Interactions of 3- and 4-Dimensional Topology

拓扑学是对抽象空间的研究，这些空间的性质通过弯曲、拉伸和扭曲等变形而不变，但不会破裂，而低维拓扑涉及局部看起来像2维、3维或4维欧几里德空间的空间(称为流形)。低维拓扑在现实世界中有着广泛的应用，包括与理论物理和量子计算的深层联系。此外，拓扑学还应用于数据科学；研究人员了解如何将大型数据集解释为拓扑空间，以便通过研究数据的拓扑形状来获得洞察力。虽然我们的世界本质上是三维的，但一个三维物体在一段时间内的演变产生了一个四维空间，在这个空间中，时间构成了第四个自由度。在过去的几十年里，第三维的拓扑学经历了爆炸性的活动，在2003年格里戈里·佩雷尔曼证明了庞加莱猜想，这是克莱数学研究所提出的七个千年问题中唯一得到解决的问题，达到了顶峰。相比之下，四维流形的拓扑学已经成为一个日益活跃的研究领域，许多基本问题仍然悬而未决。这项研究项目涉及从第三维度的想法来发现解决这些四维问题的新方法。该奖项为研究生的研究提供资金支持。该奖项还将支持大平原联盟，这是一个由PI发起的计划，旨在将研究生与附近机构的演讲机会联系起来。4-流形拓扑中的两个基本公开问题是光滑4维Poincaré猜想和切片带状猜想，前者断言与标准4-球等价的每个闭合4-流形同伦都与标准4-球同构，后者假设在4-球中限制嵌入圆盘的任何结点也限制在3-球面中浸没的带状圆盘。这个项目描述了从纽结理论、3-流形和光滑4-流形拓扑中交织思想的一组不同的问题。PI在这一领域的工作包括两个主要的合作贡献，证明了一大族同伦的4-球面与标准光滑的4-球面是微分同胚的，并发展了三维结构与4-流形剖分之间的深远联系。当前的工作主要涉及四个问题。首先，PI将表征满足介于被切片和被带状之间的中间条件的纽结族。其次，PI将寻求一种新的方法来解决Powell猜想，关于生成3-球面的Heegaard曲面不变的微分同构群。第三，PI将使用桥剖分来处理与表示光滑嵌入的2-球面的4-流形的第二同调类有关的Kirby问题，最后，PI将把3-流形的Heegaard分解的思想改编为4-流形的三分割法，以便更好地理解奇异4-流形的句柄分解。这项工作的基本主题是一种同时整合三维和四维技术的方法，以获得对长期悬而未决的问题的新见解。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。

Topology is the study of abstract spaces with properties that are unchanged by deformations such as bending, stretching, and twisting, but not breaking, and low-dimensional topology involves spaces (called manifolds) that locally look like 2-, 3-, or 4-dimensional Euclidean space. Low-dimensional topology has a wide range of real-world applications, including deep connections to theoretical physics and quantum computing. In addition, topology has applications to data science; researchers understand how to interpret a large data set as a topological space in order to gain insights by studying the topological shape of that data. While our world is 3-dimensional in nature, the evolution of a 3-dimensional object over a period of time gives rise to a 4-dimensional space, in which time constitutes a fourth degree of freedom. Topology in dimension three has seen an explosion of activity over the last several decades, reaching a peak with Grigory Perelman's proof of the Poincaré conjecture in 2003, the only resolved problem of the seven Millennium Problems posed by the Clay Mathematics Institute. In contrast, the topology of 4-dimensional manifolds has become an increasingly active area of research, with many fundamental problems still open. This research project involves adapting ideas from dimension three to discover novel approaches to these 4-dimensional problems. The award provides funds to support research by graduate students. The award will also support the Great Plains Alliance, a program initiated by the PI to connect graduate students with speaking opportunities at nearby institutions.Two foundational open problems in 4-manifold topology are the smooth 4-dimensional Poincaré Conjecture, which asserts that every closed 4-manifold homotopy equivalent to the standard 4-sphere is diffeomorphic to the standard 4-sphere, and the slice-ribbon conjecture, which posits that any knot bounding an embedded disk in the 4-ball also bounds an immersed ribbon disk in the 3-sphere. This project describes a varied set of problems interweaving ideas from knot theory, 3-manifolds, and smooth 4-manifold topology. The PI’s prior work includes two major collaborative contributions in this area, demonstrating that a large family of homotopy 4-spheres is diffeomorphic to the standard smooth 4-sphere, and developing far-reaching connections between 3-dimensional structures and 4-manifold trisections. The current work involves four main problems. First, the PI will characterize a family of knots satisfying an intermediate condition between being slice and being ribbon. Second, the PI will pursue a new approach to the Powell conjecture, concerning the generation of the group of diffeomorphisms of the 3-sphere leaving a Heegaard surface invariant. Third, the PI will use bridge trisections to address a Kirby problem related to representing second homology classes of 4-manifolds with smoothly embedded 2-spheres, and finally, the PI will adapt ideas from Heegaard splittings of 3-manifolds to trisections of 4-manifolds in order to better understand handle decompositions of exotic 4-manifolds. The underlying theme of this work is an approach that simultaneously integrates 3- and 4-dimensional techniques in order to obtain new insights into long-standing open problems.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.