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

What parts of an undergraduate curriculum in pure mathematics have been discovered since, say, $1964?$ (I'm choosing this because it's $50$ years ago). Pure mathematics textbooks from before $1964$ seem to contain everything in pure maths that is taught to undergraduates nowadays.

I would like to disallow applications, so I want to exclude new discoveries in theoretical physics or computer science. For example I would class cryptography as an application. I'm much more interested in finding out what (if any) fundamental shifts there have been in pure mathematics at the undergraduate level.

One reason I am asking is my suspicion is that there is very little or nothing which mathematics undergraduates learn which has been discovered since the $1960s$, or even possibly earlier. Am I wrong?

Solution 1:

A lot of the important basic results of complexity theory postdate 1964. The most important example that comes to mind is that the formulation of NP-completeness didn't occur until the seventies; this includes the Cook-Levin theorem, which states that SAT is NP-complete (1971) and the identification of NP-completeness as something that was important and common to many natural computational problems (Karp 1972). These results certainly appear in an undergraduate course on computability and complexity.

Ladner's theorem, which would at least be mentioned in an undergraduate course, was proved in 1975.

(In contrast, the basic results of computability theory date to Turing, Post, Church, and Gödel in the 1930s.)

Solution 2:

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.

Solution 3:

Graph theory is a relatively recent subject. Although the Seven Bridges of Königsberg problem dates back as far as 1736, one of the first accepted textbook on the subject was published by Harary in 1969, thirty years after Kőnig's work.

Many theorems were stated and proved in the last fifty years or so. For instance, the strong perfect graph theorem was conjectured in 1961 and was not proved until 2006 by Chudnovsky and al. The statement and the proof of the weaker version are often taught at the undergraduate level.

Solution 4:

A lot of things connected to what is conventionally called the chaos theory. For example, it is quite accessible for undergraduates to prove that period three implies chaos.

(Li and Yorke published their paper in 1975. However, a more general result was given by Sharkovsky in 1964 (reprinted here), which still fits your question).

Solution 5:

I taught students how to compute the Homfly polynomial in an undergrad topology course. This is an invariant for distinguishing inequivalent knots and links, and only goes back to 1985.