Modular Towers of Noncongruence Curves

弗里德教授将继续研究模块化塔。模塔是一个模空间序列，通常附属于来自有限群的一组数据。模块化塔的设计是为了在许多以前没有与此类技术相关的应用中类比模块化曲线的思想。该项目将集中于模块化曲线塔。该项目的目标之一是建立高水平的模块化曲线塔将没有合理的点，这是塔的许多应用的一个重要问题。此外，为了使模块化塔的技术更容易适用于新情况，弗里德教授将尝试建立标准，允许任意非同余曲线的塔被给定模块化塔的结构。在这个项目中，弗里德教授将继续研究与伽罗瓦逆问题相关的问题。这是一个长期悬而未决的问题，它起源于19世纪。100多年前，数学家发现，人们可以通过研究解的对称性来对方程解的性质进行分类。解决方案本身和解决方案的对称性以一种直接但非常令人惊讶的方式联系在一起。对称的简单性质常常与解的困难性质联系在一起，反之亦然。这允许简单的事情通过连接来回转移到困难的事情上。结果是非常有力的发现方法。在早期的工作中，Fried教授在这个连接中引入了一个叫做模块化塔的新物体，他希望利用这个物体加强我们对从对称到解决方案方向的连接的理解。

Professor Fried will continue research on modular towers. A Modular tower is a sequence of moduli spaces canonically attached to a set of data coming from a finite group. Modular towers are designed to allow analogies of the ideas of modular curves in many applications not previously associated with such techniques. This project will concentrate on modular curve towers. Among the goals of the project is to establish that high levels of a modular tower of curves will have no rational points, an important issue for many of the applications of the towers. Also to make the techniques of modular towers more easily applicable in new situations, Professor Fried will try to establish criteria which allow arbitrary towers of non-congruence curves to be given the structure of a modular tower. In this project, Professor Fried will continue his work on issues related to the Inverse Galois Problem. This is a long-standing unanswered question with its origins in the 19 century. Over a hundred years ago mathematicians found that one could classify properties of the solutions to an equation by studying the symmetries of those solutions. The solutions themselves and the symmetries of the solutions were connected in a direct but very surprising way. Easy properties of symmetries were often connected to difficult properties of the solutions and vice versa. This allows the easy things to be transferred back and forth to the hard things through the connection. The result is very powerful method for discovery. In earlier work, Professor Fried introduced a new object into this connection called modular towers, and he hopes to use this object strengthen our understanding of the connection in the direction from the symmetries to the solutions.