Over the last several years, Greenblatt has been working on resolution of singularities, and he recently has proved an n-dimensional local resolution of singularities algorithm for real-analytic functions. This method is explicit, elementary, and done in coordinates. In his subsequent research, he will apply his methods, using additional ideas when appropriate, to prove theorems involving oscillatory integrals, Radon transforms, and other subjects in which he has done research. In addition to these areas, he will branch out into several other of the diverse areas that relate to resolution of singularities. For example, he intends to work on multilinear generalizations of oscillatory integral operators, problems related to the stability of integrals, and associated problems in algebraic geometry such as those involving multiplier sheaves. In addition, intriguing algorithmic and computational questions arose during the development of [G1], and he plans to investigate such issues in computational algebraic geometry.
Oscillatory integral operators are a part of Fourier analysis, a field with wide application in science and engineering, such as in signal processing, cryptography, and statistics. As a result, improved understanding of oscillatory integral operators resulting from this research has the potential to help in the development of scientific applications that use Fourier analytic methods. In addition, Radon transforms are fundamental to MRI and other medical imaging applications, and also find uses in diverse fields ranging from oil exploration to homeland security. As a result, improved theoretical knowledge of Radon transforms resulting from this research may lead to advances in such fields.
