Free Analysis and Adjacent Topics

In mathematics, the order in which operations are performed often matters fundamentally: while multiplying ordinary numbers gives the same answer in either order, multiplying matrices in different orders typically yields different results. This "noncommutativity" is fundamental to the mathematical language of quantum mechanics, where, for example, it gives mathematical expression to the famous Heisenberg Uncertainty Principle. The field of "free analysis" has emerged to study functions of noncommuting variables. These are reminiscent of the ordinary functions encountered in calculus, but these functions can be evaluated not just at numbers, but at matrices of arbitrary size, and importantly, still make sense when the variables do not commute. Many of the familiar operations of calculus have analogs in this setting, but often surprising differences emerge. This project will advance the mathematical foundations of free analysis, with emphasis on connections to random matrix theory, free probability, and many other areas, both pure and applied. Of particular interest will be applications to quantum information theory. The project also supports the training of graduate students, contributing to the development of a scientifically skilled workforce. The project will, among other things, explore connections between free analysis and the statistics of large random matrices. A basic problem in this area is understanding statistical properties of noncommuting functions of random matrices (systems of random unitary matrices arise, for example, in connection with random quantum states). A key technical tool will be the invariant theory of matrix tuples, which will be studied through a new Hilbert space lens. The resulting function space is a natural object whose structure the project will systematically investigate. Additional directions include noncommutative analogs of classical limit theorems for determinants of structured matrices, and the development of some "free" analogs to the combinatorial theory of symmetric functions. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.