Pseudoholomorphic curves in low-dimensional topology

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The goal of this project is to develop new topological invariants inthe framework of Floer homology. A major part of the project is todevelop "embedded contact homology", a new invariant of contactthree-manifolds which counts embedded pseudoholomorphic curves in thefour-dimensional symplectization of the three-manifold. Embeddedcontact homology is conjecturally isomorphic to a version of theSeiberg-Witten or Ozsvath-Szabo Floer homologies. It provides abridge between the topology of smooth manifolds in three and fourdimensions, and the geometry and dynamics of holomorphic curves andcontact structures. Another part of the project is to constructFloer-theoretic invariants of families of equivalent objects fordifferent versions of Floer theory, thus obtaining homotopy invariantsof families of three-manifolds, symplectomorphisms, Legendrian knots,and other types of objects for which Floer theory can be defined.This project fits into the broad theme of developing tools tounderstand the possible global shapes of three- and four-dimensionalspaces, such as the universe that we live in. The tools used here tounderstand the shape of a space involve counting interesting geometricobjects inside the space. An important class of such objects arepseudoholomorphic curves, which are surfaces resembling soap films.By counting the number of such surfaces that exist in a space, one cangain deep information about the global structure of the space.