Grant: Discovery Grants Program - Individual

Applications provide many examples of physical and biological systems whose properties depend inherently on a multitude of spatial and temporal scales. Studies of such systems and their properties at the nanoscale are particularly intriguing and would be impossible without mathematical models leading to some of the most challenging problems in modern applied mathematics. Many such systems have brought and will be bringing tremendous changes in our everyday lives. Today, we cannot imagine our life without telecommunications systems utilizing nanostructure lasers, without nanostructure-based electron devices encountered everywhere, from satellite television systems to virtually every home as part of CD players. Solar cells incorporating such nanostructures are already used extensively in space and terrestrial programs and are essential component of green-energy development programs, while the usage of nanostructures in the areas ranging from bioimaging and biosensing to cosmetics and personal care products continue to grow.In order to use these structures efficiently, we need to know their properties which are truly fascinating. For example, the color of a piece of gold, while remains golden as it reduces to microns, changes substantially in the regime of nanometers. Decreasing the size of carbon materials to the nanoregime leads to systematic increases in their strength. We enter the entire new realm - the realm of the nanoworld where mathematics offers primary tools for its systematic studies. In this proposal we will first develop mathematical models for studies of properties of physical and biological stuctures at the nanoscale. Then, we will undertake a systematic analysis of such mathematical models for two classes of such nanostructures: low dimensional nanostructures, such as quantum dots or ""artifical atoms"", and Ribonucleic Acid (RNA) nanostructures. Finally, based on this analysis, we will study properties of these nansostructures, answering a number of natural questions that remain unexplored today.