The study of symmetries in the setting of group actions via “invariant” polynomials is called invariant theory. Throughout the 20th century, invariant theory has had a profound influence in several fields in mathematics, most notably those that fall under the broad purview of algebra. Over the last two decades, new directions in invariant theory have emerged out of connections to computational complexity. Particularly exciting is the Geometric Complexity Theory (GCT) program that has uncovered connections between invariant theory and foundational problems in complexity such as identity testing and the celebrated P vs NP problem.

In this talk, I will discuss recent advances in invariant theory concerning matrix invariants and semi-invariants, (non-commutative) identity testing, null cones and orbit closures. Towards the end, I will discuss some very promising directions for the future in this rapidly expanding field. Based on several joint works with Harm Derksen, Ankit Garg, Rafael Oliveira, and Avi Wigderson.