The proposed research is at the intersection of Geometric Measure Theory and Harmonic Analysis. The main objective of Geometric Measure Theory is to find structures in seemingly unstructured, fractal-like patterns. The classical Harmonic Analysis studies wave propagation, and investigation of singular integral operators is a crucial part of the modern approach. Our continuing research, as well as the work of several other groups of mathematicians in the US and abroad, has demonstrated that new knowledge can be obtained by exploring the interaction between these two areas.
As a result of our proposal we expect to solve several important problems in Geometric Measure Theory as well as in Harmonic Analysis. The pattern recognition (i.e., problems in Geometric Measure Theory) would be advanced by using methods originating in Harmonic Analysis, and vice versa. We also expect to develop new methods to study both patterns and waves. It is expected that these newly developed techniques will have impact to adjacent areas of engineering and computer sciences such as image processing and data compression.
