Independence of l and local terms

这一建议的重点是在代数变异及其上同调的研究中出现的问题，其中许多有算术应用。PI和其他人的早期工作导致了我们对由对应引起的上同调运算的理解的实质性发展，也导致了PI将研究的一系列新问题。该PI将研究各种上同调理论（特别是表、交和结晶上同调）上对应作用的l的独立性问题，继续他在局部陈氏类上的工作，并使用迹公式研究全局问题。这些研究路线是由动机理论驱动的，该理论预测了某些算子在上同调上的结果的独立性，并表明它们的痕迹应该具有几何意义。此外，PI将继续早期的工作，研究代数堆栈，阿贝尔变分，对数几何，傅里叶- mukai变换和模空间。PI将继续为从事与拟议研究相关项目的博士生提供建议。建议的工作涉及基本的几何结构，如上同调，模空间和堆栈，这是代数几何及其与其他领域的相互作用的核心，包括数论，表示论，组合学和实际应用。粗略代数几何是研究代数变量的几何性质的学科，代数变量是多项式方程组的解。上同调理论为研究这类变异提供了一些最有力的技术。例如，Lefschetz迹公式可以用来估计有限域上多项式的解的数量，更一般地说，代数变体的上同群是伽罗瓦表示的主要来源之一，是现代数论的基本对象。该提案解决了在此背景下产生的上同调理论问题。另一个研究代数变异的基本方法是通过它们的分类和模空间。PI在模空间和堆栈方面的工作将继续推进我们对关键模空间的理解，并提供广泛适用的基础工具。

This proposal focuses on questions that arise in the study of algebraic varieties and their cohomology, many of which have arithmetic applications. Earlier work of the PI and others has led to substantial recent developments in our understanding of operations on cohomology arising from correspondences, and also to a new suite of problems which the PI will investigate. The PI will study questions of independence of l for actions of correspondences on various cohomology theories (specifically etale, intersection, and crystalline cohomology), continue his work on localized chern classes, and study global questions using trace formulas. These lines of investigation are motivated by the theory of motives, which predicts independence of l results for certain operators on cohomology and suggests that their traces should have geometric significance.In addition, the PI will continue earlier work researching algebraic stacks, abelian varieties, log geometry, Fourier-Mukai transforms, and moduli spaces. The PI will continue to advise Ph.D. students working on projects related to the proposed research.The proposed work concerns basic geometric structures, such as cohomology, moduli spaces, and stacks, which lie at the core of algebraic geometry and its interactions with other fields, including number theory, representation theory, combinatorics, and practical applications. Roughly algebraic geometry is concerned with the study of geometric properties of algebraic varieties, which are solutions of systems of polynomial equations. Cohomology theories provide some of the most powerful techniques for the study of such varieties. For example, the Lefschetz trace formula can be used to estimate numbers of solutions of polynomials over finite fields, and more generally cohomology groups of algebraic varieties are one of the main sources of Galois representations, which are fundamental objects in modern number theory. The proposal addresses problems on cohomology theories arising in this context. Another fundamental approach to the study of algebraic varieties is through their classification and moduli spaces. The PI's work on moduli spaces and stacks will continue to advance our understanding of key moduli spaces and provide broadly applicable foundational tools.