Building knowledge on top of the huge algorithmic base of Mathematica, gives this a fighting chance of doing something significant.

Why? Mathematica does not have access to magical algorithms that none of the rest of us know about, nor does it have any particular performance benefits when it comes to implementing specific algorithms. It is a great exploratory tool and an exceptional "generalist" in computer mathematics, but beyond that it does not offers much that can't be done better/faster by a couple of smart people writing some tight application-specific code.

> Mathematica does not have access to magical algorithms that none of the rest of us know about

Yes it does. Quite a lot of them, especially w.r.t. symbolic equation-solving and integration. They're notoriously protective of their trade secrets.