dickmwong

By Rudin I assume you mean the book "Real and Complex Analysis". It is a very nice text given that it discusses the essential topics in real analysis (measure theory), basic functional analysis and complex analysis in 400 pages in a very clear and concise manner. It is surely a rewarding experience to go through the whole book if you manage to do so. That said, whether this is necessary or not is another story. (Oh, and I am also applying to PhD programs this year. So this is not an advice from somebody who is in his PhD.)

[Current Part III Student] I do not study the topics you mentioned, so my words may not be accurate. There are often reading groups at Cambridge, and as far as I know there is one on homotopy type theory (aka Univalent Foundations) this term. These reading groups are open to Part III students (in fact Part III students are invited to join them). But do note that things vary from year to year. This is particularly true for topics courses, as the choice of special topics depend very much on availability of faculty and visitors.