Intellectual Merit: As an NSF Mathematical Sciences Postdoctoral Research Fellow at Brandeis University, I made considerable progress on a variety of problems in low-dimensional topology, particularly concerning knots and surfaces in three- and four-dimensional spaces (manifolds). The focus of my research was the properties and applications of Heegaard Floer homology, a package of tools for studying three- and four-dimensional topology that has been at the center of the field since its introduction by Ozsvath and Szabo over a decade ago. A few of the most significant results are as follows: In my dissertation, I produced examples of links in 3-space that cannot occur as a cross-section of a disjoint union of smoothly embedded spheres in 4-space. During my postdoctoral fellowship, I converted these results into a pair of published papers. In joint work with John Baldwin (Boston College), we developed a new description of the knot Floer homology of a knot in 3-space that can be computed algorithmically from a planar diagram of the knot. This work is related to our continuing project that seeks to relate knot Floer homology and Khovanov homology, two knot invariants introduced in the early 2000s that have intriguing similarities despite their disparate origins. Matthew Hedden (Michigan State University) and I proved that a 3-manifold obtained by gluing together the complements of nontrivial knots in ordinary 3-space can be distinguished from ordinary 3-space by its Heegaard Floer homology, part of an ongoing effort to understand the classification of 3-manifolds whose Heegaard Floer homology groups are as simple as possible. In a joint paper with Daniel Ruberman (Brandeis) and Saso Strle (University of Ljubljana), we proved new results about the minimal complexity of non-orientable surfaces embedded in various 4-dimensional spaces. Our paper also included an appendix by Ira Gessel (Brandeis), a combinatorialist who assisted us with some number-theoretic aspects of our investigation. Broader Impacts: While at Brandeis, I benefited greatly from the proximity to other schools in the Boston area, frequently attending seminars and lectures at Harvard, MIT, Boston College, and Tufts. I created a unified online calendar of all of the topology-related seminars at these schools to facilitate interaction between researchers. I also worked closely with several of the graduate students at Brandeis, including supervising reading courses in advanced topics and offering suggestions on their research. I used my NSF funding to travel widely, both to visit my collaborators and to attend conferences around the country and the world. Finally, I am particularly grateful to have developed an outstanding relationship with my postdoctoral sponsor, Daniel Ruberman, who was extremely generous with his time and with his depth and breadth of knowledge; I look forward to further collaboration with him in the future.
