M 392C Gauge Theory (54355)

 

Tim Perutz

TTh 11am – 12:30pm

RLM 11.176

Course Description: 

This will be a course about the Seiberg-Witten equations over 4-dimensional manifolds. We will use these equations to give examples of 4-dimensional homotopy types that admit no smooth manifold structure, and others that admit infinitely many: examples that show that 4-dimensional manifolds do not play by the rules that govern smooth manifolds of any other dimensions. Besides measuring the difference between homotopy theory and smooth topology, we shall also use the Seiberg-Witten equations to detect differences between smooth and symplectic topology in dimension 4.

 

What rules do in fact govern smooth 4-manifolds? The answer is, to date, the greatest mystery in geometric topology.

 

Methods used in the course will be from algebraic topology and differential geometry - for which the two topology prelims provide appropriate background - and geometric analysis - for which I will cover background material on elliptic operators.  The course will complement the 4-manifolds course taught regularly by Prof. Gompf (which has a quite different flavor), and will provide training for students interested in Heegaard Floer and related theories for 3-manifolds.

Reference: S. K. Donaldson, The Seiberg-Witten equations and 4-manifold topology, Bulletin of the AMS, 1996; http://www.ams.org/journals/bull/1996-33-01/S0273-0979-96-00625-8/