Mulmuley and Sohoni observed that the permanent versus determinant problem can be interpreted as a specific orbit closure problem in the space of homogeneous forms, and they proposed to attack it by methods from geometric invariant and representation theory.

In the lecture, I will explain how these ideas apply in the related, but simpler setting of trilinear forms. This setting is of considerable interest by itself, since it captures the tensor rank problem, and in particular the complexity of matrix multiplication.

The analysis of the symmetries via representions leads to a bunch of challenging mathematical questions. A main idea is to understand, which irreducible representations occur in the coordinate rings of the orbit closures under question. We explain some of the known mathematical tools to adress these questions and the current limitations for advancing further.