Conference: Georgia Topology Conference

该奖项为每年5月下旬在佐治亚州雅典乔治亚大学举行的下两届佐治亚拓扑学会议提供参与者支持。一年一度的佐治亚州拓扑学会议自1961年第一次举行以来，一直是拓扑界的一项重要活动。2023年会议的焦点将是研究三维和四维的微分同胚、辛同态射和接触同态空间。2024年版将重点放在光滑的4维空间中的表面。在这两种情况下，我们都有兴趣研究局部看起来像我们生活的空间或时空的空间的性质，在这种空间中，我们可以将微积分的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对象(曲面)如何位于空间中来研究这些空间。这些都是最近取得了一些戏剧性成果的热门话题，会议的目的是将先进的和初级的研究人员聚集在一起，了解最新结果的细节，了解下一个需要解决的问题，并启动合作来解决这些问题。2023年的会议将集中在3维和4维的微分同胚、辛同构和接触同构的空间，将由共同PI David Gay，Gordana马季奇，Akram Alishahi和Michael Usher共同组织，UGA博士后Eduardo Fernandez Fuertes，Feride Ceren Kose和Lev Tovstopyat-Nelip帮助。4维拓扑的许多工作都集中在分类和区分对象，即4-流形，但同样重要的是分类和区分{态射，即4-流形之间的微分同胚。为了说明我们在光滑环境中所知的是多么的少，直到最近，我们还不知道作为边界上的恒等式的4球的微分同胚群是否可压缩。在一个戏剧性的发展中，Watanabe在2018年证明了这个群是不可压缩的，他证明了某些同伦群是非平凡的(从而反驳了光滑的四维斯梅尔猜想)，但我们仍然不知道这个群是否甚至是路连通的。考虑到4维辛结构的重要性，将这与Gromov关于4球的辛同构空间是可压缩的结果以及3维接触同构的类似结果进行比较是很有趣的。2024年的会议将集中于嵌入4维流形的曲面的光滑拓扑，作为对一般光滑4维拓扑的探索。存在着许多基本的公开问题，例如4-球面中的一个光滑嵌入的2-球面是否有一个光滑嵌入的3-球的问题。与此同时，最近也有了戏剧性的发展，例如Gabai对4维灯泡定理的证明，在某些情况下，在存在对偶球的情况下，光滑的2-球完全分类为光滑的同伦。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Morse理论和显式四维可视化的工具。更多信息可在会议网站上找到：https://topology.franklinresearch.uga.edu/georgia-topology-conferencesThis奖反映了美国国家科学基金会的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.