implicit functions

spectral sequences

I've thought about this a lot. You have to reconcile the traditional ideas of ritual and repetition with the fact that higher maths disregards a lot of that.

The followers could be scheduled to wake and do problems, could be understood to give tutoring (always free, or in exchange for lodging/food) when asked.

hidden markov models

Significant results and cultural popularity in short. With respect; i believe secrecy would diminish the effective influence of a modern math group unless you can collaborate with a group of elite mathematicians pursuing a genuinely unique and important style/field of study that would cause significant cultural or political change. (such as greek pythagoreans)
A group that developed tools and systems able to bridge the gap between right and left brain hemispheres modes of analysis and could produce interactive works of art and mathematical creative tools could marshall the power of two types of creative geniuses. In the spirit of steve jobs, (just as he used the sophistication of technology to create elegant devices that enhance the lives of people) the artists/animators and mathematicians/philosophers would be able to use systems created by engineers to illustrate and elaborate the mathematical intuitions of the interested public and math "cult" members.
In summary, apps and interactive events that would allow any mathematical insight, musical or visual artistic event to be transformed easily and powerfully into a "complex" that in turn could inform and inspire further mathematical insights, or philosophical, musical riffs.
I do not necessarily favor one type of program tthat is improved upon and anyone can use; but more so a set of connection protocols that are determined by the community