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.
Major outcomes are: we applied the probabilistic methods (one of them is Bellman function method, another one is randome geometric constructions method) to solve several outstanding problems in Harmonic Analysis such as: A_2 conjecture, bump conjecture, David--Semmes conjecture. The objectives were to solve several outstanding problems in Harmonic Analysis and to attract young researchers to learning the new probabilistic techniqies to do that. Several summer schools and conferences organized with my participation served this goal. Theose are: 1) Fall school at lake Arrowhead (with Christoph Thiele), CA, Oct. 2010. 2) AMS section meeting at Los Angeles (with Christoph Thiele), CA, Oct. 2010. 3) American Institute of Mathematics: Workshop on weighted singular integrals (with Maria Reguera and Svitlana Mayboroda), Oct. 2011, Palo Alto, CA. 4) Summer School in Antibes, France (with Laurent Barachard and Vasily Vasyunin), June 2011. 5) Summer School in Mittag-Leffler Institute, Stockholm, Sweden (with H^okan Hedenmalm and Vasily Vasyunin), July, 2012. During the implementation of grant research the opportunities for training and professional development were provided. Many young people got involved in the pilot programs, including Ignacio Uriarte-Tuero, but also SChun Yen Shen, Maria Reguera, Nick Boros, Prabhu Janakiraman, Nick Pattakos, Alexander Reznikov, Matt Bond, Maria Carro, Ben Jaye. Several summer and winter schools have been organized by me (with other people) as well as one AIM conference. Young people mentioned above got several important results and defended their PhD in the framework of PIs research in this grant. The project solved two outsatnding problems in Harmonic Analysis: David-Semmes problem and bump conjecture, it also paved the road to solving two other outstanding problems: A_2 conjecture (by T. Hytonen) and two weight Hilbert transform conjecture (finished by M. Lacey). The new methods introducing random geometric constructions and Bellman function were developed by PIs and played the major role in solving all of these 4 problems. We would like to mention that David-Seemes problem concerns connectiveness of disordered patterns. Such or analogous problems are pervasive in pattern recognition and its applications to many applied questions from security to brain research.
