Multiplicity theory and related topics in commutative algebra

联系，或联络，已被用作分类理想和投影品种的工具。寻找两种理想属于同一联系类的充分必要条件，是这一理论的中心问题。研究者对零维单项式理想给出了这个问题的答案。连杆理论的局部代数分支和投影几何分支是并行发展的，它们之间的联系很少。作者计划通过研究局部连锁和同质连锁之间的差异来研究这种微妙的相互作用。由于经典连杆产生的等价关系对于某些目的可能过于严格，因此更具包容性的戈伦斯坦连杆概念成为一种受欢迎的替代方案。研究者有一个更广泛的目标，证明在给定的正则局部环上，固定协维的所有Cohen-Macaulay理想都属于同一个Gorenstein连杆类。作者计划利用j多重性的最新研究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组解析集的成员彼此“等价”的数值条件。研究者试图通过证明基于j-多重性的一般“积分依赖的专门化原则”来提供这样一个标准。研究者还提出了一个模块j-多重性的交点理论表达式，该表达式将产生计算该多重性的递归公式。理想的核心是对理想所有可能约简的信息进行编码的子域，它与Briancon-Skoda型定理和Kawamata关于线束截面的猜想密切相关。研究者希望探索核心和乘法器理想之间的联系，这是一个来自复杂代数几何的概念。他期望对这种相互作用的更好理解将为这两个概念带来新的启示，并可能导致对单项情况下核心的组合描述。在研究出一种计算零维单项式理想核心的算法之后，研究者现在希望找到一种计算任意维单项式理想的算法。关于理想核的已知结果需要对理想发生器的局部数以及残差或深度条件的限制，这些条件是任何零维理想所满足的假设。研究者建议将理论推进到满足这些条件的理想类别之外，使用单项案例作为试验场。作者的研究方向是交换代数，这是一个研究多变量多项式方程组定性研究的领域。这样的系统出现在代数几何中，但也出现在数学以外的许多应用中。在过去的二十年里，交换代数学者对计算方面越来越感兴趣，从而建立了与计算机代数、机器人、密码学、编码理论、统计学和生物学等应用领域的联系。这位研究者的研究也有很强的计算成分，他在等奇点理论方面的工作尤其包含了具体的几何应用。

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.