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