The proposed project involves studying geometric structures on 3- and 4-dimensional manifolds. Understanding these spaces and structures is central to understanding the shape of our macroscopic universe, and many of the theories the PI intends to study have applications and origins in physics. A major goal of the PI's proposal is to develop methods which might be used to unify several existing and important tools for studying 3- and 4-dimensional manifolds. Such a unification would likely reveal interesting connections between quite disparate fields of mathematics. The PI also plans to use some of these tools to study contact structures and knots in new ways. Both serve as probes by which to study 3- and 4-dimensional manifolds. Contact structures are also important in classical mechanics, thermodynamics, dynamical systems, and in the study of liquid crystals, while knot theory has found many recent applications in the study of DNA knotting. This proposal also contains several explicit directions for student research, at both undergraduate and graduate levels, and will support the PI's teaching and mentoring activities.
Over the last couple decades, Floer theory has revolutionized the study of the topology and geometry of smooth manifolds in dimensions 3 and 4. There are multiple distinct Floer theories for 3-manifolds defined using techniques from quite different branches of mathematics. Amazingly several of these theories appear to give the same information, indicating deep connections between fields like symplectic geometry and gauge theory. These connections have lead to beautiful results like Taubes' proof of the Weinstein conjecture. Despite remarkable progress in understanding these connections, one wonders whether there is a simpler, unifying explanation. Indeed, a fundamental open problem and a distant lodestar for the PI's proposal is to axiomatize Floer theory. One approach to this and to the related problem of computing Floer theories is to develop invariants of bordered 3-manifolds and a means of computing the invariant of a closed manifold from those of its pieces. The first such invariant was defined by Lipshitz, Ozsvath, and Szabo in Heegaard Floer homology. The PI's proposed bordered monopole Floer theory is novel in that it may provide methods for computing invariants of 4-manifolds as well (it is defined in the HM-from, "master" version, of monopole Floer homology), and can likely be ported to other Floer theories (an important ingredient for an approach to axiomatization). This proposal will support the further development of this theory. A closely related aspect of the proposal involves fleshing out connections between Floer theory and Khovanov homology, an invariant of links motivated by representation theory. Such connections were instrumental, for example, in Kronheimer and Mrowka's celebrated proof that Khovanov homology detects the unknot. Often these connections arise in the form of a spectral sequence from Khovanov homology and Floer thoery. A goal of this project is to better understand these spectral sequences. The PI proposes a means for combinatorially computing their terms and for proving that the sequences are functorial with respect to link cobordisms. The hope is to use these combinatorial methods and functoriality to better understand Floer invariants in dimensions 3 and 4. A final, complementary aspect of this proposal involves defining new invariants of contact structures. The PI has done so in recent work, using instanton and monopole Floer homology. An exciting future project is to use the instanton Floer invariant to uncover hitherto unexplored links between the Stein and symplectic fillings of a contact 3-manifold and its fundamental group.
