Is the Matrix Multiplication Exponent $\omega$ too famous?
Recall that $\omega$ is the inf of all real numbers $k$ such that two $n \times n$ matrices can be multiplied in $O(n^k)$ steps. It is currently known that $2 \leq \omega < 2.3728639$.
Regarding lower bounds, there are several new ideas over the last few years from the field of Geometric Complexity Theory; see, e.g., Grochow's survey. Computer scientists aren't always too expert in the area of representation theory, so it could be a good chance for cross-area collaboration.
Regarding upper bounds, there have been several improvements over the last few years (Stothers, Vassilevska-Williams, and now Le Gall) pushing hard using the "traditional methods", a new paper on limits of the traditional methods, and the very interesting alternate approach of Cohn-Kleinberg-B.Szegedy-Umans based on group theory and arithmetic combinatorics.