Knot concordance, periodic ends, and Rohlin's invariant

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Daniel Ruberman will carry out research in geometric topology, using Seiberg-Witten gauge theory, Heegaard-Floer homology, and more traditional topological ideas. The first part of the project, in collaboration with Nikolai Saveliev and Tomasz Mrowka, studies the relation between Seiberg-Witten theory and the analysis of the Dirac operator on periodic manifolds. A second project, joint with Saso Strle and Elisenda Grigsby, is to carry out further calculations of new Heegaard-Floer knot invariants that were introduced in our study of concordance. Related work with Matthew Hedden and Hee Jung Kim will explore other aspects of knot theory and embedded surfaces in 4-dimensional manifolds.The understanding of the structure of the 4-dimensional universe in which we live is a key topic of investigation in modern mathematics. Many of the questions posed by geometers and topologists have to do with the nature of 2-dimensional surfaces sitting in a 4-dimensional space, and with the singularities present on such surfaces. The research in this proposal uses modern tools of analysis and geometry to shed light on the local nature of such singularities, including new methods for showing that such singularities cannot be smoothed. Related analytical techniques will be used to explore the global topology of 4-dimensional spaces.