Ramification, Multiplicity, and Volume

这个项目将研究代数几何和相关数学领域中的几个关键问题。代数几何是研究多项式方程的解；它在数学和其他科学的其他领域有许多应用。该项目将专注于任意域上的代数变换，这在计算机科学中具有实际应用的潜力。这项研究涉及密苏里大学的研究生和博士后研究人员。该项目研究的领域包括交换代数、代数几何、赋值理论和奇点理论。所研究的问题有四个基本方面：局部环上分次理想族的渐近重数和Hilbert多项式，更好地理解模数值等价(内径、外径和体积)的有效因子的锥形，态射的环面化(爆破非奇异子簇使态射局部具有单项式结构)和分支，亏格和局部一致化。这最后一部分涉及到更好地理解沿着正特征赋值的分支，以期解决奇点。

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.