Abstract: Recent progress in fabrication technologies has stimulated a growing interest in a number of phenomena, sometimes quite counterintuitive, which take place on the nanometer scale. Research in this field has helped to advance a broad variety of applications, including materials science, medicine, electronics, and photonics. It has also helped to explain many already well known phenomena such as the gecko lizard's ability to climb upside down, hummingbird and butterfly colors, the colors of ancient Egyptian glass, and many others. An essential component of this research is mathematical modeling, which includes both computer modeling and analysis. The talk will describe the main challenges that need to be addressed in mathematical modeling in nanoscience and will give a brief overview of the main achievements in this field. Several examples will be presented in which mathematical modeling played a critical role in the understanding of nanoscale phenomena.

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