In the preseminar, I will focus on the group isomorphism problem, which is to decide whether two ﬁnite groups are isomorphic (given their Cayley tables). I will introduce necessary definitions about finite groups and show a 80-year-old reduction from a bottleneck case of the group isomorphism to the pseudo-isometry test of alternating matrix tuples. (No group-theoretic background is needed.)
In the seminar, we present an average-case efficient algorithm which, for almost all alternating matrix tuples, tests pseudo-isometry with any other alternating matrix tuples. Our algorithm is based on a linear-algebraic analogue of the individualization and refinement technique used in [Li-Qiao 2017], combined with concepts from isomorphism of low-genus groups [Brooksbank-Maglione-Wilson 2017]. Notably, the algorithm has been implemented in MAGMA, which in experiments improves over the default (brute force) algorithm for this problem.