Recent developments in Seiberg-Witten theory

Abstract：

In 1994, Witten [12] introduced a non-linear partial differential equation on a 4-manifold, called the Seiberg-Witten equations today. This PDE has brought significant progresses in 4-dimensional topology and geometry. In this series of lectures, I shall start with the basics of Seiberg-Witten theory and survey some of rather recent developments in Seiberg-Witten theory.

The first lecture is devoted to explain the foundation of Seiberg-Witten theory for closed 4-manifolds and some of classical major applications of it. Most results in this first lecture have been obtained in the last century, but they are explained from a relatively modern viewpoint: more concretely, my exposition shall be based on the technique of finite-dimensional approximations of the Seiberg-Witten equations, along Furuta [6] and Bauer-Furuta [5].

In the second lecture, I present what one can say about the diffeomorphism groups of 4-manifolds using Seiberg-Witten theory. This lecture contains much recent study on Seiberg-Witten theory for families of 4-manifolds. I shall explain (perhaps some of) results in [1, 2, 3, 4, 7, 8, 11] of this kind.

The third lecture will be a crash course in Seiberg-Witten Floer homotopy theory, started by Manolescu [10]. Amongst various versions of Floer theory known in low dimensional topology, this theory has a big advantage that one may consider any kind of generalized cohomology theory, such as K-theory. I shall sketch Manolescu's construction of Seiberg-Witten Floer homotopy theory, and explain a recent application in our paper [9] of this framework to knot theory.

Prerequisite:

No specific advanced knowledge is needed, but it would be helpful to be familiar with the basics of both of topology and differential geometry.