CAREER: Moduli Spaces, Fundamental Groups, and Asphericality

NSF CAREER奖为代数几何和拓扑学界面的研究项目以及旨在提高美国数学教育质量的推广工作提供支持。代数几何可以被描述为对多项式方程组及其解的研究，而拓扑学是研究诸如“形状”和“空间”等概念并发展数学技术来区分和分类这些对象的数学学科。在这些领域中，一个非常重要的概念是“模空间”——这是一个数学上的“世界地图”，它给出了一个特定数学对象的所有实例的完整清单和分类。该项目的主要研究目标是更好地理解这些空间的结构，并通过从邻近的数学领域引入技术来探索新的现象。虽然主要目标是推进纯数学知识，但这些领域的发展在物理学、数据科学、计算机视觉和机器人技术方面也有长期的成功应用记录。教育部分包括一个由“数学圈研究所”（MCI）组成的外联倡议。MCI的目的是培训来自全国各地的K-12教师如何开展被称为数学圈的数学丰富活动。这个为期一周的年度项目将把教师与经验丰富的讲师配对，共同开发新的材料和方法，并将其带回他们的家乡社区。此外，还将组织一次研究会议，目的是吸引国际社会的研究人员和学生，并传播与提案的研究目标有关的发展情况。研究部分的总体目标是通过拓扑学和几何群论开发新的方法来研究各种模空间，特别是(1)阿贝尔微分层和(2)多项式族。一个主要的目标是建立这些空间的“非球性”（高同伦的消失）。第二个目标是发展它们的基本群的几何理论。非球面在代数几何空间中以惊人的频率出现，并且通常具有深远的影响。几十年来，Arnol 'd， Thom和kontsevic - zorich的非球面猜想在很大程度上仍未得到解决，并且它已被视为一个非常具有挑战性的话题。这个项目的目标是确定有前途的进展。PI开发了一种叫做“阿贝尔-雅可比流”的方法，他建议用这种方法来建立一些特殊的阿贝尔微分层的非球面性。这个纲领的成功解决将构成康采维奇-佐里奇猜想的重大进展；还描述了其他潜在的应用。第二个重点是多项式族。这包括代数曲面上的线性系统；一个程序，以更好地了解基本组概述。我们还讨论了两个单变量多项式族，并着眼于非球面猜想：(1)等临界分层和(2)少多项式的空间。它们简单到可以具体理解，同时又复杂到需要新的技术。除了拓扑学，这里提出的工作有望为几何群论注入新的例子。该领域的许多中心对象（辫群、映射类群、Artin群）都与代数几何密切相关。PI在这里研究的空间的基本群应该同样丰富，该项目的一个主要目标是实现这一目标。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This NSF CAREER award provides support for a research program at the interface of algebraic geometry and topology, as well as outreach efforts aimed at improving the quality of mathematics education in the United States. Algebraic geometry can be described as the study of systems of polynomial equations and their solutions, whereas topology is the mathematical discipline that studies notions such as “shape” and “space" and develops mathematical techniques to distinguish and classify such objects. A notion of central importance in these areas is that of a “moduli space” - this is a mathematical “world map” that gives a complete inventory and classification of all instances of a particular mathematical object. The main research objective of the project is to better understand the structure of these spaces and to explore new phenomena, by importing techniques from neighboring areas of mathematics. While the primary aim is to advance knowledge in pure mathematics, developments from these areas have also had a long track record of successful applications in physics, data science, computer vision, and robotics. The educational component includes an outreach initiative consisting of a “Math Circles Institute” (MCI). The purpose of the MCI is to train K-12 teachers from around the country in running the mathematical enrichment activities known as Math Circles. This annual 1-week program will pair teachers with experienced instructors to collaboratively develop new materials and methods to be brought back to their home communities. In addition, a research conference will be organized with the aim of attracting an international community of researchers and students and disseminating developments related to the research objectives of the proposal.The overall goal of the research component is to develop new methods via topology and geometric group theory to study various moduli spaces, specifically, (1) strata of Abelian differentials and (2) families of polynomials. A major objective is to establish “asphericality" (vanishing of higher homotopy) of these spaces. A second objective is to develop the geometric theory of their fundamental groups. Asphericality occurs with surprising frequency in spaces coming from algebraic geometry, and often has profound consequences. Decades on, asphericality conjectures of Arnol’d, Thom, and Kontsevich–Zorich remain largely unsolved, and it has come to be regarded as a significantly challenging topic. This project’s goal is to identify promising-looking inroads. The PI has developed a method called "Abel-Jacobi flow" that he proposes to use to establish asphericality of some special strata of Abelian differentials. A successful resolution of this program would constitute a major advance on the Kontsevich–Zorich conjecture; other potential applications are also described. The second main focus is on families of polynomials. This includes linear systems on algebraic surfaces; a program to better understand the fundamental groups is outlined. Two families of univariate polynomials are also discussed, with an eye towards asphericality conjectures: (1) the equicritical stratification and (2) spaces of fewnomials. These are simple enough to be understood concretely, while being complex enough to require new techniques. In addition to topology, the work proposed here promises to inject new examples into geometric group theory. Many of the central objects of interest in the field (braid groups, mapping class groups, Artin groups) are intimately related to algebraic geometry. The fundamental groups of the spaces the PI studies here should be just as rich, and a major goal of the project is to bring this to fruition.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.