Multiplicity theory and related topics in commutative algebra

联系，或联络，已被用来作为一种工具，分类理想和投射品种。寻找两个理想属于同一个联系类的充分必要条件是该理论的核心问题，尽管这是一个很大的问题。研究人员建议零维单项式理想这个问题的答案。连锁理论的局部代数和射影几何分支是并行进行的，它们之间只有很少的含义。提议者计划通过研究局部链接和同质链接之间的差异来研究这种微妙的相互作用。由于经典连锁所产生的等价关系对于某些目的来说可能限制性太大，因此更包容的Gorenstein连锁概念已经成为一个受欢迎的替代方案。调查人员有更广泛的目标表明，在一个给定的定期本地环的所有科恩-麦考利理想的一个固定的余维属于相同的Gorenstein连接类。提议者计划利用最近在j重数这个话题上的进展来解决等奇异性理论中一些长期存在的问题。等奇异性理论致力于寻找一个解析集合族中的成员彼此“等价”的数值条件。调查人员试图提供这样一个标准，证明了一般的“原则的专业化的整体依赖性”的基础上，j-多重性。调查员还提出了一个交叉理论表达式的j-多重性的一个模块，将产生一个递归公式计算此多重性。理想的核心是一个子理想，它编码了关于理想的所有可能约化的信息，同时与Briancon-Skoda型定理和Kawamata关于线丛截面的猜想密切相关。研究者希望探索核心和乘数理想之间的联系，这是一个来自复代数几何的概念。他希望更好地理解这种相互作用将揭示这两个概念的新的光，并可能导致在单项情况下的核心的组合描述。研究者已经给出了计算零维单项理想的核的算法，现在希望找到计算任意维单项理想的算法。关于理想核的已知结果需要对理想的生成元的局部数目以及剩余或深度条件进行限制，这些条件是任何零维理想都满足的假设。调查人员提出，以推进理论超越类的理想满足这些条件，使用单项式的情况下，作为一个测试ground.The proposer工程交换代数，一个领域关注的定性研究系统的多项式方程在几个变量。这种系统出现在代数几何中，但也出现在数学之外的许多应用中。在过去的二十年里，交换代数学家对计算方面越来越感兴趣，从而与计算机代数，机器人，密码学，编码理论，统计学和生物学等应用领域建立了联系。这位调查员的研究也有很强的计算组成部分，他的工作equisingularity理论，特别是包括具体的几何应用。

Linkage, or liaison, has been used as a tool for classifying ideals and projective varieties. Finding necessary and sufficient conditions for two ideals to belong to the same linkage class is the central, though wide open, problem in the theory. The investigator suggests an answer to this question for zero-dimensional monomial ideals. The local-algebraic and the projective-geometric branch of linkage theory have proceeded in parallel, with only few implications known between them. The proposer plans to investigate this subtle interplay by studying the differences between local linkage and homogeneous linkage. Since the equivalence relation generated by classical linkage might be too restrictive for some purposes, the more inclusive notion of Gorenstein linkage has become a welcome alternative. The investigator has the broader goal of showing that in a given regular local ring all Cohen-Macaulay ideals of a fixed codimension belong to the same Gorenstein linkage class. The proposer plans to exploit recent advances on the topic of j-multiplicities to address some long standing problems in equisingularity theory. Equisingularity theory strives to find numerical conditions for when the members of a family of analytic sets are `equivalent' to one another. The investigator seeks to provide such a criterion by proving a general `principle of specialization of integral dependence' based on j-multiplicities. The investigator also proposes an intersection theoretic expression for the j-multiplicity of a module that would yield a recursive formula for computing this multiplicity. The core of an ideal is a subideal that encodes information about all possible reductions of the ideal, while being closely related to Briancon-Skoda type theorems and a conjecture of Kawamata about sections of line bundles. The investigator wishes to explore the connection between cores and multiplier ideals, a notion from complex algebraic geometry. He expects that a better understanding of this interplay will shed new light on both concepts and may lead to a combinatorial description of the core in the monomial case. Having worked out an algorithm for computing the core of zero-dimensional monomial ideals, the investigator now wishes to find such an algorithm for monomial ideals of arbitrary dimension. The known results about cores of ideals require restrictions on the local numbers of generators of the ideal as well as residual or depth conditions, assumptions that are satisfied by any zero-dimensional ideal. The investigator proposes to advance the theory beyond the class of ideals satisfying these conditions, using the monomial case as a testing ground.The proposer works in Commutative Algebra, an area concerned with the qualitative study of systems of polynomial equations in several variables. Such systems arise in algebraic geometry, but also in numerous applications outside of mathematics. Over the past two decades commutative algebraists have become increasingly interested in computational aspects, thereby establishing connections to applied areas such as computer algebra, robotics, cryptography, coding theory, statistics, and biology. This investigator's research too has a strong computational component, and his work on equisingularity theory in particular encompasses concrete geometric applications.