Local Geometry of Real and Complex Analytic Mappings

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My work is at the interface of analytic geometry and commutative algebra, two major branches of modern mathematics with applications in science and engineering. It deals with the analysis of polynomial and analytic functions and the geometry of spaces on which they live. The main subject of my research program is the study of local degeneracies (or singularities) of mappings. Such minute degeneracies of mappings can completely destroy the geometric regularity of objects to which they are applied. One of the main goals of my program is to establish a computable classification of singularities of analytic and polynomial mappings, with the emphasis on applicability of computer algebra. Theory of singularities has applications beyond mathematics, for example in forecasting of earthquakes. Another ramification of my work is a theory that will give us a better understanding of the geometry of partial differential equations in complex analysis. Such equations are used in science and engineering to describe electromagnetism and many other natural phenomena. This research provides opportunities for the training of graduate students and postdoctoral fellows and involves international collaboration with other mathematicians.