CAREER: Stacks, moduli spaces, and log geometry

PI研究堆栈和对数几何的基础问题和应用。该项目大致分为两部分。 第一部分是关于模空间和对数几何的研究。 由于对数几何和对数结构与退化密切相关，对数几何在模空间紧化的研究中起着自然的作用。 PI研究了与对数几何和模空间相关的几个问题，还研究了对数Grothendieck-Riemann-Roch定理，推广了经典的方案定理。该项目的第二部分涉及代数栈上的层。项目这一部分的中心主题是跟踪公式及其对堆栈的推广。该项目所有部分的核心是PI早期工作中发现的代数堆栈和日志几何之间的密切关系。给定定义代数簇的多项式方程组，有两种自然的方法来研究该系统。 一种方法是研究方程的对称性，第二种方法是改变方程的系数（“模量”）并将其“退化”为一个更简单的（尽管可能是奇异的）系统。 由Artin、Deligne、Grothendieck和Mumford在20世纪60年代后期引入的堆栈是代数几何中研究具有额外对称性的空间的主要工具。近年来，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起着重要的作用，栈的许多令人兴奋的新应用也被发现。 例如，栈在布劳尔群的研究中以及在弦理论和代数几何之间的丰富相互作用中发挥着核心作用。对于退化的研究，一个关键的工具是对数几何理论，由加藤，方丹，和Illusie在20世纪80年代后期开发。 如上所述，在早期的工作中，PI将这个理论与堆栈联系起来，正在进行的项目使用这种关系来进一步理解退化和代数簇的模。

The PI studies both foundational questions and applications of stacks and logarithmic geometry. The project roughly breaks up into two parts. The first part concerns the study of moduli spaces and log geometry. Since log geometry and logarithmic structures are closely related to degenerations, log geometry plays a natural role in the study of compactifications of moduli spaces. The PI studies several problems related to log geometry and moduli spaces, and also studies a logarithmic Grothendieck-Riemann-Roch theorem, generalizing the classical theorem for schemes. The second part of the project concerns sheaves on algebraic stacks. A central theme in this part of the project is trace formulas and their generalizations to stacks. Central to all parts of the project is the close relationship between algebraic stacks and log geometry discovered in earlier work of the PI.Given a system of polynomial equations defining an algebraic variety, there are two natural ways to study the system. One approach is to study the symmetries of the equations, and a second approach is to vary the coefficients of the equations ("moduli") and "degenerate" them to a simpler (though possibly singular) system. Stacks, introduced by Artin, Deligne, Grothendieck, and Mumford in the late 1960's, are the main tool used in algebraic geometry to study spaces with additional symmetries. 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. For example, stacks play a central role in the study of Brauer groups and in the rich interaction between string theory and algebraic geometry. For the study of degenerations, a key tool is the theory of logarithmic geometry developed by Kato, Fontaine, and Illusie in the late 1980's. As mentioned above, in earlier work the PI related this theory to stacks, and the ongoing projects use this relationship to further our understanding of degenerations and the moduli of algebraic varieties.