Heegaard Floer homology is a collection of invariants of knots, 3-manifolds, and smooth 4-manifolds, defined using symplectic geometry. It has changed the landscape of low-dimensional topology over the last decade, and has, for example, been instrumental in many of the recent advances in contact geometry. This project is devoted to better understanding the internal workings of this theory and its relationships to other link homology theories like Khovanov and RozanskyÂ¢s categorification of the HOMFLY polynomial. One of the PIÂ¢s broad goals is to use these relationships to develop more computable invariants of knots, 3-manifolds, and contact structures. Another goal involves using Heegaard Floer homology to better understand the correspondence between topological features of open books and geometric properties of contact structures. Specific aims include finding obstructions to a contact structure having support genus one, determining whether Legendrian surgery preserves tightness, understanding the relationship between fractional Dehn twist coefficient and fillability, and developing a combinatorial spanning tree model for knot Floer homology. The PI plans to use the latter to find an axiomatic description of knot Floer homology, in part to show that it agrees with the monopole and instanton knot homologies defined by Kronheimer and Mrowka using gauge theory.
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 some of the theories the PI intends to study have applications and origins in physics. Contact structures, in particular, can serve as a probe with which to study these spaces, and are also important in classical mechanics, thermodynamics, dynamical systems, and in the study of liquid crystals. Another goal of the PIÂ¢s proposal is to develop new methods for understanding knots. Knots arise naturally in molecules like DNA. Recent work, for example, has applied Floer homology to determine how various enzymes in the body alter the knottedness of DNA. Understanding these mechanisms is important for developing certain drugs.
The PI's project involved studying geometric structures on 3- and 4-dimensional shapes called "manifolds" as well as "knots" in such manifolds. Briefly, a manifold is a shape that looks "locally" like flat Euclidean space. For example, the surface of the earth---a sphere---is considered to be a 2-dimensional manifold, or "2-manifold", because it looks locally like a flat plane. It isn't until we zoom out that we discover that the Earth's surface has an interesting global structure. One can think of a knot in a 3-manifold as an embedded piece of string in the manifold with no loose ends---tie your shoelace into a knot and then glue the ends together. Knot theory has found many recent applications to the study of DNA knotting and how certain enzymes change the knottedness of DNA strands. This may be important in developing certain drugs. Understanding knots, manifolds, and geometric structures on them is also central to understanding the shape of our macroscopic universe. For example, the language of manifolds and "differential geometry" is central to Einstein's General Theory of Relativity, which describes how gravity works and without which, our GPS's would not function properly. The PI's work is concerned with understanding these objects using tools which fall broadly under the umbrella of "Floer theories". Many of these tools have applications and origins in physics. Among the geometric objects the PI studied in this project are "contact structures" on 3-manifolds. In addition to their intrinsic interest and use in understanding 3- and 4-dimensional shapes and knots, contact structures are important in classical mechanics, thermodynamics, dynamical systems, and in the study of liquid crystals. The PI developed several novel Floer theoretic "invariants" for studying contact structures and knots in his proposal. These invariants have led to new results and opened new lines of inquiry with the promise to unite different subjects within "topology", the PI's main field. For example, his "instanton Floer" invariant of contact manifolds suggests hitherto unexplored connections between the contact geometry of a 3-manifold and the "fundamental group" of the 3-manifold, a more classical object defined in the 19th century. (The reader may remember the story of Perelman, who solved the famous Poincare Conjecture a few years ago (but refused to accept the Fields Medal). This conjecture was ultimately about the fundamental group of 3-manifolds.) The PI in addition developed "combinatorial" means for understanding some previously existing invariants of knots and manifolds, which are "easy" to define but exceedingly difficult to compute in practice (both in principle and in practice); see his "spanning tree model for knot Floer homology" and work on tangle invariants. One of the PI's main goals is to provide a unifying explanation for many of the connections and patterns we see from studying the tools themselves that we use for understanding manifolds. These connections are important as they indicate deep connections between the disparate areas of mathematics that these various tools come from. The PI's work on invariants of "bordered manifolds" is an approach to understanding these connections and providing a unified theory. As part of this project, the PI has mentored 2 graduate students, Adam Saltz and Diana Hubbard at Boston College. They are working on thesis topics related to the project. He in addition led summer projects in 2011 for Jack Sempliner, a high school student, and two Princeton undergrads, Daniel Kriz and Mike Wong, who have gone on to study math in grad school. In the Summer of 2014, he mentored BC undergrads Andrew Ferdowsian, Cynthia Chen, and Champ Davis on a computational project related to the themes above. He has traveled extensively in the United States to spread his ideas to other mathematical audiences, in conference and seminar talks, and has additional co-organized 3 conferences with the purpose of bringing together both graduate students and more established researchers around related problems (2013 Special Session of the AMS at Boston College; 2014 ICERM graduate workshop; 2014 William Rowan Hamilton Conference).