Geometric Structures in Functional Analysis - Quantization of Infinite-Dimensional Manifolds

CNCSIS grant PNII - Programme `Idei' (code 1194)

Brief description of the research project

The usefulness of the methods and ideas with a geometric flavor is well known to all of the mathematicians interested in areas like complex analysis, functional analysis, or mathematical physics. Examples in this connection are manifold: just recall the notions of convexity in the several variable complex analysis, the Krein-Milman theorem in functional analysis, or the properties of the moment map in hamiltonian mechanics and in the method of geometric quantization. The significance of geometric structures in many areas employing functional analytic methods is however much deeper than just convexity properties of subsets of various topological vector spaces. Geometric features, and particularly the symmetry properties of the geometric objects that underlie the problems we wish to address, are often reflected in properties of certain function spaces or even in the spectral theory of operators on these function spaces. This is a central theme in several areas of contemporary research in mathematics and physics, including the representation theory of finite- or infinite-dimensional Lie groups, the theory of spaces of analytic functions, harmonic analysis on groups and on bounded symmetric domains, spectral geometry of Riemannian manifolds etc. Our aim in the present research project is to point out new geometric features of certain functional analytic objects such as

  • reproducing kernels,
  • spectral functions of operators,
  • embeddings of Hilbert and Krein spaces,
  • generalized Toeplitz operators,
  • invariant subspaces of Hilbert space operators.

    Each of these objects plays an important role in the method of geometric quantization of the symplectic manifolds in Hamiltonian mechanics, and the results obtained in the first stages of the present project should lay the foundations for methods of geometric quantization applying to wide classes of infinite-dimensional symplectic manifolds.

    The main lines of research in our project are the following ones:

    Project manager

    Research team

    Ph.D. student

    Research financially supported in 2009 - 1st phase of the project

    Research financially supported in 2010 - 2nd phase of the project

    Research financially supported in 2011 - 3rd phase of the project