Independence of l and local terms

这项建议集中在研究代数簇及其上同调中出现的问题，其中许多都有算术应用。PI和其他人的早期工作导致了我们对上同调运算的理解最近有了实质性的发展，并导致了PI将调查的一系列新问题。PI将研究L在各种上同调理论(特别是Etale、交和晶体上同调)上的对应作用的独立性问题，继续他在局部化陈氏类上的工作，并使用迹公式研究全局问题。这些研究方向是由动机理论驱动的，该理论预测了某些上同调算子的L结果的独立性，并建议它们的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、交换簇、对数几何、傅立叶-Mukai变换和模空间。PI将继续为从事与建议研究相关的项目的博士生提供建议。建议的工作涉及基本几何结构，如上同调、模空间和堆栈，这些结构位于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。大致来说，代数几何是研究代数簇的几何性质的，代数簇是多项式方程组的解。上同源理论为研究这类变种提供了一些最强大的技术。例如，Lefschetz迹公式可以用来估计有限域上多项式的解的个数，更广泛地说，代数簇的上同调群是Galois表示的主要来源之一，而Galois表示是现代数论中的基本对象。该提案解决了在这一背景下出现的上同调理论问题。研究代数簇的另一种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.