RUI: Diverse Manifolds in Random Media

这个项目涉及随机介质中的标度特性，涉及热波动和无序诱导的粗糙化之间的相互作用。具体的问题是随机介质中的定向聚合物（DPRM）和磁铁中杂质固定的畴壁。DPRM中的全局优化问题与流域三角洲、毛细血管网络和大脑神经元阵列中的问题类似，都表现出一种超微树结构。DPRM的Cole- Hopf变换将其与噪声Burgers方程中的动态标度问题联系起来，后者与远离平衡驱动的晶格气体的统计力学性质有关。这个项目是通过(a)专注于随机介质中的定向聚合物问题，(b)展示该问题与其他问题（如流域三角洲、毛细血管网络和大脑中的神经元阵列）之间的数学等价性来研究各种问题的共同特性。* * *

9528071 Halpin -Healy This project involves scaling properties in random media involving interplay between thermal fluctuations and disorder induced roughening. The specific problems are directed polymers in random media (DPRM) and impurity-pinned domain walls in magnets. The global optimization problem in DPRM is similar in exhibiting an ultrametric tree structure to the problems in river basin deltas, capillary blood vessel networks and neuronal arrays in the brain. A Cole- Hopf transformation of the DPRM relates it to dynamic scaling problem in noisy Burgers equation, the latter relevant to the statistical mechanical properties of a far- from-equilibrium driven lattice gas. %%% This project is to study the common properties of a variety of problems by (a) focusing on the problem of directed polymers in a random media and (b) demonstrating the mathematical equivalence between this problem and others such as river basin deltas, capillary blood vessel networks and neuronal arrays in the brain. ***