This list is not exclusive, and the Institute welcomes scientists pursuing other fields in the theoretical sciences, including mathematics and quantitative biology.

• Algebraic number theory
• Complex networks
• Computational topology
• Crystals ex nihilo
• Discrete dynamics
• Fundamental laws of biology
• Genetic regulatory networks
• Low dimensional topology
• Mechanical fractals
• Persistent homology
• Quantum criticality
• Quantum information
• Self-assembly and nanotechnology
• Soft condensed matter theory
• Sphere packing
• Strongly correlated systems
• Superconductivity

For LIMS Scientists — Each summary describes an area of work that is more coarse-grained than an individual paper and is meant for a broad scientific audience, so abstracts are not appropriate here. The maximum length for each summary is 500 characters.

Apollonian packings

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

Fractals for mechanical efficiency

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.

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.

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.

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.