Low dimensional topology: A survey

For the past 50 years it has been understood that from the topological point of view dimensions 3 and 4 are special, and especially difficult to understand.
It has also become clear over this period of time that ideas and techniques from geometry, from geometric analysis, and from theoretical physics have proved crucial in advancing our understanding of topological phenomena in these exceptional dimensions. It has also become clear that, while there is a natural interaction between the phenomena in these dimensions and some common sources of invariants, the natures of their topology are very different. This talk will give an overview of the state of knowledge in these two exceptional dimensions, summarize the ideas and techniques that have proven useful in their study, and discuss major open questions.