Stability Patterns in the Homology of Moduli Spaces

模空间同调中的稳定性模式

• 批准号: 2202943
• 负责人: Jeremy Miller
• 金额: $ 26.99万
• 依托单位: Purdue University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-08-01 至 2025-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2202943&HistoricalAwards=false
• 关键词: Stability; Patterns; Homology; Moduli; Spaces

Homology is a mathematical tool that was introduced over a hundred years ago to measure features of shape that remain invariant under deformation. This helped make rigorous many calculations in calculus and physics involving integrals of functions defined on spaces with “holes.” Homology gives an algebraic measure of holes in geometric objects, allowing algebraic techniques to give geometric information, for example, showing that a given geometric object cannot be deformed into another one. The aim of this project is to study patterns in the homology of families of spaces coming from number theory and geometry. These patterns simplify homology calculations by reducing infinite calculations to finite calculations. In this project, new methods involving the use of computers in theoretical algebraic topology and number theory will be developed. The project will enhance graduate and postdoctoral training in algebraic topology through mentoring, seminars, and conferences. The PI will promote diversity and inclusion through participation in a summer bridge program designed for students from underrepresented racial and ethnic groups to better prepare them for graduate school.This project aims to improve understanding of the homology of arithmetic groups, a central concept in number theory, algebraic K-theory, and even the theory of manifolds. In high dimensions, the homology is known to vanish, and, in low dimensions, the homology is known to stabilize. These stable homology groups have been completely calculated in many cases. The project focuses on two ranges of dimensions: just below where the homology is known to vanish and just above the stable range. Conjecturally, the highest degree homology groups should exhibit a pattern called “extremal stability” and the homology near the stable range should exhibit a different pattern called “secondary stability.” Highly connected simplicial complexes and operadic cells will be used to try to establish these conjectures. The connection with algebraic K-theory will be a key point of emphasis when studying the homology of arithmetic groups. Similar patterns will be investigated in spaces coming from geometric topology such as various moduli spaces and configuration spaces.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.

同源性是一百多年前引入的数学工具，用于测量形状的特征，而形状的特征在变形下保持不变。这有助于在计算和物理学中进行严格的计算，这些计算涉及在带有“孔”空间上定义的功能积分的积分。同源性给出了几何对象中孔的代数测量，从而允许代数技术提供几何信息，例如，表明给定的几何对象不能变形为另一个。该项目的目的是研究来自数量理论和几何形状的空间家族的同源性模式。这些模式通过将无限的计算减少到有限计算来简化同源性计算。在该项目中，将开发出在理论代数拓扑中使用计算机和数字理论的新方法。该项目将通过心理，半手和会议来增强代数拓扑的研究生和博士后培训。 PI将通过参加夏季桥梁计划，旨在为来自代表性不足的赛车和族裔群体的学生提供更好的准备，以更好地为研究生做准备。在高维度中，已知同源性消失，并且在低维度中，同源性是稳定的。在许多情况下，这些稳定的同源组已经完全计算出来。该项目着重于两个维度范围：在同源性消失并在稳定范围之上的同源性范围之下。可以想象的是，最高程度的同源组应执行一种称为“极端稳定性”的模式，并且稳定范围附近的同源性应执行一种称为“次要稳定性”的不同模式。高度连接的简单复合物和操作细胞将用于尝试建立这些概念。在研究算术组的同源物时，与代数K理论的联系将是重点的关键点。在来自各种模量空间和配置空间等几何拓扑的空间中将研究类似的模式。该奖项反映了NSF的法定任务，并通过使用基金会的知识分子优点和更广泛的影响评估标准来评估值得支持。