Aperiodic order and almost periodic measures

• Applicant: Dr. Timo Spindeler
• Subject Area: Mathematics
• Term: from 2018 to 2021
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 415818660

Since the discovery of quasicrystals by Dan Shechtman in 1982, scientists of various disciplines have been interested in understanding the different kinds and degrees of order and disorder in solids that are not as well-ordered as crystals. Mathematicians describe the outcome of the diffraction experiment, which produces a characteristic pattern on a screen, using the so-called diffraction measure, which measures the internal order of the quasicrystal. The two essential components of this measure, the pure point part and the continuous part, are indicators of order and disorder. Meanwhile, systems that lead to a pure point diffraction, are well understood. However, little is known when (additionally or exclusively) a continuous proportion occurs. The goal of the planned project is therefore twofold. On the one hand, concrete systems containing continuous components are to be examined in detail. On the other hand, the continuous part shall be better understood by means of abstract methods. Here, it is known that it consists of two components, the absolutely continuous part and the singular continuous part. However, it is generally unknown how to determine or characterise these two components of the diffraction measure. For this purpose, the properties of almost periodic measures required for the construction of the diffraction measure are studied closely in the project. The results will be proven for more general locally compact Abelian groups, since this is a natural framework from a mathematical point of view. In addition, this project leads one step further to understanding the inverse problem: Is it possible to uniquely determine the structure to be examined on the basis of the diffraction measure (or diffraction pattern)? The answer is generally 'no', since different arrangements can lead to the same diffraction measure. Therefore, the goal is to extract as much information as possible about the underlying system from the diffraction measure.

• DFG Programme: Research Fellowships
• International Connection: Canada
• Host: Professor Dr. Nicolae Strungaru