Algebraic Stacks and Applications

PI建议继续他在堆栈及其在算术几何、代数几何和非交换代数中的应用方面的工作。其中一个项目涉及研究全局域上几何有理变元的哈斯原理和函数域的布劳尔群的周期指数问题之间的联系。这类似于Artin关于整数上适当格式的Brauer群的有限性的猜想和Tate-Shafarevich猜想之间的联系，并且建立在PI关于扭轮的模数的早期工作的基础上。第二个项目将继续PI和Kovács关于非等平凡曲线族的Shafarevich猜想的高维推广的联合工作。这个项目的关键是更好地理解堆栈对堆栈之间的态射进行参数化，其系统研究最近才开始。第三个项目旨在加深对堆栈及其内在几何的理解。PI和他的合作者将研究堆栈的混血修饰和分析的性质，以及堆栈的范畴信息内容。这项研究的最终目的是扩大代数几何和堆栈理论方法在纯代数及相关领域的应用。广义地说，代数几何是研究与代数对象相关的几何。历史上最有成果的例子之一是圆的方程；方程的二次型性质与直线通常与圆相交于两点这一事实密切相关。在过去的几千年里，数学家们已经认识到，代数和几何之间的联系比人们想象的要深得多，这导致几何方法逐渐侵蚀到纯代数的遥远领域，并使几个看似不同的学科实现了深刻的统一。代数几何中的每一项新进展最终都会应用到数学的其他领域；该领域的几个抽象领域已经被证明是非常便于计算机使用的，没有代数几何，现代密码学就不可能实现。堆栈理论相对年轻，但它一直是最近关注的焦点，并正在迅速成熟。PI的研究将致力于将堆栈理论应用于代数和几何中的广泛问题，为代数问题的几何分析带来新的工具。

The PI proposes to continue his work on stacks and their applications to arithmetic geometry, algebraic geometry, and noncommutative algebra. One project involves studying the connection between the Hasse principle for geometrically rational varieties over global fields and the period-index problem for Brauer groups of function fields. This is analogous to the connection between Artin's conjecture on the finiteness of the Brauer group for schemes proper over the integers and the Tate-Shafarevich conjecture, and builds on earlier work by the PI on the moduli of twisted sheaves. A second project will continue the joint work carried out by the PI and Kovács on higher-dimensional generalizations of the Shafarevich conjecture on non-isotrivial families of curves. Crucial to this project will be a greater understanding of stacks parametrizing morphisms between stacks, whose systematic study was only recently begun. A third project aims to deepen the understanding of stacks and their intrinsic geometry. The PI and his collaborators will study the nature of birational modifications and analytification of stacks, and the categorical information content of stacks. The ultimate goal of the proposed research is to broaden the applications of algebro-geometric and stack-theoretic methods in pure algebra and related fields.Broadly speaking, algebraic geometry is the study of the geometry associated to algebraic objects. One of the most fruitful historical examples is the equation for a circle; the quadratic nature of the equation is closely related to the fact that a line generally intersects a circle in two points. Over the last several millennia, mathematicians have come to understand that the connections between algebra and geometry run far deeper than one might imagine, and this has led to the gradual encroachment of geometric methods into far-flung areas of pure algebra and a profound unification of several seemingly-different subjects. Each new advance in algebraic geometry ultimately finds applications to other areas of mathematics; several abstract areas of the field have turned out to be very computer-friendly, and modern cryptography would be impossible without algebraic geometry. The theory of stacks is relatively young, but it has been the focus of recent attention and is rapidly maturing. The PI's research will be directed toward applying the theory of stacks to a wide range of problems in algebra and geometry, bringing new tools to the geometric analysis of algebraic problems.