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

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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 shallcontinue a line of research initiated several years ago on the physics of fibre arrays such as hair. Begun originally tounderstand 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