Stability Patterns in the Homology of Moduli Spaces

同调是一种数学工具，在一百多年前被引入，用于测量在变形下保持不变的形状特征。这有助于使严格的许多计算微积分和物理学涉及积分的功能定义的空间与“洞”。同调给出了几何对象中孔洞的代数度量，允许代数技术给出几何信息，例如，表明给定的几何对象不能变形为另一个。这个项目的目的是研究来自数论和几何的空间族的同调模式。这些模式通过将无限计算减少到有限计算来简化同源性计算。在这个项目中，将开发涉及在理论代数拓扑学和数论中使用计算机的新方法。该项目将通过指导、研讨会和会议加强代数拓扑学的研究生和博士后培训。PI将通过参加一个夏桥项目来促进多样性和包容性，该项目旨在为来自代表性不足的种族和民族的学生更好地准备研究生院。该项目旨在提高对算术群的同源性的理解，这是数论，代数K理论甚至流形理论的核心概念。在高维度中，已知同源性消失，并且在低维度中，已知同源性稳定。这些稳定的同调群在许多情况下已经被完全计算出来。该项目关注两个维度范围：刚好低于已知同源性消失的地方和刚好高于稳定范围。从理论上讲，最高程度的同源群应该表现出一种称为“极端稳定性”的模式，而接近稳定范围的同源群应该表现出一种称为“次级稳定性”的不同模式。高度连接的单纯复形和运算细胞将被用来尝试建立这些结构。与代数K-理论的联系将是研究算术群同调的一个重点。类似的模式将在来自几何拓扑的空间中进行研究，例如各种模数空间和配置空间。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。

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.