Analytical and geometric methods in low-dimensional topology

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Daniel Ruberman will carry out research in geometric topology, using Seiberg-Witten gauge theory, Heegaard-Floer homology, and techniques of knot theory. The first part of the project, joint with Nikolai Saveliev and Tomasz Mrowka, studies the smooth topology of 4-manifolds that homologically resemble a product of a 3-dimensional manifold with a circle. The central questions center around the interpretation of the classical Rohlin invariant in terms of gauge theory. Techniques involve Seiberg-Witten theory and a new index theorem for manifolds with periodic ends. A second part of the project proposes joint work of the PI with Saso Strle and Jae Choon Cha in knot theory, applying new invariants from Heegaard-Floer theory to classical problems about knot and link concordance. The results from this part will help measure the difficulty of smoothing a singularity of a surface sitting in a 4-dimensional manifold. A final portion, joint with Paul Melvin and David Auckly, is concerned with the topology of the diffeomorphism group of a 4-dimensional manifold and how it is affected by stabilization of the manifold. 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, including an investigation of their symmetries.