Algebraic Stacks and Applications

PI建议继续他的工作栈及其应用算术几何，代数几何和非交换代数。 一个项目涉及研究全球领域的几何合理品种的哈塞原则和周期指数问题的布劳尔群的功能领域之间的联系。 这类似于阿廷关于整数上的Brauer群的有限性猜想和泰特-沙法列维奇猜想之间的联系，并建立在PI关于扭曲层模的早期工作之上。 第二个项目将继续联合工作进行的PI和Kovács对高维推广的Shafarevich猜想的非isotrivial家庭的曲线。 这个项目的关键将是更好地理解堆栈参数化堆栈之间的态射，其系统的研究只是最近才开始。 第三个项目旨在加深对堆栈及其内在几何结构的理解。 PI和他的合作者将研究栈的双有理修改和分析的性质，以及栈的分类信息内容。本研究的最终目的是拓宽代数几何和栈理论方法在纯代数及相关领域的应用。广义上讲，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;方程的二次性质与直线通常与圆相交于两点的事实密切相关。 在过去的几千年里，数学家们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到纯粹代数的广泛领域，并将几个完全不同的学科统一起来。 代数几何的每一个新进展最终都能应用到数学的其他领域;这个领域的一些抽象领域已经被证明是非常适合计算机的，没有代数几何，现代密码学是不可能的。 堆栈理论相对年轻，但它一直是最近关注的焦点，并迅速成熟。 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.