Today, there are 101 papers on mathscinet using notion of [planar algebra][1], created by [Vaughan Jones][2] in 1999. 
As pointed out to me by [VS Sunder][3], this notion admits some fundation open problems, see for example the post:  
 [Why the composition of planar tangles is well-defined?][4]


  [1]: http://arxiv.org/abs/math/9909027
  [2]: http://en.wikipedia.org/wiki/Vaughan_Jones
  [3]: http://en.wikipedia.org/wiki/V._S._Sunder
  [4]: http://mathoverflow.net/q/207650/34538