Optimization, Complexity and Invariant Theory

Abstract: Symmetries are a main source of unifying principles in mathematics and physics and groups are the appropriate mathematical concept for describing symmetries. Representation theory studies linear transformations in the presence of symmetries and, e.g., plays a major role in quantum mechanics. In computer science, group symmetries are behind several fast algorithms, like the Fast Fourier transform, fast arithmetic of numbers and polynomials, matrix multiplication, construction of expanders etc. Naturally, the fundamental graph isomorphism problem is intimately connected to group theory. Geometric complexity theory postulates that symmetries and representation theory will play a key role in unveiling the mysteries of computational complexity. The talk is planned as a gentle introduction to representation theory, explaining the motivations, the key concepts and some of the key theorems. The main focus will be on the representations of the tori (C^*)^n and the general linear groups (with some explanations on the connections with the symmetric groups). In particular, we aim at explaining the basics of the powerful theory of highest weight vectors, which leads to a concrete description of representations, and which is the starting point of many investigations in geometric complexity theory and quantum information theory.