Moduli Spaces for Rings and Ideals

研究有限的，平坦的覆盖和这些覆盖的线丛的模空间。 重要的基本例子包括数域中的序（整数的有限平坦覆盖）和复射影线的有限覆盖。 这些方法涉及到对任意基方案的工作，因此例如人们得到关于上面的数论和几何示例的结果。 该项目是找到模空间，有明确的描述和合理的几何，使他们可以具体工作。 自从高斯在1801年的工作以来，多项式一直被用来研究比通常的计数数1，2，3. 例如，一个更大的数字系统可能还包括2的平方根，它不能在1，2，3.中找到。 当我们包含2的平方根时，它是通常数的二次扩展，如果我们包含2的立方根，它将是通常数的三次扩展。 这项工作试图了解什么是可能的低程度的扩展通常的数字是明确的工作与多项式有关的扩展。 这就允许人们通过将它们简化为关于多项式的更容易的计算来进行关于更大的数字系统的计算。

The investigator studies moduli spaces of finite, flat covers and line bundles of those covers. Important basic examples include orders in number fields (finite flat covers of the integers) and finite covers of the complex projective line. The methods involve working over an arbitrary base scheme, so for example one gets results about both the number theoretic and geometric examples above. The project is to find moduli spaces that have explicit descriptions and reasonable geometry so that they can be worked with concretely. Since the work of Gauss in 1801, polynomials have been use to study number systems that are bigger than the usual counting numbers 1,2,3.., For example, a larger number system might also include the square root of 2, which cannot be found among 1,2,3... When we include the square root of 2, it is a quadratic extension of the usual numbers, and if we had included the cube root of 2 it would have been a cubic extension of the usual numbers. This work tries to understand what the possible low degree extensions of the usual numbers are by working explicitly with polynomials that are related to the extensions. This then allows one to make computations regarding the larger number systems by reducing them to easier computations about polynomials.