Knots, Disks, and Exotic Phenomena in Dimension 4

流形是一种局部看起来简单，但可以具有有趣的全局形状和属性的对象。拓扑学领域的一个指导性问题是在给定等价概念的范围内，对固定维上所有可能的流形进行分类。长期存在的悬而未决的问题涉及到4维中奇异对的检测。这些流形是同胚的，但不是微分同胚的，也就是说它们只有一个很好的等价概念来区分。Fox和Milnor的技术，以及最近由Marengon、Manolescu和Piccirillo推广的技术表明，可以通过利用结点或纠缠圈的某些性质来识别奇异对，这些结或纠缠圈位于4流形的边界上，移除一个开放的球后。PI将探索这种范式，它有可能导致我们区分光滑四维流形的能力发生根本性的转变。另一组目标是研究结本身的性质，其中某些老问题可以通过PI与Blair等人一起引入的组合工具来解决。总的来说，这些方法的发展可以对拓扑学领域和应用科学产生持久的影响，例如信号处理，数据科学和控制理论，其中流形以多种形式出现-例如，作为动力系统的状态空间。PI还致力于为代表性不足的社区创造教育和研究机会。PI将继续使用组合工具来解决拓扑学中的开放性问题；她的方法将为研究生和高级本科生提供一个进入该领域前沿研究的切入点。该项目概述了低维拓扑中几个不同目标的路径，其中代数和几何现象之间的相互作用是统一原则；其中组合方法已被证明有效；其中3球面的结点和嵌入4流形中的曲面是我们感兴趣的中心对象。PI将研究的具体问题包括Cappell和Shaneson的子向秩猜想，Fox和Milnor的切片带猜想，以及结理论在4维奇异光滑结构检测中的应用。PI将使用最先进的工具，其中一些工具是她之前帮助开发和应用的。工具包括：结群的考克斯特商；结图的Wirtinger数；4-流形的三切分；在此背景下，PI和合作者提取了4维的奇异分支覆盖和带状障碍物。该奖项反映了美国国家科学基金会的法定使命，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。

Manifolds are objects which look locally simple but can have interesting global shape and properties. A guiding question in the field of topology is the classification, up to a given notion of equivalence, of all possible manifolds in a fixed dimension. Long-standing open questions concern the detection of exotic pairs in dimension 4. These are manifolds which are homeomorphic but not diffeomorphic, meaning that they are only distinguished by a very fine notion of equivalence. Techniques going back to Fox and Milnor and recently extended by Marengon, Manolescu and Piccirillo demonstrate that exotic pairs can be identified by using certain properties of knots, or entangled circles, which lie in the boundary of a 4-manifold after removing an open ball. The PI will explore this paradigm, which has the potential to lead to a fundamental shift in our ability to distinguish smooth 4-dimensional manifolds. An additional set of goals concerns studying the properties of knots in their own right, where certain old questions can be attacked by combinatorial tools introduced by the PI along with Blair and others. Overall, the development of these methods can have a lasting impact on the field of topology and on the applied sciences it informs, such as signal processing, data science and control theory, where manifolds appear in many guises - for one example, as state spaces of dynamical systems. The PI is also engaged in creating education and research opportunities for underrepresented communities. The PI will continue a program to employ combinatorial tools to address open questions in topology; her approach will provide graduate and advanced undergraduate students with an entry point to cutting-edge research in the field.The project outlines paths toward several distinct goals in low-dimensional topology for which the interaction between algebraic and geometric phenomena is the unifying principle; in which a combinatorial approach has proven effective; and in which knots in the 3-sphere and surfaces embedded in 4-manifolds are a central object of interest. Specific problems the PI will study include the Meridional Rank Conjecture of Cappell and Shaneson, the Slice-Ribbon Conjecture of Fox and Milnor and applications of knot theory to the detection of exotic smooth structures in dimension 4. The PI will employ state of the art tools, several of which she has previously helped develop and apply. The tools include: Coxeter quotients of knot groups; the Wirtinger number of a knot diagram; trisections of 4-manifolds; singular branched covers in dimension 4 and a ribbon obstruction extracted by the PI and collaborators from this context.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.