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I had an interesting thought where I wondered that if I had a grandparent who had been a mathematician (or at least studied mathematics), how much of modern undergraduate mathematics would they understand?
I think in other sciences this gap would be much greater than in mathematics. So I guess my question is, how much does mathematics, that is taught to students, change in a generation? In particular, how much has it changed in the last generation?
Mathematics advances rapidly. A generation ago, ultrafiltres and category theory were pretty new and niche, now they are pretty close to foundational as concepts in algebra and topology. Theoretical Computer Science has changed massively in the past 30 years. Some of the greatest results in the field are recent discoveries. PCP Theorem, the Regularity Lemma, and the Triangle Removal Lemma are all pretty new. The classification of finite simple groups is another thing that has massively changed mathematics. Almost all of these things caused massive changes at the undergraduate level, and a few are first year graduate topics.
By comparison, I think quantum computing and M-theory are the main changed in physics that would percolate to the undergraduate level (physicists please double check me though!).
I don't think that the undergraduate curriculum has changed very much in the past generation (for the sake of concreteness, let's say that a generation is 40 years).
Suppose we take a look at the requirements for a mathematics major at a top liberal arts college such as Williams College. We see discrete mathematics, differential equations and vector calculus, mathematical methods for scientists, statistics, linear algebra, real analysis, and abstract algebra. Contra Stella Biderman, I don't think that there have been "massive changes" in these topics at the undergraduate level caused by ultrafilters, category theory, the PCP theorem, the regularity lemma, the triangle removal lemma, or the classification of finite simple groups. At most, there might be some category theory in the abstract algebra course which would not have been in there 40 years ago. The other topics would merit at most a passing mention in an undergraduate course.
All the above topics, except possibly for discrete mathematics, are classical, and the core content would likely be familiar to your hypothetical grandfather. The one thing that I think has changed substantially in the past 40 years is the importance of computers. Discrete mathematics is now a common (though not ubiquitous) requirement for an undergraduate mathematics major, whereas it would have been unusual 40 years ago, and clearly it is computers that have made discrete mathematics more important. Many of the other topics will likely show some influence of the computer revolution, especially linear algebra, differential equations, and statistics; algorithms for handling large problems would probably not have been discussed 40 years ago at the undergraduate level, but might get a significant amount of class time now. Real analysis and abstract algebra are less likely to have changed much, although even abstract algebra may show some influence from the existence of computer algebra packages.
