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学2018会议的美国与会者提供差旅支持。会议的重点是规范理论和Floer同调在三维和四维拓扑、协调和接触拓扑中的最新应用。它包括12个研究讲座，以及一个问题会议和一个关于美国和欧洲数学研究生项目和职业的小组讨论。这一主题，即三维和四维空间的大规模几何，是纯数学基础研究的一个关键领域，具有高度国际化的研究界。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，其中大多数可以在arxiv.org上免费获得。这样的会议促进了不同地理位置的研究小组之间的面对面讨论和合作，而美国的参与对于保持该领域的领导地位至关重要。这次会议也是一个机会，可以确定并可能从其他国家招募顶尖的年轻研究人员。在现代数学中，Floer理论不变量、协调性、光滑四维拓扑和接触拓扑学等主题已经密不可分地交织在一起。这些领域继续取得戏剧性进展。最近取得进展的一个领域，源于百年三角剖分猜想的反证，是使用等变Floer理论来研究调和和同调协边中的问题。这些技术使得关于同源共生的问题变得容易理解，而这些问题在十年前似乎遥不可及。会议关注的另一个活跃领域是Floer同调的拓扑意义。与协调相关的是光滑的、闭的4-流形的拓扑，其中最近的进展包括漂亮的新障碍和构造。在一个相关的方向上，但在接触类别中，过去五年通过Floer理论、接触拓扑学和开卷在理解Legendrian纽结的拉格朗日填充方面也取得了突破。会议将提出这些领域的新成果和解决这些问题的创新思路。会议网站是http://math.ipm.ir/gt/TT2018.This奖，反映了国家科学基金会的法定使命，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。

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.