Topology and group theory

群是编码对称性的基本数学对象；例如，晶体材料的对称性。它们出现在许多领域，包括几何、算术，甚至数学物理。它们在数学对象的代数属性和几何属性之间架起了一座桥梁。主要研究人员将研究跨越这座桥的两个方向的问题，应用代数来揭示低维空间的几何性质，并使用几何来阐明似乎严格意义上的代数问题。一个重要的主题是上同调的概念，它粗略地度量了存在于几何和代数环境中的高维洞。主要研究者将研究算术群的拓扑、映射类群和自由群的自同构群。计划中的研究包括两个部分。首先，研究人员将研究这些群的对偶性质，这将被用来执行一系列上同调计算。在第二部分中，研究者将研究Torelli群的拓扑和几何。这些项目的重点包括Torelli群的有限性质，表示稳定性理论，以及与同调三球不变量的关系。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Groups are fundamental mathematical objects that encode symmetries; for instance, those of a crystalline material. They arise in many fields including geometry, arithmetic, and even mathematical physics. They provide a bridge between algebraic and geometric properties of a mathematical object. The principal investigator will study problems that traverse this bridge in both directions, applying algebra to uncover geometric properties of low-dimensional spaces and using geometry to shed light on problems that appear to be strictly algebraic. An important theme is the concept of cohomology, which roughly speaking measures high-dimensional holes that exist in both geometric and algebraic settings.The principal investigator will study the topology of arithmetic groups, the mapping class group, and the automorphism groups of free groups. The planned research has two parts. In the first, the investigator will study duality properties of these groups, which will be used to perform a series of cohomological calculations. In the second part, the investigator will study the topology and geometry of the Torelli group. Foci of these projects include finiteness properties of the Torelli group, the theory of representation stability, and relationships with invariants of homology 3-spheres.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.