# Colloquium: Gauge theory and infinite cyclic coverings

Danny Ruberman

The infinite cyclic cover of a circle by the real numbers appears early on in one's mathematical career. In analysis, it appears as part of basic Fourier analysis; in complex variables when discussing winding number, while in topology it appears when one computes the fundamental group of the circle. A philosophical view (going back to work of Milnor, Novikov, Rochlin, and Pontrjagin) is that passing to an infinite cyclic cover of an (n+1)-manifold makes it look in some ways like an n-dimensional manifold.

This approach leads to interesting invariants of smooth 4-dimensional manifolds. I will talk about how the classical Rochlin invariant of 3-manifolds becomes a 4-dimensional invariant and how that 4-dimensional invariant is related to Seiberg-Witten gauge theory. The intermediary between these is the index theory of the Dirac operator on the infinite cyclic cover.

The talk is based on papers with Jianfeng Lin, Tom Mrowka, and Nikolai Saveliev.

DR seminar

Tuesday, October 26 at 2:30pm

CW 122, CW 122

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- Mathematics, Department of
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