Geometric, Topological, and Statistical Dynamics in Soft Matter and Mathematical Biology

This is an intradisciplinary proposal to study both long-standing and new problems within Fluid and Continuum Mechanics and Mathematical Biology, and which, due to their dynamical or statistical nature, need to be addressed with new tools from Geometry, Topology and Statistics. Uniquely for a proposal in Mathematics, the research will be carried out with strong support from experimental work.

Topological ideas have played an essential role in science ever since Kelvin's Vortex theory of atoms. Though short-lived, in part because its knotted vortices are mostly unstable (one of the root causes of turbulence) the theory nevertheless served to initiate Tait's monumental work on topological properties of knots. Since then, despite recognition of the great importance of dynamic reconfigurations of topology in fields ranging from fluid dynamics to developmental biology, progress has been limited, in part due to the nonlinear character of the processes involved and lack of suitable model systems. The proposed research divides into three sub-projects.

1. Statistics of physical fibre bundles. In collaboration with scientists at Unilever Research (Port Sunlight), we shall
continue a line of research initiated several years ago on the physics of fibre arrays such as hair. Begun originally to
understand how to improve such products as shampoos and conditioners, the research uses ideas from statistical physics and fluid mechanics to construct a theory for the elastic properties of physical fibre bundles. We will extend this work to address fundamental mathematical issues related to the entanglement of physical bundles, including the statistics of contacts and crossing within bundles, the relationship between true knottedness, physical knottedness, and entanglements, and the dynamics of bundles with quenched random intrinsic curvature.

2. Topological rearrangements and singularities of soap films. From magnetohydrodynamics to proteins there are fundamental open questions regarding how knotted configurations relax to energetic minima under dissipation, subject to topological constraints. The experimental challenges in visualization and control of these processes motivate the search for laboratory realizations in which the dynamics occurs reproducibly and on demand. We have identified such an example: interconversions of minimal surfaces (represented by soap films) triggered by slow boundary deformation.
In spite of the large amount of existing work on minimal surfaces, little, if anything, has been studied regarding the dynamics of interconversion between them. We aim to develop a classification scheme for these processes and predictive PDEs describing the dynamical evolution and location of singularities for each universality class.

3. Topological inversion of embryonic algae. When Lewis Wolpert said [i]t is not birth, marriage, or death, but gastrulation which is truly the most important time in your life, he was referring to the process by which an animal embryo changes its topology from simply connected to non-simply connected by developing an inward passage that eventually becomes its gastric system. While animal embryos are notoriously difficult to study systematically because of difficulties to control and visualize, it has recently become clear that a process that occurs in multicellular algae, and which is analogous to gastrulation, provides a much simpler but faithful alternative that is amenable to careful experimentation. This process of embryonic inversion operates in an entire class of multicellular algae, with systematic variations determined by the organism size. In spite of the experimental and empirical information currently available, there is as yet no mathematical description of how such an object can turn itself inside out. Our goal is to utilize the principles of Continuum Mechanics and Differential Geometry to understand this fascinating phenomenon in Mathematical Biology

The three sub-projects within this proposal will have different types of potential impact, as summarized below.

The work on statistics, entanglements, and dynamics of fibre array bundles grew out of industrial interest (Unilever) associated with very basic questions about the properties of hair. Our work thus far has had direct impact within Unilever through the development of imaging and analysis methods to quantify the properties of hair, as needed to assess personal care products. We expect continued impact on this industry as we seek to address problems involving friction within bundles and their movement. The longstanding work with Unilever is also relevant to our work on Plateau border dynamics, which is very important to the wetting of fibres, a problem of central importance to the action of surfactants on hair. Continued collaboration with Unilever scientists will assure close alignment with their research goals. As well, the wealth of materials that consist of fibre arrays is enormous, from felt to insulation, and we anticipate that development of mathematical models for their gross properties, based on the particular details of the microstructure, will be of broad interest. The intense worldwide interest generated by our first paper on this subject serves to indicate how the research area can be used as a case study for the wider public in the application of mathematics to real-world problems.

The work on dynamical transformations that take one minimal surface to another is based on a comprehensive approach, involving theory, numerics and experiments. Here too there has already been impact on the wider public, through the BBC Four program "POP! The Science of Bubbles" that aired in 2013. We are in discussion with the producer of that program about possible future programs on the science of everyday phenomena. With these studies we have shown again how mathematics (and laboratory experiments) can be used to address universal questions on a broad range of length scales. Our work is regularly used in the classroom to demonstrate singularities and bifurcations, and we will work to create a standardized module for physics education that illustrates these points. Although the mathematical details of the work in this proposal may be technical and abstract, the problem that we shall investigate is so easily demonstrated with little equipment, and has such a natural beauty, that it is a perfect vehicle to introduce abstract mathematical concepts to a lay audience. In particular, the fact that a simplified version of the experiment can be done in one's kitchen means that children will naturally be drawn to the subject. There are many opportunities to present this research to the general public, for DAMTP and the University of Cambridge have in place programs specifically for this. These include the annual Cambridge Summer Science Festival, lectures for school-age children through the Millenium Mathematics Project, and annual Open Houses in DAMTP.

Our work on embryonic inversion will have obvious impact on developmental biology, and the PI has forged links with scientists and laboratories worldwide interested in these problems. As developmental processes analogous to inversion are so ubiquitous, we envision that the theoretical and computational advances made in this sub-project will be of use to biomedical researchers studying development. We will conduct public outreach activities through various learned societies, as we have done in the past (including the Royal Society Summer Science Exhibition), and also through an upcoming Royal Society Discussion meeting on Bio-Inspiration in 2015. But we also envision interesting avenues for the design of new materials with tunable shapes through the incorporation of shape-changing elements. Development of a theoretical framework for the optimality of such shape changes may lead to interesting developments in materials science.