Gauge Theory, Floer Homology, and Topology

规范理论、弗洛尔同调和拓扑

• 批准号: 1830070
• 负责人: Robert Lipshitz
• 金额: $ 0.6万
• 依托单位: University of Oregon Eugene
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-07-15 至 2019-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1830070&HistoricalAwards=false
• 关键词: Gauge; Theory; Floer; Homology; Topology

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.

该奖项为参加2018年德黑兰拓扑会议的美国与会者提供旅行支持，该会议于2018年6月26日至28日在伊朗德黑兰的基础科学研究所举行。会议的重点是最近的应用规范理论和弗洛尔同源三维和四维拓扑结构，和谐，接触拓扑结构。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课程和职业的小组讨论。该主题，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，具有高度的国际研究界。该领域所有已完成的研究都可通过同行评审期刊公开获得，大多数都可在arXiv.org上免费获得。此类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。这次会议也是一个机会，以确定和潜在的招募来自其他国家的顶级年轻研究人员。弗洛尔理论不变量、和谐性、光滑四维拓扑和接触拓扑等主题在现代数学中已密不可分。在这些领域继续取得巨大进展。最近的一个进展领域，源于百年历史的三角剖分猜想的反证，是使用等变弗洛尔理论来研究协调和同调配边问题。这些技术使得关于同源配边的问题变得容易理解，而这在十年前似乎是遥不可及的。另一个活跃的领域，会议的目标，是拓扑意义的弗洛尔同源性。与和谐性相关的是光滑闭四维流形的拓扑，最近的进展包括美丽的新障碍和建设。在一个相关的方向，但在接触类别，过去五年也看到了突破，在理解拉格朗日填充勒让德结，通过弗洛尔理论，接触拓扑学，和开放的书籍。会议将提出这些领域的新成果和解决这些问题的创新想法。会议网站是http://math.ipm.ir/gt/TT2018.This奖反映了NSF的法定使命，并已被认为值得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的支持。