Algebraic stacks and their applications

该研究项目集中在代数堆栈及其应用的广泛问题上。本项目分为两部分。第一部分讨论了近年来在堆的重要应用中出现的一些基础问题，如几何朗兰规划、稳定映射的模理论、高维变换的模空间的构造等。该计划的第二部分涉及理论的应用。特别是在Fontaine和Illusie意义上的对数几何与PI在早期工作中发现的堆栈之间关系的新应用，扭曲稳定映射理论的推广，以及在构造和研究一般类型、阿贝尔类型和曲线上向量束的模空间方面的应用。堆栈的概念是用于处理数学对象的内部对称性以及组的动作的工具。例如，当试图对几何对象进行分类时，人们自然会被迫处理所讨论对象的对称性。近年来，叠理论在代数几何、算术几何和数学物理的几乎每一个领域都发挥着重要的作用，并发现了许多令人兴奋的叠新应用。考虑到对称性在数学和其他科学领域的重要性，这并不奇怪。这种对堆栈的极大兴趣揭示了许多关于堆栈的重要问题。该研究项目旨在扩大我们对堆栈理论的基础方面以及许多应用的理解。

The research project is focused on a broad range of questions concerning algebraic stacks and their applications. The project is divided into two parts. The first part concerns foundational problems which have arisen in recent important applications of stacks such as the geometric Langland's program, the theory of moduli of stable maps, and the construction of moduli spaces for higher dimensional varieties. The second part of the program concerns applications of the theory. In particular new applications of the relationship between log geometry in the sense of Fontaine and Illusie and stacks discovered by the PI in earlier work, generalizations of the theory of twisted stable maps, as well as applications to the construction and study of moduli spaces for varieties of general type, abelian varieties, and vector bundles on curves.The notion of stack is a tool used to deal with internal symmetries of mathematical objects, as well as actions of groups. For example, when trying to classify geometric objects one is naturally forced to deal with the symmetries of the objects in question. In recent years, the theory of stacks has come to play an important role in almost every part of algebraic geometry, arithmetic geometry, and mathematical physics and a great number of exciting new applications of stacks have been found. This is not surprising considering the importance of symmetries in mathematics and other fields of science. This great interest in stacks has brought to light a number of important problems about stacks. The research project aims to broaden our understanding of both the foundational aspects of the theory of stacks and as well as the many applications.