Border Rank Lower Bounds of the Matrix Multiplication Tensor - Part 2

Learn advanced techniques for establishing lower bounds on the border rank of matrix multiplication tensors in this mathematical lecture. Explore two powerful methods that have proven successful for analyzing matrix multiplication complexity: Koszul flattenings, which associates matrices to tensors of interest and relates their respective ranks, and border apolarity, which refutes the existence of auxiliary data required for border rank decompositions. Discover how the natural symmetry of matrix multiplication problems plays a crucial role in both techniques, with border apolarity particularly leveraging the large symmetry group of matrix multiplication tensors to normalize and eliminate potential auxiliary data. Examine the connection between these lower bound techniques and the fundamental question of understanding ω (the matrix multiplication exponent) through rank and border rank analysis of matrix multiplication tensors, building on foundational results from Strassen and Bini. Gain insight into how symmetry invariance under basis changes in tensor factors makes these methods both theoretically sound and practically applicable to one of the most important open problems in computational complexity theory.