Moduli spaces for wildly ramified covers of curves

具有给定分支轨迹的复投影线（黎曼球）的伽罗瓦覆盖理论已被很好地理解；即可以用黎曼存在定理来描述这种盖的惯性群，并通过它们的模空间来研究这种盖的族，这些模空间称为Hurwitz空间。这些技术延伸到在一个代数闭域k上的投影线的覆盖，只要分支是驯服的。然而，如果分支是野生的，即如果场k的特征分了一些惯性群的阶，就会出现人们知之甚少的新现象。野分枝盖层的惯性群和变形理论是目前尚未解决的主要问题。PI建议研究广泛分叉的曲线覆盖，目的是回答其中的一些问题；特别是，PI期望根据其分支不变量来计算覆盖的变形空间的维度。本文还提出了构造特征为p的曲线的广分枝伽罗瓦覆盖的模空间，并研究了它们的一些性质。伽罗瓦理论对广大读者具有很高的吸引力。这个领域的几个开放问题可以向非数学家解释，这个话题连接了数学的不同领域。伽罗瓦理论是作为理解方程对称性和对有理数的扩展进行分类的一种经典方法而产生的。一些早期的应用是不可能对一个角进行三等分，将一个立方体的体积翻倍，或者解一个五次方程。学院建议开设一门新的伽罗瓦理论课程，向研究生介绍这一课题的基本背景和当前的研究问题。PI希望继续领导学生的研究项目；PI在2002年领导的项目激励了一些学生（包括来自弱势群体的学生）攻读数学研究生课程。最近发现了伽罗瓦特性理论在数据传输码中的应用。这些码可以用在有限域上定义有许多点的低次曲线来构造。PI希望进一步发展伽罗瓦理论与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.