As someone who's interested in pure math, I think about that last question a lot. It seems really likely that the only significant advances in math these days will be made by (and understood by) people with years of specialized technical training, but of course we don't know that for sure.,(reply>), In number theory in particular there are a few open problems that are readily understood by your typical non-mathematician like twin primes and Collatz. Granted there's no way they'll be able to understand the mechanisms behind the proof and it might be difficult to get across why it's a big deal that either was solved, but it's something.,(reply>), I think the answer is: maybe not quite when it comes to your average Joes, but definitely to your average undergraduate. In areas like physics and especially mathematics tons of the stuff has been around for ages, but in areas like statistics and computer science a lot of the developers and pioneers are still around. Bradley Efron , who invented bootstrapping in statistics, and Gordon Moore, or Moore's Law fame in computer science, are still alive despite their fundamental roles, and with quantum computers just now being commercially available it is possible that major discoveries that will be the bread and butter of their fields will be made in the not-too-distant future.,(reply>), I was reminded of one of the few examples that break that mould a while ago. Remember superpremutations? The guy who advanced it as far as it is today was just a weeb on 4chan who wanted to argue about an anime that was airing at the time. <br>Even now, several years later, his discovery is still the close to the farthest mathematicians gotten in the area, with his formula only having been improved upon by one digit at its minimum value.,(reply>), If you think about it, it's hard to come up with any one "significant advance" in math. There's a few that do really stand out, but even those were just new and gradual ways of doing something that was already mostly discovered. Like imaginary numbers for example - a major leap forward if you imagine it just came about, but really it was an elegant innovation over the course of 2 centuries of doing something mathematicians had previously figured out. I'd be interested to hear what you consider a significant advance in math that didn't come about as many small advances (like the ones really anyone today could contribute, given the time and inclination).,(reply>), Bleem.