Topology is often described as the study of “space”, in which we work to understand abstract spaces in which we may not have a notion of distance or angle. Typical applications include approximating a large data set as one connected object or a long molecule as a knotted arc. Many topologists are particularly interested in 4-dimensional space, where many foundational questions remain open. I aim to address some of the most significant open questions in this area: How do we characterize spheres? How many ways are there to subdivide standard real space? When do knotted loops bound simple disks? My recent research has dealt with these questions, with focus on developing explicit geometric constructions to improve our understanding of the forefront of these problems. I aim to extend this approach and push the limits of what is realizable, and thus motivate the application of recent tools (often coming from mathematical physics) that have been used to distinguish topological objects.
Fellow
