Conference: Georgia Topology Conference

University Of Georgia Research Foundation Inc, Athens GA

Investigators

Abstract

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-conferences This 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.