Knots, Disks, and Exotic Phenomena in Dimension 4

流形是看起来局部简单但可能具有有趣的全局形状和性质的对象。拓扑学领域的一个指导性问题是对固定维度上所有可能的流形进行分类，直到给定的等价概念。长期以来，悬而未决的问题涉及到4维奇异对的检测。这些流形是同胚的，但不是微分同胚的，这意味着它们只通过一个非常精细的等价概念来区分。技术可以追溯到Fox和Milnor，最近由Marengon、Manolescu和PicCirillo扩展，证明了可以通过使用结或纠缠圆的某些属性来识别奇异对，这些结或纠缠圆在移除开放球后位于4-流形的边界中。PI将探索这一范例，它有可能导致我们区分光滑4维流形的能力发生根本转变。另一组目标涉及研究纽结本身的性质，其中某些老问题可以用PI与Blair和其他人一起引入的组合工具来攻击。总体而言，这些方法的发展可以对拓扑学领域及其所提供的应用科学产生持久的影响，如信号处理、数据科学和控制理论，在这些领域中，流形以多种形式出现--例如，作为动力系统的状态空间。国际和平协会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问题；她的方法将为研究生和高级本科生提供该领域前沿研究的入口点。该项目概述了通向低维拓扑学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的原则；其中组合方法已被证明是有效的；其中3-球面中的结点和嵌入4-流形中的曲面是一个中心感兴趣的对象。PI将研究的具体问题包括Cappell和Shaneson的子午线等级猜想，Fox和Milnor的切片带状猜想，以及纽结理论在检测4维奇异光滑结构中的应用。PI将使用最先进的工具，她之前帮助开发和应用了其中的几个工具。这些工具包括：结点群的Coxeter商；结点图的Wirtinger数；4-流形的三分点；维度4中的单一分支覆盖和PI及其合作者从本文中提取的带状障碍。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。

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.