Algebraic stacks and their applications

代数栈及其应用

• 批准号: 0714086
• 负责人: Martin Olsson
• 金额: $ 9.69万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-07-31 至 2010-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0714086&HistoricalAwards=false
• 关键词: Algebraic; stacks; their; applications

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.

该研究项目集中在与代数堆栈及其应用有关的广泛问题上。该项目分为两个部分。第一部分是关于最近在堆栈的重要应用中出现的基本问题，如几何朗兰德程序、稳定映射的模理论和高维簇的模空间的构造。该计划的第二部分涉及该理论的应用。特别是在Fontaine和Illusie意义下的对数几何与PI在早期工作中发现的堆栈之间的关系的新应用，扭曲稳定映射理论的推广，以及在构造和研究一般类型的变种、阿贝尔簇和曲线上的矢丛的模空间中的应用。堆叠的概念是用于处理数学对象的内部对称性以及群的作用的工具。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，堆栈理论在代数几何、算术几何和数学物理的几乎每一个部分都扮演着重要的角色，并发现了大量令人兴奋的新应用。考虑到对称性在数学和其他科学领域的重要性，这并不令人惊讶。对堆栈的极大兴趣揭示了有关堆栈的许多重要问题。该研究项目旨在扩大我们对堆栈理论的基本方面以及许多应用的理解。