Contact topology and automorphisms of surfaces

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Fundamental work of Giroux established a one-to-one correspondence between contact structures on closed three-manifolds and automorphisms of surfaces up to stabilization via compatible open book decompositions. It is this correspondence between two classically studied, fundamentally important, objects that the PIs propose to study. Since positive stabilization of surface automorphisms generates an almost intractable equivalence relation, it is important to discern the properties of a contact structure from just a single representative automorphism. An example of such a result is the fact that automorphisms which are compositions of positive Dehn twists induce contact structures that are Stein fillable, i.e. that arise as natural boundaries of Stein manifolds. We are working to understand questions like: what property of an automorphism guarantees symplectic fillability, what property implies existence of Giroux torsion. Investigating these questions will have applications to the study of contact invariants in Heegaard-Floer homology theory in both the bounded and unbounded cases.Contact topology or geometry and its even dimensional counterparts, symplectic topology or geometry, were born out of the study of questions arising in classical mechanics and thermodynamics. Three-dimensional manifolds are mathematical objects modeled on the space in which we live. Contact structures on such spaces arise naturally in the study of fluid flows as the family of planes perpendicular to the flow. A familiar example of a contact structure occurs in the design of DLP front projection televisions where they dictate the use of literally millions of tiny mirrors rather than one large curved mirror. Considerable progress has been made in the last several decades on the three-dimensional contact topology and four-dimensional symplectic topology. Recent progress has allowed researchers to apply two-dimensional techniques to the inherently three-dimensional study of contact topology.