Moduli spaces for wildly ramified covers of curves

普利斯DMS-0400461奖项目摘要具有给定分支轨迹的复射影线（黎曼球面）的伽罗瓦覆盖理论是很好理解的;也就是说，人们可以用黎曼存在定理描述这种覆盖的惯性群，并通过它们的模空间（称为Hurwitz空间）研究这种覆盖的族。这些技巧可以推广到代数闭域k上的投影线的覆盖，只要其分支是驯服的。然而，如果分支是任意的，即如果场k的特征线划分了某个惯性群的阶数，则会出现新的现象，而这些新的现象却很少被理解。有主要的开放性问题，涉及广泛分歧覆盖的惯性群和变形理论。PI建议研究曲线的广泛分歧覆盖，目的是回答其中的一些问题;特别是，PI希望根据其分歧不变量计算覆盖的变形空间的维数。PI还提出为特征p中曲线的广义分歧Galois覆盖构造模空间，并研究它们的一些性质。几个开放的问题，在这方面可以解释非数学家和主题连接不同领域的数学。伽罗瓦理论出现经典的一种手段，了解对称性的方程和分类扩展的有理数。一些早期的应用是，它是不可能的三等分一个角度，一个立方体的体积增加一倍，或解决一个五次方程。PI建议开发一门关于伽罗瓦理论的新课程，向研究生介绍这一主题的基本背景和活跃的研究问题。 PI希望继续领导学生的研究计划; PI在2002年领导的计划激励了几名学生（包括来自代表性不足群体的学生）攻读数学研究生。最近发现了伽罗瓦理论在特征$p$的数据传输码中的应用。 这些代码可以构造使用低次曲线，其中有许多点定义在有限域上。PI希望进一步发展伽罗瓦理论和编码理论之间的联系。最后，PI将继续与德国和纽约的数学家合作，并将通过会议和论文传播这项研究的结果。

Project Summary for award DMS-0400461 of PriesThe theory of Galois covers of the complex projective line (Riemann sphere) with given branch locus is well-understood; namely, one can describe the inertia groups of such a cover with Riemann's Existence Theorem and study families of such covers via their moduli spaces, which are called Hurwitz spaces. These techniques extend to covers of the projective line over an algebraically closed field k as long as the ramification is tame. However, if the ramification is wild, i.e. if the characteristic of the field k divides the order of some inertia group, new phenomena occur which are much less understood. There are major open problems involving the inertia groups and deformation theory of wildly ramified covers. The PI proposes to study wildly ramified covers of curves, with the goal of answering some of these problems; in particular, the PI expects to compute the dimension of the deformation space of a cover in terms of its ramification invariants. The PI also proposes to construct moduli spaces for wildly ramified Galois covers of curves in characteristic p and to investigate some of their properties.Galois theory has high appeal to a broad audience. Several open problems in this area can be explained to non-mathematicians and the topic connects diverse areas of math. Galois theory arose classically as a means of understanding symmetries of equations and of classifying extensions of the rational numbers. Some of the early applications were that it is impossible to trisect an angle, double the volume of a cube, or solve a quintic equation. The PI proposes to develop a new course on Galois theory, to introduce graduate students to the fundamental background and active research problems of this topic. The PI would like to continue to lead research programs for students; the program which the PI led in 2002 motivated several students (including students from underrepresented groups) to pursue graduate study in mathematics. An application of Galois theory in characteristic $p$ to data-transfer codes was recently discovered. These codes can be constructed using curves of low degree which have many points defined over finite fields. The PI would like to develop the connection between Galois theory and coding theory further. Finally, the PI will continue collaborating with mathematicians in Germany and New York and will disseminate the results from this research through conferences and papers.