Kolloquien und Oberseminare Übersicht Detail

Vortragende Person/Vortragende Personen:
Yeongrak Kim

The rank of a tensor T is the minimum number of decomposable tensors whose sum equals to T which extends the notion of the matrix rank. Understanding the rank of a given tensor has great theoretical and practical applications, however, the rank of a tensor of high order is very hard to determine in most cases. For instance, Strassen's algorithm for matrix multiplication tells us that we only need 7 multiplications (not 8) when we multiply two 2 by 2 matrices, in other words, the 2 by 2 matrix multiplication tensor has rank 7. Usually, the study of rank complexities of a tensor is based on a flattening method that derives a certain matrix from the given tensor. The Koszul flattening method, introduced by Landsberg and Ottaviani, is a simple and powerful method that works for a tensor of order 3 using the exterior product. It has several applications in the study of lower bounds of tensor ranks and Waring ranks for various tensors (of order 3) appearing in algebra and geometry, including the matrix multiplication tensor and the determinant/permanent polynomial for 3 by 3 matrices. 

Motivated by their observations, I will introduce a recursive Koszul flattening method, a successive usage of Koszul flattening for tensors of higher orders. As applications, I will discuss some observations on the lower bounds on tensor ranks of the determinant and permanent as tensors of order n. These results greatly improve lower bounds on the border ranks of those tensors for n at least 4.
This is a joint work in progress with Jong In Han and Jeong-Hoon Ju.