The research area of Geometric Analysis historically has grown out of the study of calculus and differential equations involving curves and surfaces or domains with curved boundaries. This has been the origin of many mathematical disciplines such as Differential Geometry, Lie Group Theory and the Calculus of Variations, which are subareas of Geometric Analysis. The development of techniques to handle the mathematical intricacies of doing analysis on manifolds (the higher dimensional and intrinsic analogues of curves, surfaces and domains) is fundamental to much of Engineering and Physics. It is therefore not surprising that much of the impetus for current research in Geometric Analysis comes from these disciplines. Physics in particular, through theoretical developments in relativity theory, quantum mechanics and string theory, has spurred many exciting new research directions in geometric analysis. For instance, as a consequence of these motivations, tremendous progress has been made on topological classification problems for manifolds.
Problems in Applied Mathematics or Engineering can also give rise to deep and fascinating geometric questions. An example is the following: by carefully listening to a drum being played can you reconstruct the drum? More specifically do the characteristic frequencies found in a Fourier decomposition of the sounds coming from the drum allow you to mathematically recover the particulars of the drum. Mark Kac put it succinctly, "Can you hear the shape of a drum"? One can pose a similar inverse problem on a general Riemannian manifold, which is a manifold endowed with a local metric (distance function); namely, from a knowledge of the frequencies of the standing solutions of the wave equation on a Riemannian manifold, how much of the manifold and its metrical properties can one recover?
This kind of broad interplay between physical motivations, geometric reasoning and various analytical methods is characteristic of the activities of our faculty and graduate students doing research in Geometric Analysis.