Motivated by problems in algebraic complexity theory (e.g., matrix multiplication) and extremal combinatorics (e.g., the cap set problem and the sunflower problem), we introduce the geometric rank as a new tool in the study of tensors and hypergraphs. We prove that the geometric rank is an upper bound on the subrank of tensors and the independence number of hypergraphs. We prove that the geometric rank is smaller than the slice rank of Tao, and relate geometric rank to the analytic rank of Gowers and Wolf in an asymptotic fashion. As a first application, we use geometric rank to prove a tight upper bound on the (border) subrank of the matrix multiplication tensors, matching Strassen's well-known lower bound from 1987.

翻译：受代数复杂度理论（如矩阵乘法）和极值组合学（如帽集问题和向日葵问题）中问题的启发，我们引入几何秩作为研究张量和超图的新工具。我们证明了几何秩是张量的次秩和超图的独立数的上界。我们证明了几何秩小于陶哲轩的切片秩，并以渐近方式将几何秩与高尔斯和沃尔夫的分析秩联系起来。作为首个应用，我们利用几何秩证明了矩阵乘法张量的（边界）次秩的紧上界，与斯特拉森1987年著名的下界相匹配。