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.

这个NSF CAREER奖为代数几何和拓扑学接口的研究计划提供支持，以及旨在提高美国数学教育质量的推广工作。代数几何学可以描述为研究多项式方程组及其解，而拓扑学是研究诸如“形状”和“空间”等概念的数学学科，并开发数学技术来区分和分类这些对象。在这些领域中，一个重要的概念是“模空间”-这是一个数学“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，并通过引入邻近数学领域的技术来探索新现象。虽然其主要目标是推进纯数学知识的发展，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着长期的成功应用记录。教育部分包括一个由“数学圈研究所”组成的外联倡议。MCI的目的是培训来自全国各地的K-12教师开展被称为数学圈的数学丰富活动。这个为期一周的年度计划将教师与经验丰富的教师配对，共同开发新的材料和方法，带回他们的家乡社区。此外，为了吸引国际研究人员和学生，并传播与提案的研究目标相关的进展，还将组织一次研究会议。研究部分的总体目标是通过拓扑学和几何群论开发新方法，以研究各种模空间，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“非球面性”（消失的高同伦）。第二个目标是发展几何理论的基本群体。非球面性在代数几何的空间中以令人惊讶的频率出现，并且往往具有深远的影响。几十年过去了，Arnol'd、Thom和Kontsevich-Zorich的非球面性问题在很大程度上仍然没有解决，它已经被认为是一个非常具有挑战性的话题。这个项目的目标是确定有希望的进展。PI开发了一种称为“Abel-Jacobi流”的方法，他建议使用该方法来建立阿贝尔微分的某些特殊层的非球面性。一个成功的决议，这一计划将构成一个重大进展的Kontsevich-Zorich猜想，其他潜在的应用也进行了说明。第二个主要焦点是多项式族。这包括线性系统的代数曲面;一个程序，以更好地了解基本群体的概述。本文还讨论了两类一元多项式的非球面性质：（1）等临界分层和（2）少项空间。这些都很简单，可以具体理解，同时又很复杂，需要新的技术。除了拓扑结构，这里提出的工作有望注入新的例子到几何群论。该领域的许多中心对象（辫子群、映射类群、阿廷群）都与代数几何密切相关。PI研究的空间的基本群体应该同样丰富，该项目的主要目标是实现这一目标。该奖项反映了NSF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。