Contact structures play a central role in 3- and 4-dimensional topology, in no small part due to its relationship with the various Floer homology theories - contact homology, Heegaard Floer homology, embedded contact homology, and Seiberg-Witten Floer homology - all of which have had spectacular applications over the last decade. The main goal of the PI's research program is to understand what happens to these Floer homology theories when we glue contact 3-manifolds with boundary. The cut-and-paste theory of contact structures in dimension three - developed by numerous authors including the PI - is expected to play an important role. As part of the program, the PI plans to study a certain category, called the contact category and constructed from contact structures, which is the "categorical glue" for gluing contact manifolds with boundary. The PI, in joint work with Colin and Ghiggini, also plans to establish the equivalence of two of the Floer homology theories - Heegaard Floer homology and embedded contact homology - using the framework of open book decompositions due to Giroux; this can be interpreted as a gluing result for relative embedded contact homology.
The PI proposes a study of 3- and 4-dimensional spaces using a probe called a contact structure. The 3- and 4-dimensional spaces we study will locally be similar to the standard (Euclidean) 3- and 4-dimensional spaces. These objects may be very complicated globally, but a local observer cannot tell the difference, just as an ant cannot tell whether it is sitting on a flat plane or a very large sphere. A more thorough mathematical understanding of contact structures should contribute significantly towards our understanding of 3- and 4-dimensional shapes, from the knotting of DNA at the microscopic level to the shape of the universe at the macroscopic level.
