Owens proposes to study various applications of the recently introduced Heegaard Floer invariants of Ozsvath and Szabo to three-manifolds, smooth four-manifolds, and knots. In particular, Owens proposes to study relationships between Heegaard Floer invariants and intersection pairings of four-manifolds. Ozsvath and Szabo have shown that the Heegaard Floer correction terms of a 3-manifold Y give constraints on the intersection pairings of 4-manifolds bounded by Y. Owens will study number-theoretic invariants of quadratic forms that shed light on these results, and will also investigate whether certain 4-manifolds that arise in applications may be used to calculate the correction terms of their boundary. He will also continue his study of applications of Heegaard Floer theory to knot theory. In particular he is working on new methods for computing unknotting numbers and on smooth concordance invariants of knots.
A three-dimensional manifold is a space which requires 3 coordinates to (locally) describe one's position, such as for example our physical universe. Manifolds of any dimension may be studied; perhaps surprisingly, it turns out that geometry and topology is most difficult and interesting in the dimensions most relevant to modelling the universe, namely three and four (known as "low-dimensional topology"). Heegaard Floer invariants were introduced in 2001 by Ozsvath and Szabo and have proved to be a powerful tool, leading to many important advances in our understanding of low-dimensional topology, and solutions to long-standing problems that resisted other methods. Applications include computation of various measures of complexity of knotted circles in three-dimensional space, such as the genus (which measures how complicated a surface must be which has the knot as its boundary) and the unknotting number (which measures how many times you need to pass the string through itself to untie the knot). Owens will use the Heegaard Floer techniques to study the relationship between three- and four-dimensional manifolds and to understand properties of knotted circles in three-manifolds.
