Daniel Ruberman will carry out research in geometric topology, using Seiberg-Witten and Yang-Mills gauge theory. The first part of the project, to be carried out in collaboration with Nikolai Saveliev and Tomasz Mrowka, is to understand 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; an important facet is the analysis of end-periodic Dirac operators. Solutions of the main problems posed will decide the existence of some manifolds predicted by high-dimensional topology (surgery theory) and settle a fundamental question about the homology cobordism group. This latter question is key in understanding the triangulability of manifolds of dimension 5 or more. The second part of the project proposes joint work with Saso Strle. We plan to investigate the degree to which invariants of a 3-manifold derived from the Seiberg-Witten equations constrain the topology of those 4-manifolds which it bounds. The results will be used to investigate a classical problem about group actions on the K3 surface and other algebraic varieties.
Many of the most difficult problems in topology are concerned with the structure of three and four-dimensional manifolds. Questions about phenomena in these dimensions have particular interest because those are the dimension of space and time that make up our world. The research to be carried out under this grant makes use of powerful geometric and analytical techniques that have been introduced into the field over the last twenty years, under the general rubric of gauge theory. Much of the application of four-dimensional gauge theory has been to manifolds that are simply-connected, in which any loop can be shrunk to a point. The research proposed will use Yang-Mills and Seiberg-Witten gauge theories to shed light on manifolds which are not simply-connected, where new phenomena are expected. By studying the Dirac operator (originally introduced to give a relativistic quantum theory of the electron) on manifolds with a certain type of periodicity, we seek to get restrictions on an invariant of non-simply connected manifolds, called the Rohlin invariant. This subtle invariant is connected to three-dimensional topology, and its behavior determines whether high-dimensional manifolds may be triangulated.
