Modular Towers of Noncongruence Curves

弗里德教授将继续对模块化塔楼的研究。模塔是一个模空间序列，它典型地依附于一组来自有限群的数据。模块化塔楼的设计允许在许多以前与此类技术不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一个高水平的模块化曲线塔将没有有理点，这是一个重要的问题，许多塔的应用。此外，为了使模块化塔楼的技术更容易应用于新的情况，弗里德教授将尝试建立标准，允许不一致曲线的任意塔楼被赋予模块化塔楼的结构。在这个项目中，弗里德教授将继续他在与伽罗瓦逆问题有关的问题上的工作。这是一个长期悬而未决的问题，起源于19世纪。一百多年前，数学家发现，人们可以通过研究方程解的对称性来对这些解的性质进行分类。解本身和解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。这使得容易的事情可以通过连接来回转移到困难的事情。其结果是非常有力的发现方法。在早期的工作中，Fry教授在这种联系中引入了一个新的对象，称为模块化塔，他希望利用这个对象来加强我们对从对称到解的方向上的联系的理解。

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.