The complexity of Iterated Matrix Multiplication is a central theme in Computational Complexity theory, as the problem is closely related to the problem of separating various complexity classes within $P$. In this talk, we will discuss the algebraic formula complexity of multiplying $d$ many $2x2$ matrices and show that the well-known divide-and-conquer algorithm cannot be significantly improved at any depth, as long as the formulas are multilinear. We will also give two applications of this lower bound to multilinear formula complexity (this is joint work with Suryajith Chillara and Srikanth Srinivasan (IITB)).
