Inverse Problems and Their Applications

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The project concentrates on two related areas of research: inverse problems in additive combinatorics and singularity problem in random matrix theory. As far as the inverse problems are concerned, the PI will be focusing on various settings of the inverse Littlewood-Offord problems. One of his main goals is to obtain a clear picture as to why a multilinear form of independent random variables is highly concentrated on a short interval. For the singularity problem, he proposes to study the least singular value for a number of random matrices of correlated entries such as symmetric matrices and doubly stochastic matrices. In numerous important counting problems in mathematics, understanding the underlying structure has been shown to be fundamental. As our Littlewood-Offord inverse problems connect the concentration of random walks to additive structures such as arithmetic progressions, it is expected that a systematic study of these problems would yield many sophisticated applications. A key part of our project is to develop one of these applications into random matrix theory, establishing the circular law and elliptic law for a broad range of random matrices of correlated entries. These laws support the well-known universality phenomenon observed by physicists and probabilists many years ago: many facts about the distribution of eigenvalues of random matrices seem to be universal, they do not depend on the distribution of the entries.In addition to research, the PI will develop undergraduate and graduate level courses which cover important topics in combinatorics and probability. Some of the topics in these courses will be related to the problems in this project. The PI will continue to write research papers, organize seminars, and give a wide range of talks, from research level to introductory one in order to encourage students into studying mathematics.