Ramification, Multiplicity, and Volume

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This project will investigate several key 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 and other sciences. The project will focus on algebraic transformations over arbitrary fields, which has potential for practical application in computer science. The research involves graduate students and postdoctoral researchers at the University of Missouri.This project is research in the areas of commutative algebra, algebraic geometry, valuation theory, and singularity theory. The problems that are being investigated are in four basic areas: Asymptotic multiplicities and Hilbert polynomials of graded families of ideals in a local ring, understanding better the cone of effective divisors modulo numerical equivalence (inradius, outradius and volume), toroidalization of morphisms (after blowing up nonsingular subvarieties make a morphism locally have a monomial like structure) and ramification, the defect and local uniformization. This last part involves better understanding ramification along a valuation in positive characteristic, with a view towards resolution of singularities.