RUI: Diverse Manifolds in Random Media

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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. ***