Conference: Georgia Topology Conference

该奖项为每年5月下旬在乔治亚州雅典乔治亚大学举行的下两届乔治亚拓扑会议提供参与者支持。自1961年第一届佐治亚拓扑会议召开以来，一年一度的佐治亚拓扑会议一直是拓扑学界的重要活动。2023年会议的重点将是研究三维和四维的微分同态、复形态和接触形态的空间。2024年的版本将专注于光滑的四维空间中的表面。在这两种情况下，我们感兴趣的是研究空间的性质，这些空间局部看起来像我们生活的空间或时空，我们可以将微积分工具与组合和图解工具结合起来。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）在空间中的位置来研究这些空间。这些都是热门话题，最近已经看到了一些戏剧性的结果，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维的微同态、复形态和接触形态空间，将由共同项目负责人David Gay、Gordana Matic、Akram Alishahi和Michael Usher共同组织，UGA博士后Eduardo Fernandez Fuertes、Feride Ceren Kose和Lev tovstopyat nelip的帮助下。在四维拓扑学中，很多工作都集中在分类和区分对象上，即4流形，但同样重要的一个项目是分类和区分4流形之间的态射，即4流形之间的微分同态。为了说明我们对光滑环境知之甚少，直到最近，我们还不知道边界上的恒等四球的微分同态群是否可缩。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小猜想），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是偶路径连接的。考虑到4维辛结构的重要性，将其与Gromov关于4球辛形态空间可收缩的结果以及3维接触形态的类似结果进行比较是很有趣的。2024年的会议将重点关注嵌入在4流形中的表面的光滑拓扑，作为对一般光滑四维拓扑的探索。存在着许多基本开放问题，如补具有循环基本群界的4球中的光滑嵌入2球是否为光滑嵌入3球。与此同时，最近有了戏剧性的发展，比如Gabai对四维灯泡定理的证明，在某些情况下，在双球存在的情况下，将光滑的2球完全分类为光滑的同位素。这是一个非常活跃的研究领域，结合了规范理论、Khovanov同调、高维莫尔斯理论和显式四维可视化的工具。更多信息可以在会议网站上找到：https://topology.franklinresearch.uga.edu/georgia-topology-conferencesThis该奖项反映了美国国家科学基金会的法定使命，并通过基金会的智力价值和更广泛的影响审查标准进行评估，认为值得支持。

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.