On Jung-type Behavior in II_1 Factors
In mathematics, one of the central philosophies is to try and understand structural properties of an object by looking at how it embeds into familiar and rich objects. In this talk, we will look at separable II_1 factors and try to understand their structure by looking at the space of its embeddings into ultraproduct von Neumann algebras, like R^w for instance. The main motivation is a theorem of Jung that says that if any two embeddings (provided they exist) of a tracial von Neumann algebra N into R^w are unitarily conjugate, then N is amenable. In recent joint works with S. Atkinson and I. Goldbring, we have been able to generalize this theorem in very interesting directions. We will look at some of these results in this talk.