Tight contact structures are closely related to the topology of the underlying 3-manifold. They provide bounds on the genus of the surface representing a homology class, are related to taut foliations on 3-manifolds and to the topology of symplectic 4-manifolds, to Seiberg-Witten theory and Floer homology. Matic proposes to further analyze contact manifolds using cut-and-paste techniques developed in collaboration with Honda and Kazez. The main part of cut-and-paste contact topology are gluing theorems. Matic proposes to keep studying the gluing techniques and to apply them to various open questions, like the existence of tight contact structures on Haken homology spheres. Here the techniques from foliation theory fall short. There are in fact recent examples of Haken homology spheres that do not carry taut foliations. As a result of her investigations she hopes to be able to better understand a basic question: what information about the topology of the 3-manifold can we get from existence of a tight contact structure. Three-dimensional manifolds are modeled on the three dimensional space we live in. Questions in symplectic and contact geometry were originally motivated by dynamical questions in physics and contact structures arise naturally in hydrodynamics. The proposed research has the potential to produce new results applicable to these questions.
