Non-commutative fundamental groups in Diophantine geometry

这个建议主要涉及两个变量的丢番图方程，即，多项式关系的整数系数，其中一个试图了解的集合整数解决方案。这类研究的历史可以追溯到希腊数学的传统，而在20世纪，抽象的现代机器在解决古老的难题上的惊人应用，比如怀尔斯对费马大定理的证明。研究者提出了一种新的方法来研究这些经典问题，通过将拓扑学和几何学的基本思想结合起来，这些思想超越了二十世纪的主要发展，因为相关的结构主要是非交换和非线性的。最终的目标是构建有效解决两个变量的丢番图方程的方法。

This proposal is concerned primarily with Diophantine equations in two variables, i.e., polynomial relations with integers coefficients for which one seeks to understand the collection of integer solutions. The history of such investigations reaches back to the tradition of Greek mathematics, while the twentieth century has seen spectacular applications of abstract modern machinery to the resolution of difficult old questions, such as Wiles' proof of Fermat's last theorem. The investigator proposes a new approach to studying these classical problems by incorporating fundamental ideas of topology and geometry that go beyond the principal developments of the twentieth century in that the relevant structures are, in the main, non-commutative and non-linear. An eventual goal is to construct methods for effectively resolving Diophantine equations in two-variables.