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

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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.