Moduli spaces for wildly ramified covers of curves

具有给定分支轨迹的复射影直线(黎曼球面)的伽罗瓦覆盖的理论是众所周知的，即人们可以用黎曼存在定理来描述这样的覆盖的惯性群，并通过它们的模空间来研究这种覆盖的族，称为赫尔维茨空间。这些技巧扩展到代数闭域k上投影线的覆盖，只要分支是驯服的。然而，如果分支是狂野的，也就是说，如果场k的特征划分了某个惯性群的阶数，就会出现新的现象，而这些现象却鲜为人知。涉及广泛分叉覆盖的惯性群和变形理论的主要公开问题。PI建议研究广泛分支的曲线覆盖，目的是回答其中的一些问题；特别是，PI期望根据覆盖的分支不变量来计算覆盖的变形空间的维度。PI还建议构造特征p中曲线的广义分枝Galois覆盖的模空间，并研究它们的一些性质。Galois理论具有很高的吸引力。这一领域的几个公开问题可以向非数学家解释，这个主题将数学的不同领域联系起来。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出现的。早期的一些应用是，不可能对一个角度进行三角剖分，不可能将立方体的体积扩大一倍，也不可能解五次方程。国际和平研究所建议开设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。国际学生联合会希望继续领导学生的研究项目；国际学生联合会在2002年领导的项目激励了几名学生(包括代表人数不足的群体的学生)继续攻读数学研究生课程。最近发现了伽罗瓦理论在数据传输码中的一种应用。这些码可以使用定义在有限域上的多个点的低次曲线来构造。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，国际数学家协会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.