Erhrhardt This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).
The PI plans to investigate certain concrete problems in Operator Theory, mostly centered around Toeplitz operators and matrices as well as their modifications and generalizations (Toeplitz+Hankel, Wiener-Hopf-Hankel operators). The asymptotics of the determinants of these objects is of particular interest and is related to some 30-year conjectures, which have only been partially solved, and when solved have led to new open problems. The tools for studying the asymptotics of such determinants include asymptotic spectral theory, Banach algebra techniques (non-commutative Gelfand theory), certain aspects of Numerical Analysis (Stability Theory), and also Wiener-Hopf factorization theory. The asymptotics of determinants is interest if one wants to study the asymptotic eigenvalue distribution of, for instance, Toeplitz matrices (which is one naive, but only partially solved problem). Another application are certain problems in Random Matrix Theory. There a first type of problem concerns the linear statistics for various random matrix ensembles in certain scaling limits. A second class of problem is related to the asymptotics of so-called gap probablities, again for various types of random matrices. It has turned out that both the distribution function for the linear statistics as well as the gap probabilities can be expressed in terms of the determinants of certain operators, where the underlying operator depends on the matrix ensemble under consideration. The focus of this research project is to use this link and apply the techniques from operator theory to solve those problems in Random Matrix Theory.
The proposal tries to connect two areas of Mathematics, Operator Theory and Random Matrix Theory. Operator Theory studies the analytical properties of certain mathematical objects and has a lot of hard and powerful tools available. Random Matrix Theory is a field which has already many connections to other branches of Mathematics. It has also an applied aspect in that its goal is to model and explain complex systems with an inherent random behavior. Such systems arise in models of Statistical Physics, but also in, for instance, Wireless Communication. There are very ``basic'' problems in Random Matrix Theory, which have not yet or only recently found their solution. It has turned out that some of these questions can be formulated in terms of the asymptotics of certain determinants. Operator Theory along with its sophisticated tools can be considered as being capable of tackling and solving these questions. It is the goal of the proposal to study in particular those problems in Operator Theory that arise from Random Matrix Theory. This will hopefully solve some of the basic questions there. On the other, the influence from Random Matrix Theory will likely lead to the development of new and the considerable extension of existing tools and methods in Operator Theory which are connected to asymptotic determinant problems as well as spectral theory.
