Imagine trying to understand commutative rings (or varieties) without a notion of "dimension" -- for example, without the concept of the transcendence degree of a field. Unfortunately, this is the situation that noncommutative algebraists often find ourselves in! In fact, in the noncommutative setting there are multiple definitions of the "transcendence degree" of a division ring (noncommutative field), most or all of which behave badly in some situations; in addition, the relationships between different definitions are often poorly understood. This talk will discuss some of the ways that various transcendence degrees are useful in understanding division rings, as well as some of the pitfalls of extending commutative notions to the noncommutative setting. Little noncommutative theory will be assumed.
This talk is not strictly part of the derived category topic, although derived categories will probably be mentioned.