We present several variants of the sunflower conjecture of Erdos and Rado [ER60] and discuss the relations among them. We then show that two of these conjectures (if true) imply negative answers to questions of Coppersmith and Wino grad [CW90] and Cohn et al [CKSU05] regarding possible approaches for obtaining fast matrix multiplication algorithms. Specifically, we show that the Erdos-Rado sunflower conjecture (if true) implies a negative answer to the ``no three disjoint equivoluminous subsets'' question of Coppersmith and Wino grad [CW90]; we also formulate a ``multicolored'' sunflower conjecture in $\Z_3^n$ and show that (if true) it implies a negative answer to the ``strong USP'' conjecture of [CKSU05] (although it does not seem to impact a second conjecture in [CKSU05] or the viability of the general group-theoretic approach). A surprising consequence of our results is that the Coppersmith-Wino grad conjecture actually implies the Cohn et al. conjecture. The multicolored sunflower conjecture in $\Z_3^n$ is a strengthening of the well-known (ordinary) sunflower conjecture in $\Z_3^n$, and we show via our connection that a construction from [CKSU05] yields a lower bound of $(2.51\ldots)^n$ on the size of the largest {\em multicolored} 3-sunflower-free set, which beats the current best known lower bound of $(2.21\ldots)^n$ [Edel04] on the size of the largest 3-sunflower-free set in $\Z_3^n$.