New Directions in Homology of Moduli Spaces

模空间同调的新方向

• 批准号: 1803766
• 负责人: Michael Hopkins
• 金额: $ 15.58万
• 依托单位: Harvard University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-08-15 至 2022-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1803766&HistoricalAwards=false
• 关键词: New; Directions; Homology; Moduli; Spaces

The research conducted under this grant concerns the study of moduli spaces of algebraic and geometric objects, and in particular those of manifolds. Manifolds are mathematical objects generalizing the three-dimensional space in which we live, and appear throughout mathematics and science. Locally they look like ordinary space, though possibly of a different dimension, but globally their shape may be complicated. As a consequence, they can have interesting symmetries or can be deformed in interesting ways. It is a classical problem to understand which symmetries and deformation exist, and how to distinguish these in a computationally effective manner. Moduli spaces provide a geometric method to access this information, and we will contribute to the progress of science by furthering their study.To understand of the topology of moduli spaces of algebraic and geometric interest, and in particular their homology, the Principal Investigator will apply the techniques of homotopy theory and higher algebra. Firstly, the PI shall use the existence of additional higher-algebraic structures on moduli spaces. The PI will then use higher-algebraic cellular structures to reinterpret homological stability results and obtain meta-stable results about those homology groups for which homological stability does not hold. Secondly, the PI will destabilize recent stable results in manifold theory by comparing closely related moduli 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.

根据这项补助金进行的研究涉及代数和几何对象的模空间的研究，特别是流形。流形是数学对象，概括了我们生活的三维空间，并出现在数学和科学中。在局部，它们看起来像普通的空间，虽然可能有不同的维度，但从全局来看，它们的形状可能很复杂。因此，它们可以具有有趣的对称性，或者可以以有趣的方式变形。理解存在哪些对称性和变形，以及如何以计算有效的方式区分这些对称性和变形是一个经典问题。模空间提供了一种几何方法来获取这些信息，我们将通过进一步的研究来促进科学的进步。为了理解代数和几何兴趣的模空间的拓扑结构，特别是它们的同调，首席研究员将应用同伦理论和高等代数的技术。首先，PI将使用模空间上的附加高等代数结构的存在性。PI然后将使用更高的代数细胞结构来重新解释同调稳定性结果，并获得关于那些同调稳定性不成立的同调群的亚稳定结果。其次，PI将通过比较密切相关的模空间来破坏流形理论中最近的稳定结果。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。

项目成果

The cohomology of Torelli groups is algebraic

Torelli 群的上同调是代数的

• DOI: 10.1017/fms.2020.41
• 发表时间: 2020
• 期刊: Sigma
• 作者: Kupers, Alexander;Randal-Williams, Oscar
• 通讯作者: Randal-Williams, Oscar

Intersection forms of spin 4-manifolds and the pin(2)-equivariant Mahowald invariant

自旋 4 流形与 pin(2) 等变 Mahowald 不变量的交集形式

• DOI: 10.1090/cams/4
• 发表时间: 2022
• 期刊: Communications of the American Mathematical Society
• 作者: Hopkins, Michael;Lin, Jianfeng;Shi, XiaoLin Danny;Xu, Zhouli
• 通讯作者: Xu, Zhouli

E_2-cells and mapping class groups

E_2-细胞和映射类组

• DOI: 10.1007/s10240-019-00107-8
• 发表时间: 2019
• 期刊: Publications mathématiques de l'IHÉS
• 作者: Galatius, Søren;Kupers, Alexander;Randal-Williams, Oscar
• 通讯作者: Randal-Williams, Oscar

Characteristic classes of bundles of K3 manifolds and the Nielsen realization problem

K3 流形束的特征类和 Nielsen 实现问题

• DOI: 10.2140/tunis.2021.3.75
• 发表时间: 2021
• 期刊: Tunisian Journal of Mathematics
• 影响因子: 0.9
• 作者: Giansiracusa, Jeffrey;Kupers, Alexander;Tshishiku, Bena
• 通讯作者: Tshishiku, Bena

ON THE COHOMOLOGY OF TORELLI GROUPS

论托雷利群的上同调

• DOI: 10.1017/fmp.2020.5
• 发表时间: 2020
• 期刊: Pi
• 作者: KUPERS, ALEXANDER;RANDAL-WILLIAMS, OSCAR
• 通讯作者: RANDAL-WILLIAMS, OSCAR