Mathematical Sciences: Multiplicative Structures on Free Resolutions & Local Algebra

首席研究员将研究交换代数的三个领域。第一个领域涉及自由分辨率的乘法结构。她将找到一个循环模上最小自由分辨率的可交换微分梯度代数结构存在的判据。她还将确定在完全交的连杆类中由理想定义的循环模的最小分辨率是否存在这样的结构。第二个领域是关于奇点的基本群和等价。特别地，她将确定一个数字k，使得对于幂级数环R中给定的行列式理想I，以及对于相同高度的任何其他理想J， J等于I模I的雅可比理想的k次幂，保证R的同构存在，它将I映射到J上，因此J也是一个行列式理想。第三个领域是确定，在正特征下，一个单素数的符号代数是否有限生成。这是代数几何领域的研究，是现代数学中最古老的部分之一，但在过去的25年里，它已经有了革命性的发展。在它的起源中，它处理的图形可以用最简单的方程，即多项式，在平面上定义。如今，该领域不仅使用代数的方法，而且还使用分析和拓扑的方法，相反，它在这些领域以及理论计算机科学和机器人技术中也得到了应用。

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.