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

这是一个跨学科的建议，研究流体和连续介质力学和数学生物学中的长期存在的和新的问题，由于它们的动力学或统计性质，需要用几何、拓扑学和统计学的新工具来解决。这项研究将在实验工作的大力支持下进行，这是唯一一项在数学中提出的建议。自开尔文的原子涡旋理论以来，拓扑学思想在科学中发挥了重要作用。尽管这一理论的存在时间很短，部分原因是它的结状涡大多是不稳定的(这是湍流的根本原因之一)，但该理论还是开启了泰特关于结的拓扑性质的不朽工作。从那时起，尽管认识到动态拓扑重构在从流体动力学到发育生物学等领域的重要性，但进展一直受到限制，部分原因是所涉及过程的非线性特征和缺乏合适的模型系统。这项拟议的研究分为三个子项目。1.物理纤维束统计。在与联合利华研究(阳光港)的科学家合作下，我们将继续几年前开始的一系列关于纤维阵列(如头发)物理的研究。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，利用统计物理和流体力学的思想来构建物理纤维束的弹性性能理论。我们将把这项工作扩展到解决与物理丛的纠缠有关的基本数学问题，包括丛内接触和交叉的统计，真结、物理结和纠缠之间的关系，以及具有淬灭的随机本征曲率的丛的动力学。2.肥皂膜的拓扑重排和奇异性。从磁流体力学到蛋白质，都存在一些基本的开放问题，即在受到拓扑约束的情况下，结状构型如何在耗散下松弛到能量极小值。这些过程的可视化和控制方面的实验挑战促使人们寻找动态可重现和按需发生的实验室实现。我们已经确定了这样一个例子：由缓慢的边界变形触发的极小曲面(以肥皂膜为代表)的相互转换。尽管现有关于极小曲面的大量工作，但关于它们之间的相互转换的动力学研究很少，如果有的话。我们的目标是为这些过程和描述每个普适类奇点的动态演化和位置的预测偏微分方程组开发一个分类方案。3.胚胎藻类的拓扑反转。当刘易斯·沃尔伯特说，这不是出生、结婚或死亡，而是原肠形成，这是你生命中真正最重要的时刻，他指的是动物胚胎通过发展一条向内通道，最终成为它的胃系统，将其拓扑结构从简单连接改变为非简单连接的过程。众所周知，由于难以控制和可视化，动物胚胎很难进行系统研究，但最近有一点变得清楚了，即在多细胞藻类中发生的类似于原肠受精的过程，提供了一个简单得多但可靠的替代方案，可以进行仔细的实验。这种胚胎倒置的过程在整个类别的多细胞藻类中运作，系统的变化由有机体的大小决定。尽管目前有实验和经验方面的信息，但目前还没有关于这种物体如何从里到外翻转的数学描述。我们的目标是利用连续统力学和微分几何的原理来理解数学生物学中这一迷人的现象

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