What is...?
What is Tensor Isomorphism?
Two graphs are isomorphic if they are the same up relabelling the vertices. Two matrices are equivalent if they are the same up to elementary row and column operations. Tensor isomorphism generalises these basic notions in graph theory and linear algebra. Specifically, 3-tensors (or 3-way arrays) are isomorphic, if they can be transformed into one another by multiplication with three invertible matrices along the three directions.
In this talk, we will explore how this seemingly simple notion of tensor isomorphism connects to some main objects in mathematics. For example, it captures isomorphism notions of several algebraic structures, including groups, polynomials, and (associative and Lie) algebras, via the orbit containment relation as introduced by Gelfand and Panomerav. Furthermore, tensor isomorphism over GL(n, Z) also arises naturally in contexts such as the classification of Calabi–Yau threefolds and Bhargava’s approach of Gauss composition laws.