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.

该奖项提供了参与者支持下两个格鲁吉亚拓扑会议，在雅典举行，佐治亚州每年5月下旬在格鲁吉亚大学。一年一度的格鲁吉亚拓扑会议一直是一个重要的事件拓扑社会以来，第一次这样的会议于1961年举行。2023年会议的重点将是研究三维和四维空间中的单同态、辛同态和接触同态。2024年版将专注于光滑四维空间中的表面。在这两种情况下，我们感兴趣的是研究空间的性质，这些空间在局部上看起来像我们所生活的空间或时空，并且我们可以将微积分工具与组合和图解工具联合收割机结合起来。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）如何位于空间内部来研究这些空间。这些都是热门话题，最近已经取得了一些引人注目的成果，会议的目的是将先进和初级研究人员聚集在一起，了解最近成果的细节，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年会议将重点关注微分同胚空间，symplectomorphisms和contactomorphisms在第3和第4维度，并将共同主办的共同PI大卫盖伊，Gordana马蒂奇，阿克兰Alishahi和迈克尔亚瑟，从UGA博士后爱德华多费尔南德斯Fuertes，费里德Ceren Kose和列夫Tovstopyat-Nelip的帮助。四维拓扑学中的许多工作都集中在对四维流形的分类和区分上，但同样重要的是对四维流形之间的{态射，即双态射进行分类和区分。为了说明我们在光滑环境中所知甚少，直到最近我们还不知道作为边界上的单位元的4球的微分同胚群是否是可收缩的。在一个戏剧性的发展中，Watanabe在2018年通过证明某些同伦群是非平凡的（从而反驳了光滑的四维Smale猜想）证明了这个群是不可收缩的，但我们仍然不知道这个群是否甚至是路径连通的。考虑到4维辛结构的重要性，有趣的是将其与格罗莫夫的结果进行比较，格罗莫夫的结果是4-球的辛同构空间是可收缩的，沿着的是3维接触同构的类似结果。2024年的会议将专注于嵌入在4-流形中的曲面的光滑拓扑，作为一般光滑四维拓扑的探索。许多基本的开放问题存在，如问题的顺利嵌入2球的4球的补有循环的基本群边界的顺利嵌入3球。与此同时，最近有戏剧性的发展，如加白的证明4维灯泡定理，在某些情况下完全分类顺利2球顺利合痕的存在下，双球。这是一个非常活跃的研究领域，结合了规范理论，Khovanov同调，高维莫尔斯理论和明确的4维可视化工具。更多信息可以在会议网站上找到：https://topology.franklinresearch.uga.edu/georgia-topology-conferencesThis奖项反映了NSF的法定使命，并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来支持。