I N T E R E S T S
This list is not exclusive, and the Institute welcomes scientists pursuing other
fields in the theoretical sciences, including mathematics and quantitative
biology.
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.
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. (More)
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. (More)
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. (More)
Fractals 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. (
More)
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.
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.
Materials at high pressures
We have developed methods, dubbed “ab 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'. (
More)
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.
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. (More)
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.
(
More)
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.
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. (More)
Pattern detection in microarray 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. (More)
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. (More)
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. (
More)
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.
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. (More)
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.