CAREER: Moduli Spaces, Fundamental Groups, and Asphericality

职业：模空间、基本群和非球面性

• 批准号: 2338485
• 负责人: Nicholas Salter
• 金额: $ 48.95万
• 依托单位: University of Notre Dame
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2024
• 资助国家: 美国
• 起止时间: 2024-07-01 至 2029-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2338485&HistoricalAwards=false
• 关键词: CAREER; Moduli; Spaces; Fundamental; Groups

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.

该 NSF 职业奖为代数几何和拓扑学交叉领域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多项式方程组的研究。及其解决方案，而拓扑学是研究“形状”和“空间”等概念并开发数学技术来区分和分类这些对象的数学学科，这些领域中最重要的概念是“模空间”。 - 这是一个数学“世界地图”，它对特定数学对象的所有实例进行了完整的清单和分类。该项目的主要研究目标是通过导入技术更好地理解这些空间的结构并探索新现象。虽然主要目标是增进纯数学知识，但这些领域的发展在物理、数据科学、计算机视觉和机器人技术方面也有着长期的成功应用记录。由“数学圈研究所”（MCI）组成的倡议。 MCI 的目的是培训来自全国各地的 K-12 教师开展名为“数学圈”的数学强化活动。这项为期 1 周的年度计划将让教师与经验丰富的讲师配对，共同开发新的材料和方法并带回来。此外，还将组织一次研究会议，旨在吸引国际研究人员和学生，并传播与提案研究目标相关的进展。研究部分的总体目标是开发新方法。通过拓扑和几何群论研究各种模空间，特别是（1）阿贝尔微分层和（2）多项式族。一个主要目标是建立这些空间的“非球面性”（更高同伦的消失）。其基本群中的非球面性在来自代数几何的空间中出现的频率令人惊讶，而且几十年来，阿诺德、托姆和阿​​诺德的非球面性猜想常常产生深远的影响。 Kontsevich-Zorich 问题在很大程度上仍未得到解决，它已被视为一个极具挑战性的课题，该项目的目标是确定有希望的进展，他建议使用一种名为“Abel-Jacobi flow”的方法。建立阿贝尔微分的某些特殊层的非球面性，该程序的成功解决将构成 Kontsevich-Zorich 猜想的重大进步；还描述了第二个潜在的应用。主要重点是多项式族。这包括代数曲面上的线性系统；还概述了两个单变量多项式族，着眼于非球面猜想：（1）等临界分层和(2) 少数项空间。这些空间足够简单，可以具体理解，但也足够复杂，需要新的技术。除了拓扑之外，这里提出的工作有望为几何群注入新的例子。该领域的许多重要对象（辫群、映射类群、Artin 群）都与代数几何密切相关，PI 在此研究的空间基本群应该同样丰富，并且是一个主要目标。该项目的目的是实现这一目标。该奖项反映了 NSF 的法定使命，并且通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。