LEAPS-MPS: Braids and Mapping Class Groups: Investigating Left-orders, Twisting, and Positivity

该奖项全部或部分由2021年美国救援计划法案（公法117-2）资助。该项目涉及拓扑学中的几个研究问题，即形状的研究。拓扑学可以帮助我们研究困难而有趣的问题，例如我们的宇宙是什么形状，我们如何从嘈杂的数据集中提取有意义的信息，或者人类DNA如何解开几英尺长，可以折叠起来适合细胞。 该项目研究结的数学特性，结是三维空间中的打结循环，三维流形是局部看起来像我们生活的三维世界的物体。中心主题是调查如何由二维信息，即表面上的地图，决定的结和三维流形的信息。将三维问题转化为二维问题是贯穿拓扑学和几何学的强大工具。在这个项目中，节点可以用称为辫子的对象表示，三流形可以用称为开卷分解的对象表示，这两种方法分别将曲面上的映射与节点或三流形相关联。人们可以问，这些映射的性质如何决定相应的纽结或三流形的性质。调查员还计划举办职业准备活动，在公共卫生等广泛兴趣领域的数学圈，以及她所在机构的杰出演讲者系列，目的是增加代表性不足的群体在STEM中的参与，并改善她所在机构的研究环境。研究人员将进行三个项目，调查如何由二维信息支配的三维流形和结的属性。第一个项目探讨了与左序和扭曲有关的辫子的结果是否可以扩展到曲面上更一般的映射，并可能应用于三流形和L空间猜想的研究。第二个项目问是否欧拉特征的开卷分解为一个给定的固定的三个流形是有界的，与潜在的应用接触几何中的一个开放的问题。第三个项目是对编织积极性和打结电缆概念的调查，可能应用于安井的公开问题。该奖项反映了NSF的法定使命，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。

This award is funded in whole or in part under the American Rescue Plan Act of 2021 (Public Law 117-2).This project concerns several research questions in topology, which is the study of shapes. Topology can help us investigate difficult and interesting questions, such as what the shape of our universe is, how we can extract meaningful information from noisy data sets, or how human DNA, which unravels to be several feet long, can fold up to fit inside a cell. The project investigates mathematical properties of knots, which are knotted up loops in three-dimensional space, and three-manifolds, which are objects that locally look like the three-dimensional world we live in. The central theme is to investigate how information about knots and three-manifolds is dictated by two-dimensional information, namely, maps on surfaces. Turning a three-dimensional question into a two-dimensional one is a powerful tool throughout topology and geometry. In this project, knots can be represented by objects called braids, and three-manifolds can be represented by objects called open book decompositions, both of which give ways of associating a map on a surface to a knot or three-manifold, respectively. One can ask how properties of these maps dictate properties of the corresponding knot or three-manifold. The investigator also plans to hold career preparation events, math circles in areas of broad interest such as public health, and a distinguished speaker series at her institution, with the goal of increasing the participation of underrepresented groups in STEM and enhancing the research environment at her institution. The investigator will undertake three projects that investigate how properties of three-manifolds and knots are dictated by two-dimensional information. The first project explores whether results about braids relating to left-orders and twisting can be extended to more general maps on surfaces, with potential applications to the study of three-manifolds and the L-Space Conjecture. The second project asks whether the Euler characteristics of open book decompositions for a given fixed three-manifold are bounded, with potential applications to an open question in contact geometry. The third project is an investigation of the notion of braid positivity and knot cables, with potential applications to an open question of Yasui.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.