Demands for new mathematical theories to describe nano-scale phenomena are increasing. The project to meet them and to develop mathematical models has been supported by JST CREST from 2008 to 2013. Under the projects, we set four research subjects: 1. Design of K4 Lattice (Sunada Lattice) Characteristics and Its Synthesis, 2. Development of Structured Illumination, 3. Analysis of Ultrafast Phase Separation Dynamics, 4. Achievement of High Heat-Conductivity in Heat-Resistant Polymers. In the present article and the following articles, some of our mathematical challenges are discussed.
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