Conference: Georgia Topology Conference

会议：乔治亚州拓扑会议

• 批准号: 2301632
• 负责人: Gordana Matic
• 金额: $ 3.94万
• 依托单位: University of Georgia Research Foundation Inc
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-05-01 至 2025-04-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2301632&HistoricalAwards=false
• 关键词: Conference; Georgia; Topology

The award provides participant support for the next two Georgia Topology Conferences, held in late May each year in Athens, GA at the University of Georgia. The annual Georgia Topology Conference has been an important event for the topological community ever since the first such conference was held in 1961. The focus of the 2023 conference will be the study of spaces of diffeomorphisms, symplectomorphisms and contactomorphisms in dimensions three and four. The 2024 edition will focus on surfaces in smooth 4-dimensional space. In both settings, we are interested in studying the properties of spaces which locally look like the space, or space-time, that we live in, and in which we can combine the tools of calculus with combinatorial and diagrammatic tools. In the first case, we study these spaces by thinking about their symmetries, and in the second case we study these spaces by thinking about how simpler objects (surfaces) sit inside the spaces. These are both hot topics that have seen some dramatic recent results and the purpose of the conferences is to bring advanced and beginning researchers together to learn about the details of recent results, to understand the next questions that need to be solved, and to kick start collaborations to address these questions.The 2023 conference will focus on spaces of diffeomorphisms, symplectomorphisms and contactomorphisms in dimensions 3 and 4, and will be co-organized by co-PIs David Gay, Gordana Matic, Akram Alishahi and Michael Usher, with help from UGA postdocs Eduardo Fernandez Fuertes, Feride Ceren Kose and Lev Tovstopyat-Nelip. Much work in 4-dimensional topology has focused on classifying and distinguishing the objects, namely 4-manifolds, but an equally important project is to classify and distinguish the {morphisms, namely diffeomorphisms between 4-manifolds. To illustrate how little we know in the smooth setting, until very recently we had no idea whether the group of diffeomorphisms of the 4-ball which are the identity on the boundary was contractible. In a dramatic development, Watanabe showed in 2018 that this group is not contractible by showing that certain homotopy groups were nontrivial (thus disproving the smooth 4-dimensional Smale conjecture) but we still do not know if this group is even path connected. Given the importance of symplectic structures in dimension 4, it is interesting to compare this to Gromov's result that the space of symplectomorphisms of the 4-ball is contractible, along with similar results for contactomorphisms in dimension 3. The 2024 conference will focus on the smooth topology of surfaces embedded in 4-manifolds as a probe into smooth 4-dimensional topology in general. Numerous foundational open problems exist, such as the question of whether a smoothly embedded 2-sphere in the 4-sphere whose complement has cyclic fundamental group bounds a smoothly embedded 3-ball. At the same time there have been dramatic developments recently, such as Gabai's proof of the 4-dimensional lightbulb theorem, that in certain situations completely classifies smooth 2-spheres up to smooth isotopy in the presence of dual spheres. This is a very active area of study with contributions combining tools from gauge theory, Khovanov homology, higher dimensional Morse theory and explicit 4-dimensional visualization. More information can be found on the conference website: https://topology.franklinresearch.uga.edu/georgia-topology-conferencesThis 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.

该奖项为每年五月下旬在佐治亚州雅典市佐治亚大学举行的接下来两届佐治亚拓扑会议提供参与者支持。自 1961 年首次召开以来，一年一度的乔治亚拓扑会议一直是拓扑学界的重要活动。2023 年会议的重点将是第三维和第四维微分同胚、辛同胚和接触同胚的空间研究。 2024 年版本将重点关注光滑 4 维空间中的表面。在这两种情况下，我们都有兴趣研究空间的属性，这些空间在局部看起来像我们居住的空间或时空，并且我们可以将微积分工具与组合和图表工具结合起来。在第一种情况下，我们通过考虑它们的对称性来研究这些空间，在第二种情况下，我们通过考虑更简单的物体（表面）如何位于空间内来研究这些空间。这些都是热门话题，最近已经取得了一些引人注目的成果，会议的目的是将高级和初级研究人员聚集在一起，了解最近结果的细节，了解接下来需要解决的问题，并启动合作来解决这些问题。2023 年的会议将重点关注 3 维和 4 维的微分同胚、辛同胚和接触同胚空间，并将由联合 PI David Gay、Gordana 共同组织Matic、Akram Alishahi 和 Michael Usher，在 UGA 博士后 Eduardo Fernandez Fuertes、Feride Ceren Kose 和 Lev Tovstopyat-Nelip 的帮助下。 4 维拓扑中的许多工作都集中在对象（即 4 流形）的分类和区分上，但同样重要的项目是对态射（即 4 流形之间的微分同胚）进行分类和区分。为了说明我们对光滑环境知之甚少，直到最近我们还不知道作为边界上恒等式的 4 球微分同胚群是否是可收缩的。渡边在 2018 年取得了戏剧性的进展，通过证明某些同伦群是非平凡的（从而反驳了光滑的 4 维 Smale 猜想），证明了该群是不可收缩的，但我们仍然不知道该群是否是偶数路径连通的。鉴于 4 维辛结构的重要性，将其与 Gromov 的结果进行比较是很有趣的，即 4 球的辛同胚空间是可收缩的，以及 3 维接触同胚的类似结果。2024 年的会议将重点关注光滑嵌入 4 流形中的表面拓扑，作为对一般平滑 4 维拓扑的探索。存在许多基本的开放问题，例如在 4 球体中平滑嵌入的 2 球体（其补集具有循环基本群）是否限制平滑嵌入的 3 球的问题。与此同时，最近也出现了戏剧性的发展，例如 Gabai 对 4 维灯泡定理的证明，在某些情况下，在存在双球体的情况下，将光滑的 2 球体完全分类为光滑的同位素。这是一个非常活跃的研究领域，其贡献结合了规范理论、霍瓦诺夫同调、高维莫尔斯理论和显式 4 维可视化的工具。更多信息可在会议网站上找到：https://topology.franklinresearch.uga.edu/georgia-topology-conferences该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查进行评估，被认为值得支持标准。