Matrix multiplication in $O(n^2 \log n)$
I came across this paper about a year ago, but have not gotten around to reading it closely. I can tell you that the approach is not believed to be correct. On page 36 of the same paper, there is an attached comment by Don Knuth, who points out what looks to be a serious shortcoming of the approach. 

To understand this paper, you will need to learn about group algebra and representation theory. It will be tough if you haven't seen that kind of material before.