Contact topology of 3-manifolds

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Tight contact structures are closely related to the topology of theunderlying 3-manifold. They provide bounds on the genus of the surfacerepresenting a homology class, are related to taut foliations on3-manifolds and to the topology of symplectic 4-manifolds, toSeiberg-Witten theory and Floer homology. Matic proposes to furtheranalyze contact manifolds using cut-and-paste techniques developed incollaboration with Honda and Kazez. The main part of cut-and-pastecontact topology are gluing theorems. Matic proposes to keep studyingthe gluing techniques and to apply them to various open questions, likethe existence of tight contact structures on Haken homology spheres.Here the techniques from foliation theory fall short. There are infact recent examples of Haken homology spheres that do not carry tautfoliations. As a result of her investigations she hopes to be able tobetter understand a basic question: what information about the topologyof the 3-manifold can we get from existence of a tight contactstructure. Three-dimensional manifolds are modeled on the three dimensional spacewe live in. Questions in symplectic and contact geometry wereoriginally motivated by dynamical questions in physics and contactstructures arise naturally in hydrodynamics. The proposed research hasthe potential to produce new results applicable to these questions.