Question: It is commonly believed that for all $\epsilon > 0$, it is possible to multiply two $n \times n$ matrices in $O(n^{2 + \epsilon})$ time. Some discussion is [here](https://cstheory.stackexchange.com/questions/5341/evidence-that-matrix-multiplication-can-be-done-in-quadratic-time).

I have asked some people who are more familiar with the research whether they think that there is a $k>0$ independent of $n$ such that there exists an $O(n^2 \log^k n)$ algorithm for matrix multiplication and they overwhelmingly seemed to have intuition that the answer is "no" but could not explain why. That is, they believe that we can do it in $O(n^{2.001})$ time, but not $O(n^2 \log^{100} n)$ time.

What reasons are there to believe that there is no $O(n^2 \log^k n)$ algorithm at a fixed $k>0$?

Answer: 
There's an algorithm for multiplying an $N \times N^{0.172}$ matrix with an $N^{0.172} \times N$ matrix in $N^2 \operatorname{polylog}\left(N\right)$ arithmetic operations. The main identity used for it comes from Coppersmith's paper "Rapid multiplication of rectangular matrices", but the explanation for why it leads to $N^2 \operatorname{polylog}\left(N\right)$ instead of $N^{2 + \epsilon}$ is in the appendix of Williams' [paper](https://arxiv.org/abs/1401.2444), "New algorithms and lower bounds for circuits with linear threshold gates".

This only works because Coppersmith's identity has some additional structure you can take advantage of, and the more recent MM algorithms don't seem to have this structure. That said, I'm not sure why one can't hope to extend this approach to $N \times N \times N$ matrix multiplication.
Josh Alman showed some cool lower bound results of MM, which won CCC 2019 best student paper award! 
<http://drops.dagstuhl.de/opus/volltexte/2019/10834/pdf/LIPIcs-CCC-2019-12.pdf>