You may wonder what is "geometric analysis?" Simply put, it is an area of mathematics which incorporates both differential geometry and analysis. Geometric analysis has a myriad of applications to physics, applied mathematics, numerical analysis, computer graphics, image recognition, medical technology, and more! For example, we use partial differential equations (analysis) to understand and hence predict the behavior of heat, light, and waves. The equations of general relativity are in fact Riemannian curvature equations which use differential geometry to describe the way gravity curves space and time. String theorists model quantum physics using certain Riemannian manifolds (differential geometry) and operators acting on vector bundles (analysis). Physicists develop models to describe our Universe and its natural phenomena, and understanding these models often requires both differential geometry and analysis. In computer graphics and image recognition, one uses spectral data (or moment/content data) to describe shapes: this interaction of geometry and analysis is spectral geometry.

My research interests include:

• Spectrum of generalized Laplace type operators (quantum mechanics).
• Length spectrum and dynamics (classical mechanics); dynamical zeta functions.
• Heat and wave operators, spectral invariants, and spectral zeta functions.
• Pseudodifferential operators and microlocal analysis (the ``alphabet calculus'').
• Linear and non-linear partial differential equations modeling natural phenomena, and collaborations with applied mathematics.
• Singular spaces, in particular manifolds with conic and edge singularities.
• Compact manifolds-with-boundary which have specific structure near boundary, such as b-manifolds, "edge" manifolds (manifolds with corners), and non-compact manifolds having specific geometric structure at infinity: asymptotically locally Euclidean or asymptotically conic scattering, asymptotically Euclidean, asymptotically hyperbolic, conformally compact.
• Canonical Riemannian metrics. (The "best" Riemannian metrics).
• Analytic number theory, especially in connections with spectral geometry and dynamics.