Integrality, Blowup Algebras and Multiplicities

从仿射代数或理想到积分闭包是一个基本运算。调查者打算检查这一过程的计算方面。虽然存在计算仿射域的积分闭包的算法，但对这种计算的复杂性知之甚少。为了解决这个问题，调查人员打算找到估计的代数生成器的积分闭包的数量和这些生成器的degreesof.另一方面，在理想的情况下，仍然没有有效的方法来计算积分闭包。由于缺乏算法，研究者打算用一个更容易得到的冒号理想的和来表达或至少近似一个理想的积分闭包。理想的核心是积分闭包的对应物：它编码了关于给定理想的所有可能约简的信息，即，所有的理想，其中给定的理想是完整的。本书作者打算研究一个数学公式，如果得到证明，将给出一个明确的表达核心的一大类理想。继续有点计算的主题调查proposesto建立边界的Castelnuovo-Mumford正则嵌入投影品种在其定义的equations.The已知的结果需要强有力的假设的奇异性的品种，调查希望削弱。在这方面，他打算表明，理性奇点坚持genericlinkage下的理想的联系类的一个完整的intersect.In延续他的早期工作爆破代数的调查员计划研究环代表特殊纤维的爆破.例如，这些“特殊纤维环”描述了有理映射的像，包括割线簇和高斯像。研究特殊纤维环的Cohen-Macaulay性，估计其深度并计算多重数。适当的深度估计将有一个由Mazur的猜想，起源于怀尔斯的工作变形的伽罗瓦表示。公式的多重性，另一方面将导致一个扩展的Teissier-Pluecker公式的程度certain dual varieties。调查工作在该地区的数学称为“交换代数”，其中涉及定性研究系统的多项式方程在几个变量。 这样的系统出现在数学之外的许多应用中。 在过去的二十年里，交换代数学家越来越关注计算方面，从而强调与计算机代数，机器人，密码学和编码理论等应用领域的联系。 这个项目涉及交换代数的理论和计算方面。

Passing from an affine algebra or an ideal to the integral closureis a fundamental operation. The investigator intends to examinecomputational aspects of this process. While there exist algorithmsfor computing the integral closure of an affine domain, little isknown about the complexity of such calculations. To address thisproblem the investigator intends to find estimates for the numberof algebra generators of the integral closure and for the degreesof these generators. In the case of ideals on the other hand, onestill has no efficient method for computing integral closures. Inview of this lack of an algorithm, the investigator intends toexpress, or at least approximate, the integral closure of an idealby a sum of colon ideals that are more readily accessible. The coreof an ideal is a counterpart of the integral closure: it encodesinformation about all possible reductions of a given ideal, i.e.,all ideals over which the given ideal is integral. The investigatorintends to work on a conjectural formula that, if proved, would givean explicit expression for the core of a large class of ideals.Continuing the somewhat computational theme the investigator proposesto establish bounds for the Castelnuovo-Mumford regularity ofembedded projective varieties in terms of their defining equations.The known results require strong assumptions on the singularities ofthe variety which the investigator hopes to weaken. In this vein heintends to show that rational singularities persist under genericlinkage of ideals in the linkage class of a complete intersection.In continuation of his earlier work on blowup algebras theinvestigator plans to study the rings representing special fibers ofblowups. These `special fiber rings' describe, for instance, imagesof rational maps, including secant varieties and Gauss images. Theinvestigator intends to study the Cohen-Macaulayness of special fiberrings, estimate their depth and compute multiplicities. Suitabledepth estimates would have a bearing on a conjecture by Mazur thatoriginates in Wiles' work on deformations of Galois representations.Formulas for the multiplicity on the other hand would lead to anextension of the Teissier-Pluecker formula for the degrees of certaindual varieties.The investigator works in the area of Mathematics called "Commutative Algebra," which deals with the qualitative study of systems of polynomial equations in several variables. Such systems arise in numerous applications outside mathematics. Over the past two decades commutative algebraists have become more concerned with computational aspects, thereby emphasizing connections to applied areas such as computer algebra, robotics, cryptography and coding theory. This project addresses both theoretical and computational aspects of commutative algebra.