Moduli spaces of geometric structures

最近数学中最引人注目的进步之一是佩雷尔曼对瑟斯顿的三维空间几何程序的解析。这使得理解这种空间上的几何结构集是非常重要的，这种空间被称为模空间。这个项目的挑战是开发实用的方法来计算模空间，了解它们的全局性质，并使用它们来定义新的不变量，从而深入了解空间的形状、几何和代数。这是一个关键问题，因为模空间描述了空间是刚性的还是可以变形的。它们是几个数学领域的中心对象，包括几何和拓扑学、规范理论、动力系统、数学物理和不变量理论。

One of the most spectacular recent advances in mathematics was Perelman's resolution of Thurston's geometrisation program for three-dimensional spaces. This makes it very important to understand sets of geometric structures on such spaces, called moduli spaces. The challenge of this project is to develop practical methods to compute moduli spaces, understand their global properties, and use them to define new invariants giving insights about shape, geometry and algebra of a space. This is a key problem as moduli spaces describe whether a space is rigid or can be deformed. They are a central object in several fields of mathematics, including geometry and topology, gauge theory, dynamical systems, mathematical physics and invariant theory.