look at these slides of the open dreamkit project: http://opendreamkit.org/meetings/2016-01-25-DKS/Kohlhase_sli... ... somewhere there lmfdb should be mentioned
You could look at the top n undergraduate math programs in your state and see what the flowchart of courses are and their corresponding syllabi and just go from there.
This is somewhat similar to (and significantly more detailed than) something I've wanted for a long time:
Feb. 26, 2009
What the world really needs is a poster that teachers
can hang in their classrooms with a map of the world
of math. It would essentially be a directed acyclic
graph, with nodes representing mathematical concepts,
and arrows linking those concepts to the next level
of concepts one can learn, as well as real-life
problems that can be solved with that level of math.
For example, the chart would start with basic
arithmetic, with addition and subtraction leading to
multiplication and division. The related tasks
possible with addition would be things like grocery
shopping, subtraction would be figuring change from a
purchase or determining how much time remains before
some event. Multiplication would allow one to make
simple designs, calculate taxes and tips, etc. The
arrows would then go through the concepts of algebra,
geometry, calculus, and on to things like the Fourier
transform. The things one can do at a particular
level could be represented as a bubble, with more
math leading to a bigger bubble (and, if necessary to
convince the kids, more money).
This would also be beneficial to college students
trying to convince their brains to remember all the
seemingly-useless things they are learning in one
class because they need to understand the concepts
for next semester's classes. In fact, such roadmaps
would make life a lot easier in general. "Want to
become a $140000/year contractor? These are the
steps you follow."
I think you have to be logged in, but you can see a screenshot here:
That being said, I was attending high school in the "we just got cable modems & youtube has 2 sets of lectures, one being Sussman teaching SICP" days. Those who are interested can easily find dozens of lectures from any conference/symposium/some-random-bright-grad-student-talking-to-a-room-of-8-people and then directly e-mail him/her. Knowledge is accessible really accessible these days, and those who are motivated to read further on a topic have high quality textbooks, pre-prints of papers, and open discussion amongst the communities (i.e., lurking on a Terry Tao project, or watching in real time as the classification of finite groups is underwent by a group of post-docs and grad students). For those who are motivated and learn by conversation/immersion like I do (did?).
Perhaps this explains my point a little clearer. (Note that Fermat has more than one proof, and I could easily at least the intuition behind the proof to a 2nd year undergraduate with no background in higher mathematics, given perhaps two semesters.) Conceptually, something like Poincare would be way more challenging just because at that phase everyone's used to Newtonian physics and Euclidian geometry; unteaching those 'intuitions' would take a semester alone. It could, however, be 'intuitively' (i.e., without rigor) taught fairly easily to someone with a background in higher level physics (i.e., most of the foundational maths of Riemannian geometry, etc, have been both taught and intuitively absorbed by students at that point).
 https://www.youtube.com/watch?v=6oWLIVNI6VA Here's a lecture from a Princeton professor emeritus (IIRC) that demonstrates what I'm sure I poorly explained here. Introduction ends around ~6 minutes.
 https://jeremykun.com/2013/02/08/why-there-is-no-hitchhikers... and in refutation - actually there is, sort of.
href="www.lmfdb.org" should be href="http://www.lmfdb.org"
(Submitted title was "Scientists Launched an Enormous Atlas of the Mathematical Universe", which was arguably editorialized and rather baity—please don't do that. We replaced it with a representative phrase from the article.)
I knew Duolingo had a design like that.
But this project seems to be specific to number theory, and not really concerned with proofs at all.
Where can I find the webcast mentioned?