Collaborative Research: Topics in Infinite-Dimensional and Stochastic Dynamical Systems

虽然许多物理、生物、气候和金融过程似乎受到随机或随机力量的影响，但这些过程背后也存在一些连贯的结构，这些结构提供了某种程度的可预测性。该项目为确定这些隐藏结构和分析几种应用中出现的具体情况奠定了基础。其中包括果蝇翅膀的胚胎发育及其新近发现的与细胞膜钾通道电流的关系。这个项目的一部分是开发和使用复杂的数学技术来理解离子电流。果蝇模型对哺乳动物的发育具有启示意义，并可能导致对一些严重出生缺陷的原因的理解。抽象数学研究的应用还包括理解受随机扰动影响的其他动力系统，包括高激发等离子体的密度分布或合金的精细结构，以及缺陷如何随时间分布和演变。该项目建立在主要研究者和其他人过去的工作基础上，以建立嵌入在复杂动力系统相空间中的连贯结构的存在，无论是有限维还是无限维，都是确定性的或受随机强迫的。在项目过程中发展的基本和抽象理论是在物理和生物科学中具体和观察到的现象背后，特别是在分子，微观或纳米尺度上。需要无限维的动力系统来表示受物理定律或生化过程影响的量的时空波动，例如发育中的胚胎苍蝇翅膀中骨形态蛋白的分布，细胞膜中离子通道的电流，相对论性等离子体的密度，或合金中微观缺陷的运动，仅举几个本项目中考虑的系统。此外，尽管这些可能很复杂，但必须考虑到由于环境中的热波动或其他波动以及在小尺度上对数量的不精确测量而产生的随机扰动。虽然人们不能希望给出所有受复杂空间和时间相互作用影响的状态的精确表示，但由于鲁棒（但可能是隐藏的）结构的存在，人们有时可以收集信息，例如不变流形及其不变叶形，其存在是由控制所研究过程的定律所暗示的。该项目的目标是发现这种结构存在的条件，即使是在随机强迫的情况下，并检查在特定的物理和生物系统中潜在的方程的含义。

While many physical, biological, climatological, and financial processes appear to be subject to random or stochastic forces, there are also coherent structures underlying these processes that give some measure of predictability. This project is laying the groundwork both for the determination of these hidden structures and for analyzing specific situations arising in several applications. Among these is the embryonic development of the wing of a fruit fly and its newly discovered relation to current through potassium channels in the cell membrane. Part of this project is to develop and use sophisticated mathematical techniques to understand that ionic current. The fruit fly model has implications for mammalian development and may lead to an understanding of the cause of some serious birth defects. Applications of the abstract mathematical investigations also include understanding of other dynamical systems subject to random perturbations, including the density distribution in highly excited plasmas or the fine structure of an alloy and how defects are distributed and evolve in time. This project builds upon the past work of the principal investigators and others to establish the existence of coherent structures embedded in the phase space of complex dynamical systems, both finite- and infinite-dimensional and both deterministic or subject to random forcing. The fundamental and abstract theory to be developed during the course of the project lies behind concrete and observed phenomena in the physical and biological sciences, particularly at the molecular, microscopic, or nano-scale. Infinite-dimensional dynamical systems are required to represent the temporal and spatial fluctuations of quantities subject to physical laws or biochemical processes, such as the distribution of bone morphogenic protein in a developing embryonic fly wing, the current through an ion channel in a cell membrane, the density of a relativistic plasma, or the motion of microscopic defects in an alloy, to name just a few of the systems considered in this project. Furthermore, as complex as these may be, stochastic perturbations must be considered due to thermal or other fluctuations in the environment and imprecise measurement of quantities at small scales. While one cannot hope to give exact representations of all states subject to complex spatial and temporal interaction, one can sometimes glean information due to the presence of robust, but possibly hidden, structures such as invariant manifolds and their invariant foliations whose existence is implied by the laws governing the processes under investigation. The goals of this project are to discover the conditions under which such structures exist, even when stochastically forced, and to examine the implications in the particular physical and biological systems underlying the equations.