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

这是一个跨学科的建议，研究长期存在的和新的问题，在流体和连续介质力学和数学生物学，其中，由于其动力学或统计性质，需要从几何，拓扑和统计的新工具来解决。这是一个数学上的独特提案，研究将在实验工作的有力支持下进行。自开尔文的原子涡旋理论以来，拓扑思想在科学中发挥了重要作用。虽然短暂的，部分是因为它的结涡大多是不稳定的（湍流的根本原因之一）的理论，但仍然开始泰特的不朽工作的拓扑性质的结。从那时起，尽管认识到从流体动力学到发育生物学等领域的拓扑结构的动态重构的重要性，但进展有限，部分原因是所涉及的过程的非线性特征和缺乏合适的模型系统。拟议的研究分为三个子项目。1.物理纤维束的统计。在与联合利华研究所（阳光港）的科学家合作中，我们将继续几年前开始的关于纤维阵列（如头发）物理学的研究。这项研究最初是为了了解如何改进洗发水和护发素等产品，利用统计物理学和流体力学的思想，构建了物理纤维束弹性特性的理论。我们将扩展这项工作，以解决基本的数学问题有关的纠缠的物理束，包括统计的接触和交叉内束，真正的knotedness之间的关系，物理knotedness，和纠缠，和动力学的猝灭随机固有曲率束。2.肥皂膜的拓扑重排与奇异性。从磁流体力学到蛋白质，关于打结配置如何在耗散下放松到能量最小值，受到拓扑约束，存在基本的开放问题。这些过程的可视化和控制的实验挑战激励了对实验室实现的搜索，其中动态可重复地和按需发生。我们已经确定了这样一个例子：相互转换的最小表面（由肥皂膜）触发缓慢的边界deformation.In尽管大量的现有工作的最小表面，很少，如果有的话，已经研究了它们之间的相互转换的动力学。我们的目标是开发一个分类方案，这些过程和预测偏微分方程描述的动力学演化和位置的奇异性，为每个普遍性类。3.胚胎藻类的拓扑反演。当刘易斯·沃尔珀特说不是出生、结婚或死亡，而是原肠胚形成，这才是你生命中最重要的时刻时，他指的是动物胚胎通过发展一个向内的通道，最终成为胃系统，从而改变其拓扑结构的过程，从简单连接到非简单连接。虽然动物胚胎由于难以控制和可视化而难以系统地研究，但最近已经清楚的是，发生在多细胞藻类中的一个类似于原肠胚形成的过程提供了一个更简单但可靠的替代方案，可以进行仔细的实验。这种胚胎倒置的过程在整个一类多细胞藻类中起作用，系统性的变化取决于生物体的大小。尽管目前有实验和经验的信息，但还没有数学描述这样一个物体是如何把自己从里到外的。我们的目标是利用连续介质力学和微分几何的原理来理解数学生物学中的这一迷人现象

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