Mathematical Sciences: Multiplicative Structures on Free Resolutions & Local Algebra

首席研究者将研究交换代数的三个领域。第一个领域涉及关于自由分辨率的乘法结构。她将在循环模的最小自由分解上找到一个关于结合交换微分分次代数结构存在的判据。她还将确定，对于完全交的链类中的理想所定义的循环模的最小分解，是否存在这样的结构。第二个领域涉及基本群和奇点的等价。特别地，她将确定一个数k，使得对于幂序列环R中给定的行列式理想I和任何其他相同高度的理想J，J等价于I模I的雅各布理想的第k次方，保证存在R的同构，该同构将I映射到J上，使得J也是行列式理想。第三个领域是用正特征确定单项素数的符号代数是否有限生成。这是对代数几何领域的研究，代数几何是现代数学中最古老的部分之一，但在过去的25年里，它取得了革命性的成就。在它的起源中，它处理的是可以在平面上用最简单的方程定义的图形，即多项式。如今，这个领域不仅使用代数的方法，而且使用分析和拓扑学的方法，相反，在这些领域以及理论计算机科学和机器人学中找到了应用。

The principal investigator will study three areas of commutative algebra. The first area is concerned with multiplicative structures on free resolutions. She will find a criterion for the existence of an associative commutative differential graded algebra structure on the minimal free resolution on cyclic modules. She will also determine if such a structure exists for the minimal resolutions of the cyclic modules defined by the ideals in the linkage class of a complete intersection. The second area is concerned with fundamental groups and equivalences of singularities. In particular, she will determine a number k such that for a given determinantal ideal I in a power series ring R and for any other ideal J of the same height, J equivalent to I modulo the kth power of the Jacobean ideal of I guarantees the existence of an isomorphism of R which maps I onto J so that J is also a determinantal ideal. The third area is to determine, in positive characteristics, if the symbolic algebra of a monomial prime is finitely generated. This is reasearch in the field of algebraic geometry, one of the oldest parts of modern mathematics, but one which has had a revolutionary flowering in the past quarter-century. In its origins, it treated figures that could be defined in the plane by the simplest equations, namely polynomials. Nowadays the field makes use of methods not only from algebra, but from analysis and topology, and conversely is finding application in those fields as well as in theoretical computer science and robotics.