On the face of it, a matrix is just a box of numbers. But the encoding of information in the form of matrices is one of the most powerful ideas of modern mathematics; the language of matrices underpins nearly every area of pure and applied mathematics, and anchors the frameworks of physics, engineering, and other sciences. The study of randomly generated matrices is one of the most exciting areas of mathematics today, with theoretical advances giving us an ever-growing picture of what "typical" matrices are like, and providing us with important applications in areas like image compression and the analysis of the Big Data that surrounds us. In this project, the PI will continue her research program investigating the behavior of very large random matrices, and how that behavior depends on the way they are generated; this will advance the theory, which will help better understand the applications to some of the areas mentioned above. She will also complete her monograph "The Random Matrix Theory of the Compact Classical Groups", which gives an overview of what has been learned about randomly generated matrices with a particular relationship to high-dimensional geometry; this activity will disseminate difficult technical knowledge about random matrices to a broad audience. The PI is active in teaching at all levels, and particularly in mentoring and encouraging girls and young women interested in mathematical careers; those training and outreach activities will have great impact on the diversity of those entering STEM fields.
The monograph mentioned above will describe the remarkable advances in the theory of Haar-distributed random matrices in recent years. Topics will include central limit theorems for the entries of principal sub-matrices and for general lower-dimensional projections, quantitative limit results for the empirical spectral measures, self-similarity patterns within the eigenvalue distributions, concentration of measure with geometric applications, including new proofs of the Johnson-Lindenstrauss lemma and Dvoretzky's theorem, and results on the distribution of the characteristic polynomial, with connections to the Riemann zeta function. New research projects include the study of random matrices whose distributions are invariant under rotations in matrix space, continued work on quantitative approximations of empirical spectral measures for various classes of random matrices, and in various metrics, and problems in random topology connected with topological data analysis.
