Anabelian methods in arithmetic and algebraic geometry

这一建议旨在进一步发展代数几何、数论和拓扑学之间的美好联系，尽管还没有被很好地理解。代数变量——多项式方程组的解集——在数学中无处不在，它们的基本性质部分来自于它们的研究处于这些不同数学领域的交叉点。这个提议的目标是解开这些看似不同的领域之间的联系，主要是通过对代数变量的基本群的研究，一个不变量，松散地说，抓住了变量中的循环结构。最近的发现表明，这个不变量对于理解多项式方程系统的有理解至关重要——这种所谓的“丢番图”问题几千年来一直吸引着数学家。尽管这些问题有着悠久的历史，但我们现在才开始通过截面猜想、非阿贝尔夏博蒂方法和其他最近的发展来理解这些问题与拓扑学的联系。广义地说，代数几何和数论中与基本群相关的方面被称为“伪几何”，这是该提案的主题。在我以前的工作的基础上，我计划更好地理解代数变体拓扑的可倒性方面，特别是可倒性几何与单表示之间的关系，目的是证明几何中两个众所周知的开放问题：几何扭转猜想和（猜想的）Hard Lefschetz定理的正特征。这些问题的进展将从根本上推进我们对代数变体拓扑的理解。我还计划（与Aaron Landesman合作）在Putman-Wieland猜想上取得进展，这是曲面拓扑中的一个基本问题，通过使用代数几何和拓扑技术；同样，我与Li、Salter和Srinivasan的合作表明，这种技术可以深入了解格罗腾迪克的截面猜想，这可能是anabelian几何和算术之间的基本（猜想）联系。这项工作也将产生对模空间拓扑的洞察力，这是代数几何研究的基本对象之一。最后，本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-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方法的技术，以找到坏约简曲线上的有理点，这将是使该方法作为一种实用的方法来寻找多项式方程系统的解的关键。

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.