Topics in Commutative Algebra, Singularities, and Valuations

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This project will investigate foundational problems in algebraic geometry and related areas of mathematics. Algebraic geometry is the study of solutions to polynomial equations; it has many applications in other areas of mathematics, the sciences, and engineering. The project will focus on the structure of algebraic transformations over arbitrary fields and real and complex analytic transformations. The results of this research have potential for practical application to computer science.The research in this project is in the areas of commutative algebra, algebraic geometry, valuation theory, and singularity theory. The local structure of algebraic and analytic mappings will be investigated, especially with regard to ramification and good stable forms of mappings after enough blowups along a valuation or an étoile. Motivating problems are local uniformization, resolution of singularities and equisingularity in positive characteristic, and the global structure of algebraic and analytic maps.