Colloquium| Institute of Mathematical Sciences
Time：16:00-17:00, July 26 Friday
Location：Room S408, IMS
Speaker: Brian Weber
Abstract: It is possible to hope for geometric structure theorems for 4-manifolds, provided one restricts attention to special geometries. Ka¨hler manifolds for instance, the focus of tremendous current research, has expected or conjectured structure theorems, and many powerful results already exist. Far less is known, or even conjectured, in the half-conformally ﬂat setting: aside from a few topological obstructions, almost the only thing known is that such manifolds are plentiful. In this colloquium talk, we discuss the issues with ﬁnding structure theorems for constant scalar curvature half-conformally ﬂat 4-manifolds, and look at some recent progress in understanding their instantons. Methods of potential theory—the study of solutions of 4f = 0—along with the theory of solutions of the ﬁrst order ODE dω = 0, ω ∈V2 interact to provide crucial, sometimes counter-intuitive theorems in the Ka¨hler setting. Although half-conformally ﬂat 4-manifolds lack a complex structure, these methods from Ka¨hler geometry can be transferred over, with some (but not all) of the same successes. A full structure conjecture is certainly out of reach, but the structure of CSC half-conformally ﬂat manifolds can be powerfully constrained.