Principal Investigator: Mihai Putinar The project focuses on three themes, all well connected to each other: quadrature domains for harmonic functions via hyponormal operators of rank-one self-commutator, cubature formulas in several variables for polynomials of a fixed degree via dilation theory (with prescribed commutators, for tuples of self-adjoint operators) and the spectral picture of spherical isometries in a Hilbert space. The basis for the first subject is the progress made in the last five years in the study of quadrature domains with linear algebra techniques, and specifically using determinantal functions of pairs of matrices. There are many analogies with classical perturbation theory of self-adjoint operators which point to several open problems resulting from this approach. The second theme is based on the observation made a couple of years ago by the P.I. that practically all facts connected with multivariate cubatures can be translated into familiar terms of dilation theory (of commuting Jacobi matrices). It is expected that this point of view will be successfully exploited by both sides. The third subject is a specialization of the second, for positive measures supported by odd spheres. The complex variables and the associated Toeplitz operators (which are typical examples of spherical isometries-- ubiquitous objects in modern operator theory) are expected to carry in a closed, flexible form the basic information about the original measure. Inverse problems, that is the reconstruction of complex objects from partial data, are everywhere present in modern science and technology. To give a few classical examples we can mention the vibrations of a string, the shape of a planet knowing its distant gravitation field, or the evolution of an oil spill in water. The proposal is aimed at solving some mathematical questions related to inverse problems related to shapes or to mass distributions. A history of more than one century of remarkable discoveries in this area will be comb ined with recent advances in modern mathematics. The resulting work will be significant for the present research frontier in mathematics and potentially for applied fields such as geophysics, image processing or tomography.