Multiscale models for nanostructures with geometric phases and time-dependent coupling

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Tiny systems that can be as small as 1/1000th the diameter of a human hair continue to bring tremendous revolutionary changes in our everyday lives. In this program, we are particularly interested in an important class of such systems known as low dimensional nanostructures (LDNs). They can be self-assembled and their extraordinary properties can be engineered to a specific application at hand. As a result, their potential is virtually unlimited. Since many effects, that were negligible at larger scales, cannot be ignored any longer for LDNs, the mathematical modelling is decisively becoming a major tool in their studies.****The proposed research program is aimed at further advancement in the development and applications of multiscale mathematical models for the analysis of LDNs and other systems of interest, and expanding it to a new state-of-the-art mathematical and computational framework for analyzing such systems. Specifically, the main goal is to account systematically for geometric phases and time-dependent coupling effects by developing mathematically coherent approaches to the study of the systems where such effects are essential. Since its first appearance in quantum systems, the elegantly simple mathematical concept of geometric phase has also been successfully applied to a wide range of classical and hybrid quantum-continuum systems. Nevertheless, many mathematical challenges in this field are still on only scarcely explored horizons.***The program will capitalize on the developed expertise and advances already made by the PI's group. Application-wise, major focus will be given to LDNs and to several classes of biosystems at the molecular and nanoscale levels, in particular Ribonucleic acid nanostructures and photosynthetic complexes, as well as to hybrid quantum-continuum systems.****The program will, firstly, allow a systematic study of properties of such systems where geometric phases and time-dependent couplings are essential. Although such systems are pervasive in natural and man-made environments, their systematic studies for several important classes of problems are currently absent. Secondly, it will provide a better understanding of the connection between the geometric-phase-induced forces and important dynamic phenomena in a field of great fundamental and technological interest. Thirdly, given their ubiquitous nature, it is expected that the models and tools developed in this proposal will assist in addressing other challenging problems of mathematics and its applications. Indeed, methods and tools to be developed within this program are expected to be indispensable for a quite general class of problems where the influence of geometric phases and dynamic coupling effects on the properties of the systems is significant. The results may not be restricted to just the nanostructures and can be useful in studying other important systems in science and engineering.**