POWRE: Differentials, Singularities and Applications

该POWRE奖支持东北大学数学系客座教授研究孤立超曲面奇点的微分。该奖项将允许pii与她所在领域的几位主要研究人员进行互动。作为人力资源开发活动的一部分，她还将赞助东北大学当地的研究生研讨会。本研究项目结合了循环同调和k理论的理论知识以及计算机代数技术来收集关于孤立超曲面奇点的不变量的信息。孤立超曲面奇点最显著的不变量是所谓的Tjurina数，或微分的扭转模的维数。她将微分的扭转模确定为循环同调的霍奇分量和理想商。后一种识别导致了利用Gorenstein - Artin代数中的矩阵对偶性计算生成器数量的有效算法。作为其研究活动的一部分，PI将承担以下三部分的项目：1。孤立奇点的结构：本研究的预期结果将是三维超曲面奇点的微分模长度和产生点数目的上界。2. 孤立超曲面奇点的残数和对偶性：目标将是残数的显式描述和与逆系统的联系的研究。对数微分模块的描述以及与超平面/曲面排列的联系：项目的这一部分本质上是计算性的，并将重点放在外部代数中Groebner基计算算法的使用和开发上。该power项目由MPS多学科活动办公室（OMA）和数学科学部（DMS）联合支持。

This POWRE award supports a Visiting Professorship at the Department of Mathematics at Northeastern University to study differentials of isolated hypersurface singularities. This award will allow the PIto interact with several leading researchers in her field. As part of the human resource development activities she will also sponsor the local graduate student seminar at Northeastern.This research project combines theoretical expertise from cyclic homology and K-theory and computer algebra techniques to gather information about invariants of isolated hypersurface singularities. The most salient invariant of isolated hypersurface singularities is the so-called Tjurina number, or the dimension of the torsion module of differentials. She has identified the torsion module of differentials both as a Hodge-component of cyclic homology and as an ideal quotient. The latter identification has led to an efficient algorithm for computation of the number of generators exploiting Matlis duality in Gorenstein Artin Algebras. As part of her research activities the PI will undertake the following three-part project:1. The structure of isolated singularities:The anticipated outcome of this research will be an upper bound for the number of generators and the length of the module of differentials for 3 dimensional hypersurface singularities. 2. Residues and Duality for isolated hypersurface singularities:The goal will be an explicit description of residues and an investigation of the connection with inverse systems.3. Description of the module of logarithmic differentials and connection with hyperplane/surface arrangements:This part of the project will be computational in nature and will focus on the use and development of algorithms for Groebner basis computations in exterior algebras.This POWRE project is jointly supported by the MPS Office of Multidisciplinary Activities (OMA) and the Division of Mathematical Sciences (DMS).