What parts of a pure mathematics undergraduate curriculum have been discovered since 1964?

Except for top university mathematics programs which have truly gifted undergraduates in them-such as Harvard, Yale or the University of Chicago-I seriously doubt undergraduates are exposed to truly modern breakthroughs in mathematics in any significant manner. Indeed, it's rare for first year graduate courses to contain any of this material in large doses!

This question reminds me of an old story my friend and undergraduate mentor Nick Metas used to tell me. When he was a graduate student at MIT in the early 1960's,he had a fellow graduate student who was top of his class as an undergraduate and published several papers before graduating. When he got to MIT, he refused to attend classes, feeling such "textbook work" was beneath him." This is all dead mathematics-I want to study living mathematics! Stop wasting my time with stuff from before World War I!" As a result, he had some really bizarre holes in his training. For example, he understood basic notions of algebraic geometry and category theory, but he didn't understand what the limit of a complex function was. As a result, not only did he fail his qualifying exams, his own presented research suffered greatly-he was always playing catch-up. Eventually, he dropped out and Nick never heard from him again. He always tells his students this story in order to make them understand something fundamental about mathematics-it's a subject that builds vertically, from the most basic foundations upward to not only more sophisticated results, but from the oldest to the most recent results.

This is why I think undergraduates simply can't be exposed to "recent" results-it takes until they're at least first year graduate students for a wide enough conceptual foundations to be erected in them to even begin to understand these concepts.