lol i don't understand your point "harvard doesn't have secret math - the proof is all of these publicly available math lectures from world renowned mathematicians"

the value of harvard math profs isn't reaped by the undergrads, it's reaped by the grad students that work closely/directly with these profs. i'm not 100% sure of this but benedict gross probably isn't holding several hours of office hours every week for his undergrad algebra classes (although he might for the grad sections).

now the knock-on effect, obviously, is that the very good math phd students that pursue and are admitted to harvard do provide a lot of value for the undergrads (since they're TAing/holding office hours for those undergrads). but those same harvard phd students then go get tenure track positions at mediocre schools where office hours are held by profs. so it all evens out as far as undergrads go :)

As somebody who's studied some advanced maths, I think that their point is that abstract algebra is effectively a sort of secret maths techniques, sitting in plain sight but somehow sprayed with repellent or maybe equipped with a Somebody Else's Problem field.

I have had more than one afternoon where I learned a concept wholly from this sort of lecture. The ability to ask the professor questions can be useful, but almost all of my questions are actually answered by being more careful about listening to what they are saying the first time.

>abstract algebra is effectively a sort of secret maths techniques

abstract algebra (rings, groups, fields) is taught in absolutely every undergrad math program in the country (in the world?). there's nothing unique to harvard in this aspect.

Harvard teaches a lot more than the average undergrad program.

A Harvard undergrad degree covers a depth of knowledge that matches a master's degree at an average program.

You can read the definitions of algebra in 1 day. That's not what a class is about.