (pictured) at pressures ten times those found in the centre of the earth. Our studies atttribute the origin of these mysterious atomic arrangements to the squeezing of electrons from the atoms to the gaps between them, forming what is known as an ‘electride’.

9

#### Microfluidics

Processing of multi-phase fluids through micron scale pumping & mixing devices may give much greater control over the properties of the resulting materials; e.g. controlled droplet sizes and the ability to design particles. In “lab-on-a-chip” chemical analysis, this degree of control is essential. By considering idealized, short droplets, we found a simple approximation for the interaction of droplet trains in different channel topologies, and showed that in even a simple y-shaped channel, this led to complex flow behaviours.

10

#### Network Dynamics

Much recent research concerns the structure of networks and their application to biological systems. Less is known about the dynamics on those networks and how these describe the functioning of those systems. We focus on Boolean Networks, Dynamical Systems and Stochastic Networks and are particularly interested in understanding the relationship between these as models for gene regulation.

11

#### Non-affine deformations in elasticity

Materials formed from cross-linked rod-like molecules and polymers are ubiquitous in nature, particularly in the cytoskeletons of cells. The elasticity of these materials is key to understanding how cells move and divide. This elasticity is strongly affected by non-affine deformations, that is, local strains that differ from the globally applied strain and soften the response of the material. Our work investigates the effect disorder plays in shaping non-affine responses showing that it can dramatically affect stiffness.

12

#### Non-coding DNA

How much non-coding DNA do eukaryotes require? In most eukaryotes, a large proportion of the genome does not code for proteins. The non-coding part is observed to vary greatly in size even between closely related species. Data suggest that eukaryotes require a certain minimum amount of non-coding DNA, and that this minimum increases quadratically with the amount of coding DNA. We derive a theoretical prediction of this minimum based on a simple model of the growth of regulatory networks.

13

#### Null models of Evolvability

If a system is robust, such that most mutations do not change function, how does it ever adapt to discover a better phenotype? This paradox has prompted much recent work, and strikes at the heart of the neutralist-selectionist divide. We work on null models to test properties of robustness and adapatibility in random genotype-phenotype maps. Using techniques from percolation theory, we find that robustnes can promote adaptability even in random maps.

14

#### Pattern detection in micro-array data

Over the last decade, microarrays have generated an unprecedented amount of genetic expression data. Our work introduces an approach for detecting statistically significant patterns in these datasets without making prior assumptions about the nature of the pattern. This method is based on concepts from Algorithmic Information Theory.

15

#### Robustness and Evolution

Robustness means that errors in instructions do not cause errors in function. In evolution, robustness is critical because genomes are constantly undergoing mutations. How robust can an evolving population become? What is the fitness value of robustness? Our work shows that because mutations drive random walks in sequence space, there is a limit on how robust a phenotype can become. This limit depends only on the size of the phenotype in genotype space.

16

#### Self-assembly, Modularity and Physical Complexity

Self-assembly is not just ubiquitous in biology and physics, it is also a language that can be used to describe a physical structure, and measure its complexity and modularity. We developed a versatile lattice model of self-assembly, insights from which we apply to more general structures such as molecules and protein complexes. We also show that genetic algorithms can be used in conjunction with our lattice model to answer questions about the emergence of symmetry and modularity in biological evolution.

17

#### Sintering

Sintering is a process in which individual crystals touch and then heal together, forming grain boundaries. This can transform a powder or slurry into a solid material. By studying the early stages of sintering, just after contact, we have found a solitary wave solution to the governing equations, which implies a novel scaling for the initial growth of the neck between two particles with time. This may have implications for the flow and solidification of materials such as magma and cryogenic ice slurries.

18

#### Sphere Packing

Although equal sized spheres can be packed most densely in crystalline arrangements, in practice macroscopic hard spheres can only be brought to the random close packed density of 64%. This state provides a first approximation to the structure of granular materials, such cement or oil sands. By studying polydisperse spheres, and approximately mapping the problem onto one dimension, have constructed a simple and fast algorithm to estimate the random packing density of any size distribution.

19

#### Structure of the Early Universe

A remarkable property of the large-scale structure of the universe is that its correlations show a well defined power law behavior which strongly points to complex properties in the sense of self-organized criticality. However, their present interpretation within the standard model of cosmology is that they are a sort of accident. Our goal is to understand whether gravitational clustering of mass points alone may generate complex scale-invariant structures. Preliminary results in one dimension provide a solid support to this hypothesis.