Mathematical Sciences: Topics in Commutative Algebra and Algebraic Geometry

该项目支持交换代数和代数几何的研究。主要研究人员将研究仿射环上的无挠模、射影模和Cohen-Macaulay模及其局部化；二维仿射整环的素理想结构；以及Cohen-Macaulay局部环上的素理想的Hilbert函数和约化数。他们还将考虑有理曲面的自同构群；嵌入有理曲面的理想理论问题；以及射影三维空间中曲线的集合论完全交问题。代数几何是研究由多项式方程组的零点集产生的几何对象的学科。这是最古老的，也是目前最活跃的数学分支之一。它在数学、计算机科学和物理学中有着广泛的应用。

This project supports research in commutative algebra and algebraic geometry. The principal investigators will study torsion-free, projective and Cohen-Macaulay modules over affine rings and their localizations; prime ideal structure of two-dimensional affine domains; and Hilbert functions and reduction numbers of primary ideals in Cohen-Macaulay local rings. They will also consider automorphism groups of rational surfaces; ideal theoretic questions of embedded rational surfaces; and the set-theoretic complete intersection problem for curves in projective 3-space. Algebraic geometry is the study of the geometric objects arising from the sets of zeros of systems of polynomial equations. This is one of the oldest and currently one of the most active branches of mathematics. It has widespread applications in mathematics, computer science and physics.