Symplectic manifolds arise as phase spaces in the Hamiltonian formulation of classical dynamics, and contact manifolds can be viewed as either an extension of such a setting to include how it progresses over time, or a restriction to a system of a fixed energy. Understanding the geometry of symplectic and contact manifolds is thus important to understanding the dynamics of physical systems, and has applications in many diverse fields, such as optics, thermodynamics, and hydrodynamics.
In the modern sense, though, the main motivation for the study of symplectic manifolds derives from their place in the study of smooth 4-manifolds, spaces which generalize the notion of space-time in which we live. Considering such a space along with a symplectic structure has been shown to allow one to define very powerful invariants of the spaces themselves. As they can be extremely complicated, these spaces are often studied by cutting into smaller, less complicated pieces, then gluing these back together. In the symplectic setting, the information contained in this construction is primarily encoded by contact structures, along which the cutting and pasting are performed. The relevant framework here is that of a "topological quantum field theory", a set-up which has proved a hugely powerful idea in many areas of mathematics.
In the 80's and 90's though it became clear that certain contact manifolds exhibit a kind of `flexibility', making them entirely unsuited to their place in this study. This discovery, along with the classification of such structures in 3-dimensions, was one of the major important developments leading to the growth of the field of contact and symplectic topology.
The study of flexibility is very much a topological issue, lending itself to the standard tools of low-dimensional topology, in contrast to the algebraic-geometry-based setup described above. The resulting notions of `flexible' vs `rigid' phenomena have become central to the field. The research outlined in this proposal will focus on the ``gap'' between these two notions, building connections via which the machinery and techniques of each side can be brought to bear on the problems of the other.
A first step concerns an ``algebraisation'' of flexibility. Building on previous work of the principle investigator, we aim to fully characterise the notion in the language of Floer homology, a tool closely related to the above-mentioned geometric study of 4-manifolds, and one which has proven particularly useful in the study of contact 3-manifolds. This is a multi-faceted project which involves a refinement of the so-called ``contact class'' of various flavors of Floer homology, and brings together a wide variety of tools from on the one hand the topological and combinatoric world of decompositions of 3-manifolds and on the other the more rigid and algebraic world of pseudo-holomorphic curves. This project will extend to provide tools for the algorithmic detection of flexibility, as well as defining a way of measuring rigidity.
Along with this, we will explore the relation of this framework to existing notions of `torsion', and generalizations of the framework into higher dimensions, as well as differing geometries (complex vs symplectic). We will also seek to apply the results, as well as previous work, to applications concerning classification of contact 3-manifolds, in particular finding examples of contact 3-manifolds with arbitrary ``support genus''.
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