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5月下旬在乔治亚州雅典举行的接下来的两个佐治亚州拓扑会议提供了参与者的支持。自1961年第一次会议举行以来，年度佐治亚州拓扑会议一直是拓扑界的重要事件。2023年会议的重点将是对三分之二和第四个维度的差异，同构和接触型的空间的研究。 2024年版将重点放在平滑4维空间中的表面上。在这两种设置中，我们都有兴趣研究当地看起来像我们居住的空间或时空的空间的特性，并且可以将微积分工具与组合和图形工具相结合。在第一种情况下，我们通过思考它们的对称性来研究这些空间，在第二种情况下，我们通过思考更简单的物体（表面）如何位于空间内的方式来研究这些空间。 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和Michael Usher在UGA Postdocs Eduardo Fernandez Fuertes，Feride Ceren Kose和Lev Tovstopyat-Nelip的帮助下。 4维拓扑的许多工作都集中在分类和区分对象，即4个manifolds，但同样重要的项目是分类和区分{形态学，即4个manifolds之间的差异性。为了说明在平稳的环境中我们知道的知识很少，直到最近，我们才知道4球的差异群是否是边界上的身份。在戏剧性的发展中，渡边在2018年表明，通过表明某些同型组是非平凡的（因此反驳了光滑的4维smale猜想），这一组不符合缩度，但我们仍然不知道该组是否甚至连接了路径。考虑到维度4中符号结构的重要性，有趣的是将其与Gromov的结果进行比较，即4球的符号切除型空间是可违约的，以及在维度3中的接触符号的相似结果。2024会议将集中在4-元素中的4-Manifolds As Possology in 4-nifimens中的表面拓扑的集中在4-Dimens中。存在许多基本的开放问题，例如，在4个球员中是否具有循环基本组的四个球员中平滑嵌入的2个球体的问题是否平滑地嵌入了3球。同时，最近发生了戏剧性的发展，例如Gabai证明了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，完全将平滑的2速度分类为平滑的同位素。这是一个非常活跃的研究领域，其贡献结合了量规理论，霍瓦诺夫同源性，更高维摩尔斯理论和显式4维可视化的工具。可以在会议网站上找到更多信息：https：//topology.franklinresearch.uga.edu/georgia-topology-conferencesthis奖，反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛影响的审查标准来通过评估来支持的。