The classification theorem for finite simple groups might be a good candidate though. The classification theorem takes up tens of thousands of journal pages, and mostly between the 1950s and 2004. It was a huge deal when I was in college in the 1990s. The Wikipedia article is pretty good, but of course assumes you know something about group theory to begin with. And I’ll be honest that I don’t remember most of the definitions involved, and have forgotten more than I do remember.
Simple groups are building blocks of all groups, in a loose sense similar to how prime numbers are the building blocks of the multiplicative ring of integers. The Jordan Hölder theorem makes this precise. So by classifying the finite simple groups, we have gained a massive leap in understanding how all finite groups work.
The Feit Thompson theorem, which was probably the first glimpse that such a thing was possible, is about 255 pages of very dense mathematics. It was important enough that it took an entire issue of the journal in question. This theorem says that every finite simple group of odd order is solvable, and the proof was certainly the longest up to that point in group theory, and possibly the longest in any journal to that point. It’s not the longest any more, though. One proof (Aschbacher and Smith, on quasithin groups) ends up being about 1300 pages!
But the point here is that there was no reason to believe this was even possible. Especially in such a small number of naturally occurring families — alternating, cyclic, Lie groups, derived subgroups of Lie groups, and 26 groups that don’t fit anywhere else. Pretty amazing.
As to why math problems are getting harder so solve, I’m not sure that’s true. We’ve got better tools to tackle the hard problems now, but things have gotten somewhat specialized. For a non-specialist it’s hard to even understand what some of the problems are asking about — the Hodge Conjecture, one of the millennium problems, is definitely in this category. And mathematicians want to generalize things as broadly as possible, so sometimes helpful details get lost. So the complexity may not be essential, just in the presentation and the fact that lay people, even those in other specializations of math, don’t know the jargon involved. This is, of course, a double edged sword since often the more general questions can actually be easier to tackle, having lost unnecessary details that get in the way.
However, many problems that were super difficult in the past fall down fairly easily now. Take partial differential equations, for instance. We have tons of numerical methods that didn’t exist 100 years ago, along with the computers to run them on, so engineers and scientists don’t do as much analytic equation solving. Numerical approximations are generally sufficient unless you’re working in some unstable region of an equation or are try to prove some qualitative thing about the system involved. And we can deal with many of those too.
The classification theorem for finite simple groups might be a good candidate though. The classification theorem takes up tens of thousands of journal pages, and mostly between the 1950s and 2004. It was a huge deal when I was in college in the 1990s. The Wikipedia article is pretty good, but of course assumes you know something about group theory to begin with. And I’ll be honest that I don’t remember most of the definitions involved, and have forgotten more than I do remember.
Simple groups are building blocks of all groups, in a loose sense similar to how prime numbers are the building blocks of the multiplicative ring of integers. The Jordan Hölder theorem makes this precise. So by classifying the finite simple groups, we have gained a massive leap in understanding how all finite groups work.
The Feit Thompson theorem, which was probably the first glimpse that such a thing was possible, is about 255 pages of very dense mathematics. It was important enough that it took an entire issue of the journal in question. This theorem says that every finite simple group of odd order is solvable, and the proof was certainly the longest up to that point in group theory, and possibly the longest in any journal to that point. It’s not the longest any more, though. One proof (Aschbacher and Smith, on quasithin groups) ends up being about 1300 pages!
But the point here is that there was no reason to believe this was even possible. Especially in such a small number of naturally occurring families — alternating, cyclic, Lie groups, derived subgroups of Lie groups, and 26 groups that don’t fit anywhere else. Pretty amazing.
As to why math problems are getting harder so solve, I’m not sure that’s true. We’ve got better tools to tackle the hard problems now, but things have gotten somewhat specialized. For a non-specialist it’s hard to even understand what some of the problems are asking about — the Hodge Conjecture, one of the millennium problems, is definitely in this category. And mathematicians want to generalize things as broadly as possible, so sometimes helpful details get lost. So the complexity may not be essential, just in the presentation and the fact that lay people, even those in other specializations of math, don’t know the jargon involved. This is, of course, a double edged sword since often the more general questions can actually be easier to tackle, having lost unnecessary details that get in the way.
However, many problems that were super difficult in the past fall down fairly easily now. Take partial differential equations, for instance. We have tons of numerical methods that didn’t exist 100 years ago, along with the computers to run them on, so engineers and scientists don’t do as much analytic equation solving. Numerical approximations are generally sufficient unless you’re working in some unstable region of an equation or are try to prove some qualitative thing about the system involved. And we can deal with many of those too.