Gauge theory and Floer homology in low-dimensional topology

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The understanding of the structure of the 4-dimensional universe in which we live is a key topic of investigation in modern mathematics and physics. 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 project 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.Daniel Ruberman will carry out research in geometric topology, using Seiberg-Witten gauge theory, Heegaard-Floer homology, and more traditional topological techniques. The first parts of the project, joint with Nikolai Saveliev and Tomasz Mrowka, are concerned with 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; solutions of the main problems will decide the existence of manifolds predicted by high dimensional surgery theory. The PI will work with Adam Levine and Saso Strle on applications of new invariants from Heegaard Floer theory to torsion homology classes in 4-manifolds, with Levine and Matthew Hedden on ends of 4-manifolds, and with Selman Akbulut on the construction of new smooth 4-manifolds. A final portion, joint with David Auckly, Hee Jung Kim, and Paul Melvin, is concerned with the topology of the diffeomorphism group of a 4-dimensional manifold and how it is affected by stabilization of the manifold.