Algebraic Stacks and Applications

PI 建议继续从事堆栈及其在算术几何、代数几何和非交换代数方面的应用方面的工作。 其中一个项目涉及研究全局域上几何有理簇的哈斯原理与布劳尔函数域群的周期指数问题之间的联系。 这类似于 Artin 对整数固有方案的布劳尔群有限性的猜想与 Tate-Shafarevich 猜想之间的联系，并且建立在 PI 早期关于扭转滑轮模量的工作之上。 第二个项目将继续 PI 和 Kovács 就非等平凡曲线族 Shafarevich 猜想的高维推广开展的联合工作。 该项目的关键是更好地理解堆栈之间的参数化态射，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形状的理解。 PI 和他的合作者将研究堆栈的双理性修改和分析的本质，以及堆栈的分类信息内容。 本研究的最终目标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。 从广义上讲，代数几何是对与代数对象相关的几何的研究。 历史上最有成果的例子之一就是圆的方程。该方程的二次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，数学家逐渐认识到代数和几何之间的联系比人们想象的要深刻得多，这导致几何方法逐渐侵入纯代数的遥远领域，并导致几个看似不同的学科的深刻统一。 代数几何的每一项新进展最终都会在数学的其他领域得到应用。事实证明，该领域的几个抽象领域非常适合计算机，如果没有代数几何，现代密码学就不可能实现。 堆栈理论相对较年轻，但它一直是最近关注的焦点，并且正在迅速成熟。 PI 的研究将致力于将栈理论应用于广泛的代数和几何问题，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.