Anabelian methods in arithmetic and algebraic geometry

这项建议旨在进一步发展代数几何、数论和拓扑学之间的美丽联系，尽管这些联系还没有被很好地理解。代数变体--多项式方程系统的解的集合--在数学中无处不在，它们的基本性质部分来自于这样一个事实，即它们的研究处于数学的不同领域的交集。这一提议的目标是解开这些看似不同的领域之间的联系，主要是通过研究代数簇的基本群，不变量粗略地说捕捉到了簇中循环的结构。最近的发现表明，这种不变量对于理解多项式方程组的有理解是至关重要的--这种所谓的“丢番图”问题几千年来一直吸引着数学家。尽管它们有很长的历史，但我们现在才开始通过截面猜想、非阿贝尔Chabauty方法和其他非常新的发展来理解这些问题与拓扑学的联系。广义地说，代数几何和数论与基本群相联系的方面被称为“阿纳贝尔几何”，这是提议的主题。在我以前工作的基础上，我计划更好地理解代数簇拓扑的再交换方面，特别是再交换几何和单列表示之间的关系，目的是证明几何学中的两个著名的公开问题：几何扭转猜想和正特征中的(猜想)难Lefschetz定理。在这些问题上的进展将从根本上促进我们对代数簇拓扑学的理解。我还计划(在与Aaron Landesman的联合工作中)通过运用代数几何和拓扑技术，在Putman-Wieland猜想上取得进展，Putman-Wieland猜想是曲面拓扑中的一个基本问题；同样，我与Li、Salter和Srinivesan的联合工作表明，这些技术可以洞察Grothendieck截面猜想，也许是Anabelian几何和算术之间的基本(猜想)联系。这项工作还将深入了解模空间的拓扑，这是代数几何的基本研究对象之一。最后，这项建议将建立在算术几何的最新发展--特别是非阿贝尔Chabauty方法--的基础上，以开发解决算术问题的实用方法。特别是，与Eric Katz的合作将产生运行非阿贝尔Chabauty方法以在不良归约曲线上寻找有理点的技术，这将是使该方法作为求解多项式方程组的一种方法实用的关键。

This proposal aims to further develop the beautiful, though not yet well-understood, connections between algebraic geometry, number theory, and topology. Algebraic varieties -- the sets of solutions to systems of polynomial equations -- are ubiquitous in mathematics, and their fundamental nature comes in part from the fact that their study lies at the intersection of such varied fields of mathematics. The goal of this proposal is to unwind the connections between these seemingly disparate fields, primarily through the study of the fundamental group of an algebraic variety, an invariant which loosely speaking captures the structure of loops in the variety. Recent discoveries have shown that this invariant is crucial to understanding, for example, rational solutions to systems of polynomial equations -- such so-called "Diophantine" questions have fascinated mathematicians for millenia. Despite their long history, we are only now beginning to understand the connections of such questions to topology, via the section conjecture, the non-abelian Chabauty method, and other extremely recent developments. Broadly speaking, the aspect of algebraic geometry and number theory connected to the fundamental group is called "anabelian geometry," which is the subject of the proposal. Building on my previous work, I plan to better understand anabelian aspects of the topology of algebraic varieties, and in particular the relationship between anabelian geometry and monodromy representations, with the goal of proving two well-known open questions in geometry: the geometric torsion conjecture and the (conjectural) Hard Lefschetz theorem in positive characteristic. Progress on these questions would fundamentally advance our understanding of the topology of algebraic varieties. I also plan to make progress (in joint work with Aaron Landesman) on the Putman-Wieland conjecture, a fundamental question in the topology of surfaces, by bringing to bear algebro-geometric and topological techniques; similarly, my joint work with Li, Salter, and Srinivasan shows that such techniques can yield insight into Grothendieck's section conjecture, perhaps the fundamental (conjectural) connection between anabelian geometry and arithmetic. This work will also yield insight into the topology of moduli spaces, one of the fundamental objects of study in algebraic geometry. Finally, this proposal will build on very recent developments in arithmetic geometry -- in particular, the non-abelian Chabauty method -- to develop practical methods for solving arithmetic questions. In particular, joint work with Eric Katz will yield techniques for running the non-abelian Chabauty method to find rational points on curves of bad reduction, which will be crucial to make the method practical as a way to find solutions to systems of polynomial equations.