Advances in algebraic stacks and applications

代数堆栈是代数和物理现象的几何表示。堆栈为研究几何、拓扑和数论中的问题提供了一个基本的数学结构，这些问题在弦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，这将具有广泛的意义。特别是，在代数堆栈上获得的新结果将被应用于解决代数几何中一个长期悬而未决的问题。具体地说，该项目将为最有趣和最受研究的代数族之一--光滑曲线的模空间--的双曲几何提供新的描述。

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.