At the scale of atomic physics, the classical laws of physics pioneered by Newton must be adjusted to the laws of quantum mechanics discovered by Heisenberg, Schrodinger, Dirac and others. This process is called quantization. Within quantum mechanics, as within classical mechanics, the laws describing the mechanical systems usually display a high degree of symmetry. This is a reflection of the nineteenth century discovery that symmetries correspond to the conserved quantities like energy and momentum. The symmetry related to the law of conservation of momentum expresses itself in the fact that the laws of physics are presumed to be the same everywhere in the universe. In this project the principal investigator will use noncommutative geometry tools such as equivariant index theory to study several interesting problems about quantization involving various symmetries.
The principal investigator will pursue three research projects in geometry quantization and equivariant index theory: (1) applying index theory to geometric quantization of various generalization of symplectic manifolds such as Hamiltonian loop group spaces and log sympelctic manifolds (2) geometrically realizing an irreducible representation of a reductive Lie group as an equivariant analytic index on corresponding coadjoint orbit, and studying its restriction to its maximal compact subgroup (3) generalizing equivariant index theory to noncompact manifolds or groups, based on the KK-theory developed by Kasparov.
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.
