## Grants and Contracts Details

### Description

Status | Finished |
---|---|

Effective start/end date | 6/1/07 → 5/31/12 |

### Funding

- National Science Foundation: $139,859.00

- Perry, Peter (PI)

Intellectual Merit. The Principle Investigator will continue his research
in three areas of geometric analysis: (1) resonances in geometric
scattering, (2) nonlinear dispersive equations, and (3) harmonic analysis on
two-step nilpotent Lie groups.
Asymptotically real- and complex-hyperbolic manifolds are complete,
non-compact Riemannian manifolds with "simple geometry at infinity." We
will study the fundamental questions related to the existence, distribution,
and geometric content of the scattering resonances, exploiting recent
progress in scattering theory for these geometric classes. We will analyze
invariants of the scattering data such as determinants of Laplacians and
Selberg's zeta function.
In the past decade, global well-posedness has been shown for nonlinear
dispersive equations such as the Korteweg-de Vries (KdV) equation for initial
data which are highly singular or unbounded at infinity. By studying
the inverse problem for Schrodinger operators with singular and unbounded
potentials, we will extend the inverse scattering method to the KdV equation
and the closely related mKdV equation with these classes of initial data.
Our aim is to obtain a more fully geometric picture of the KdV flow and
develop new tools for qualitative analysis of the solutions.
Two-step nilpotent Lie groups play an important role in spectral geometry
(compact nilmanifolds provide interesting examples of non-isometric
Riemannian manifolds with the same spectral data), in control theory (certain
two-step nilpotent Lie groups are model spaces for control problems),
and in harmonic analysis (the sub-Laplacian is a natural model for hypoelliptic
differential operators). We will study the sub-Riemannian geometry of
certain of these spaces and the associated behavior of heat and wave kernels
for geometric operators on these spaces.
Broader Impact. Inverse spectral theory is the mathematical discipline
that underlies important applications of mathematics to medical imaging,
geophysical prospection, and non-destructive testing. In these applications,
the "geometry" of a physical system (a human body, the earth, or an industrial
product) is reconstructed from the scattering of electromagnetic, seismic,
or acoustic waves. Advances in geometric scattering theory clarify the
relationship between scattering data and geometry. The inverse scattering
method has produced powerful tools to study the propagation of nonlinear
waves in hydrodynamics and nonlinear optics; extending these methods to a
richer set of initial conditions will deepen our understanding of mathematical
models for nonlinear waves.

Status | Finished |
---|---|

Effective start/end date | 6/1/07 → 5/31/12 |

- National Science Foundation: $139,859.00

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