Knots, Disks, and Exotic Phenomena in Dimension 4

流形是看起来局部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的等价概念下，对固定维中所有可能的流形进行分类。长期悬而未决的问题涉及检测奇异对4维。这些流形是同胚的，但不是非同胚的，这意味着它们只能通过一个非常精细的等价概念来区分。技术可以追溯到福克斯和米尔诺，最近由Marengon，Manolescu和Piccirillo扩展，证明可以通过使用结或纠缠圆的某些性质来识别奇异对，这些结或纠缠圆位于4-流形的边界上。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。另外一组目标涉及研究结本身的性质，其中某些老问题可以通过PI沿着Blair等人引入的组合工具来解决。总体而言，这些方法的发展可以对拓扑学领域及其所告知的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将继续一个项目，采用组合工具来解决拓扑学中的开放问题;她的方法将为研究生和高年级本科生提供该领域前沿研究的切入点。该项目概述了通往低维拓扑学几个不同目标的道路，其中代数和几何现象之间的相互作用是统一的原则;其中组合方法已被证明是有效的;并且其中3-球面中的结和嵌入4-流形中的表面是感兴趣的中心对象。PI将研究的具体问题包括Cappell和Shaneson的子午线秩猜想，Fox和Milnor的切片丝带猜想以及结理论在4维奇异光滑结构检测中的应用。PI将采用最先进的工具，其中一些她以前曾帮助开发和应用。这些工具包括：结群的Coxeter数;结图的Wirtinger数; 4-流形的三等分; PI和合作者从这一背景下提取的4维奇异分支覆盖和带状障碍。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。

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.