The project is aimed at unifying three distinct matrix models (one quantum, the other random/thermodynamic and the third purely algebraic) known today in connection with planar elliptic growth, an interface dynamics covering a large variety of natural and theoretical phenomena. The existence of exact solutions to the equations governing such growth dynamics, their complete integrability, at least in the case of a flat metric, and the impressive amount of converging recent research on these topics make the project timely and unavoidable. As mathematical tools, the proposal will rely on and develop new facets of complex orthogonal polynomials, moment matrices and potential theoretic operators arising in the study of planar or spatial shapes. Particular emphasis will be put on the maximum entropy method for recovery of shade functions from indirect measurements, such as partial geometric tomographic data and the asymptotics of the spectra of random non-hermitian matrices.
Following a decade of intense and vibrant discoveries in fluid mechanics and quantum physics, the present proposal addresses a series of mathematical questions of high interest for these specific areas of modern science. The tradition of encoding planar shapes and the geometry of volumes into numbers or algebraic symbols goes back to the XVII-century landmark contributions of Rene Descartes and his followers in what is today called ?analytic geometry?. Much later development and applications of electricity, magnetism and atomic science we all benefit today was possible by a second major step into the same direction, that is the representation of complex physical entities as large arrays of numbers or symbols called matrices. The proposed project focuses on the study of matrix models arising in the current research of moving interfaces of fluids or more sophisticated but similar media. Such moving boundaries phenomena are illustrated by cancer growth, crystal formation, ice melting, oil reserves, carbon monoxide sequestration and plasma dynamics. The PI is assisted by a group of enthusiastic doctoral students and he is connected, via several collaborative works, with experts in other fields of mathematics, physics or engineering.
A number of modern technological advances depend on a refined analysis of signals, including large data, as for instance images. The mathematical tools developed for this aim are known as structured function systems. The codification and reading of data from representations on such structured systems (for instance wavelets) was lying at the foundation of the present project. The PI has contributed to the construction and understanding of the structure of multivariate tight wavelet frames, solving a long standing problem in this area, in two dimensions, the major frame for image analysis. Also he has continued his work on inverse problems, with an emphasis this time on a variational method based on entropy estimates. In such a way, a variety of hard to analyse real life data (related to chaotic dynamical systems) can be conditioned for classical inversion methods. The theoretical work of the PI is represented by a final and definitive step, after some thirty years of continuous activity on the subject, referring to the classification of multivariate systems of commuting linear operators. The PI edited two volumes containing high quality pure, respetively applied, mathematical works. He has also written a couple of survey articles, disseminating for a larger public recent advances in polynomial optimization.
