Floer homology and monodromy

We inhabit a three-dimensional universe, and so the classification of three-dimensional shapes (or 3-manifolds) has been a major area of research since at least the 19th century. Poincaré famously conjectured in 1904 that the simplest 3-manifold, the three-dimensional sphere, could be identified in purely algebraic terms through an object called the fundamental group. This question motivated nearly a century of progress until its celebrated resolution by Perelman in 2003, as part of his proof of the even more general Geometrisation Conjecture. Even so, many important questions about 3-manifolds remain unanswered to this day. One important approach to the construction and understanding of 3-manifolds is through the use of knotted circles in ordinary three-dimensional space. Knots form crucial building blocks for 3-manifolds, because every 3-manifold can be built from knots by a process called Dehn surgery: we cut several knots out of the three-dimensional sphere and glue them back in by some sort of twist, and different knots and choices of twisting can lead to radically different spaces. Understanding these knots allows us to describe both the geometry of the resulting spaces -- how are they naturally curved? -- as well as their topology, meaning the properties of their shapes that don't change when we bend or stretch them. This means that understanding knots gives us an important window into the world of 3-manifolds. Fundamental questions about these building blocks include the construction of knot invariants, algebraic tools which help us to distinguish different knots; the effectiveness of these invariants at telling knots apart; and discerning what these invariants have to say about the 3-manifolds created by surgery on a given knot. This proposal seeks to answer such questions through the use of Floer homology, a package of powerful invariants of knots and 3-manifolds that traces its origins back to mathematical physics in the 1980s. These invariants are generally hard to work with, because their definitions involve solving partial differential equations, but one incarnation, Heegaard Floer homology, is unusually computable and can in many cases be described purely combinatorially. It comes with an associated knot invariant, called knot Floer homology, that can even be computed quickly by existing software. Due to their accessibility, these invariants have led to an astounding number of breakthroughs in low-dimensional topology since their introduction in the early 2000s. In the proposed work, I plan to extend and make use of a deep relationship between Heegaard Floer homology and periodic Floer homology, an invariant of symmetries of surfaces that arises in a very different way from the world of symplectic geometry. This relationship was established by Lee and Taubes in 2012, but its usefulness in attacking topological questions has only started to become apparent in the last few years, and I will build technical tools that will greatly extend its applicability. I will apply these tools to show that knot Floer homology can positively identify many knots of arbitrary complexity. I will also use them to answer long-open questions about the geometric structures that can be realised by surgeries on knots, and to determine the extent to which surgeries on many knots are unique. Each of these goals will greatly enhance our understanding of both the geometric meaning and the strength of knot Floer homology, and its relationship to fundamental questions about topology in three dimensions.