Advances in algebraic stacks and applications

代数堆栈是代数和物理现象的几何表现。堆栈提供了一个基本的数学结构来研究几何，拓扑和数论中的问题，这些问题在弦理论和复杂性理论中有着深刻的应用。这个项目将证明关于代数堆栈的新的基本定理，这将具有广泛的影响。特别是，代数堆栈上获得的新成果将被应用，以解决一个长期存在的开放问题，代数几何。具体来说，该项目将提供一个新的描述双有理几何的一个最有趣的和研究的代数品种，模空间的光滑曲线。

Algebraic stacks are a geometric manifestation of algebraic and physical phenomena. Stacks provide a fundamental mathematical structure to study questions in geometry, topology and number theory having deep applications to string theory and complexity theory. This project will prove new fundamental theorems about algebraic stacks that will have broad implications. In particular, the new results obtained on algebraic stacks will be applied in order to resolve a long-standing open problem in algebraic geometry. Specifically, the project will provide a new description of the birational geometry of one of the most interesting and studied algebraic varieties, the moduli space of smooth curves.