Speaker
Yinan Li, Wuhan University
Abstract
The Tensor Isomorphism problem asks whether two tensors lie in the same orbit under the natural action of general linear groups. In quantum information theory, this corresponds to determining whether two multipartite pure states are SLOCC/LU-equivalent. In the setting of finite fields, many fundamental isomorphism problems—including those of graphs, groups, algebras, and polynomials—reduce to testing isomorphism between structural tensors. In this talk, I will first survey recent progress on tensor isomorphism from a complexity-theoretic perspective. I will describe average-case polynomial-time algorithms for structural tensor isomorphism problems, which in turn suggest potential attacks on certain post-quantum digital signature schemes based on isomorphism assumptions. I will then present a proof that the average order of automorphism groups of 3-tensors is a constant. This result yields nearly tight upper bounds on the number of p-groups of Frattini/nilpotency class 2 and resolves several open questions posed by Blackburn, Neumann, and Venkataraman. The main mathematical tools involved include random matrix theory and fixed-point analysis over finite fields.