Gauge Theory, Floer Homology, and Topology

该奖项为参加2018年6月26日至28日在伊朗德黑兰基础科学研究所举行的“德黑兰拓扑2018”会议的美国参与者提供旅行支持。会议重点讨论了规范论和Floer同调在三维和四维拓扑、协调和接触拓扑中的最新应用。它的特色是12个研究讲座，以及一个关于美国和欧洲数学研究生课程和职业的问题会议和小组讨论。该主题，即三维和四维空间的大尺度几何，是纯数学基础研究的关键领域，具有高度国际化的研究社区。该领域所有已完成的研究都可以通过同行评议的期刊公开获得，而且大多数都可以在arXiv.org上免费获得。诸如此类的会议促进了地理位置不同的研究小组之间的面对面讨论和合作，美国的参与对于保持该领域的领导地位至关重要。该会议还提供了一个机会，以确定并潜在地招募来自其他国家的顶尖年轻研究人员。花理论不变量、协调、光滑四维拓扑和接触拓扑等主题在现代数学中已经成为不可分割地纠缠在一起的问题。在这些领域继续取得显著进展。最近取得进展的一个领域，源于对已有百年历史的三角测量猜想的反驳，是使用等变弗洛尔理论来研究协调和同调配合问题。这些技术使十年前似乎遥不可及的关于同源配位的问题变得容易理解。会议针对的另一个活跃领域是花同源性的拓扑意义。与和谐相关的是光滑闭合4流形的拓扑结构，其中最近的进展包括美丽的新障碍物和结构。在一个相关的方向，但在接触范畴，在过去的五年里，通过Floer理论，接触拓扑和打开的书籍，我们也看到了对Legendrian结的拉格朗日填充的突破。会议将展示这些领域的新成果和解决这些问题的创新思路。该奖项反映了NSF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，认为值得支持。

This award provides travel support to U.S. participants in the conference Tehran Topology 2018, held at the Institute for Research in Fundamental Sciences in Tehran, Iran, on June 26-28, 2018. The conference focuses on recent applications of gauge theory and Floer homology to three- and four-dimensional topology, concordance, and contact topology. It features twelve research lectures, as well as a problem session and a panel discussion on graduate programs and careers in mathematics in the U.S. and Europe. The topic, namely, the large-scale geometry of three- and four-dimensional spaces, is a key area of basic research in pure mathematics, with a highly international research community. All completed research in the area is publicly available through peer-reviewed journals, and most is freely available world-wide on arXiv.org. Conferences such as this facilitate in-person discussions and collaborations between geographically disparate research groups, and U.S. participation is essential to maintain leadership in the field. The meeting also serves as an opportunity to identify and potentially recruit top young researchers from other nations. The topics of Floer-theoretic invariants, concordance, smooth four-dimensional topology, and contact topology have become inextricably entwined in modern mathematics. There continues to be dramatic progress in these areas. One area of recent progress, stemming from the disproof of the hundred-year-old Triangulation Conjecture, is using equivariant Floer theory to study problems in concordance and homology cobordism. These techniques have made accessible questions about homology cobordism that seemed far out of reach ten years ago. Another active area, targeted by the conference, is the topological meaning of Floer homology. Related to concordance is the topology of smooth, closed 4-manifolds, in which recent progress includes both beautiful new obstructions and constructions. In a related direction but in the contact category, the last five years have also seen breakthroughs in understanding Lagrangian fillings of Legendrian knots, via Floer theory, contact topology, and open books. The conference will present new results in these areas and innovative ideas for approaching these problems. The conference website is http://math.ipm.ir/gt/TT2018.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.