Ramification, Multiplicity, and Volume

本专题将探讨代数几何及相关数学领域的几个关键问题。 代数几何是研究多项式方程的解的学科;它在数学和其他科学的其他领域有许多应用。 该项目将侧重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。 该研究涉及密苏里州大学的研究生和博士后研究人员。该项目是交换代数、代数几何、赋值理论和奇点理论等领域的研究。正在调查的问题是在四个基本领域：渐近多重性和希尔伯特多项式的分级家庭的理想在一个地方环，更好地了解锥的有效因子模数值等价（inradius，antridius和体积），toroidalization态射（后炸毁非奇异子品种使态射当地有一个单项式的结构）和分歧，缺陷和当地的一致化。这最后一部分涉及到更好地理解分支沿着一个估值的积极特点，以期对决议的奇异性。

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.