In this talk, we show how the algorithmic problem of algebra isomorphism naturally leads to questions about random matrices over finite fields. To settle these questions, we need to extend both the generating function approach as in Neumann and Praeger (J. London Math. Soc., 1998) and the characteristic sum method of Gorodetsky and Rodgers (Trans. Amer. Math. Soc., 2021). Several open questions remain for further exploration.

Joint work with Tiange Li, Yinan Li, Dacheng Tao, Yingjie Wang. Paper available at https://arxiv.org/pdf/2604.00591.