Geometry Seminar

Date: January 20, 2023

Time: 4:00PM - 5:00PM

Location: BLOC 302

Speaker: Tim Seynnaeve

Title: Matrix product states, geometry, and invariant theory

Abstract: Matrix product states play an important role in quantum information theory to represent states of many-body systems. They can be seen as low-dimensional subvarieties of a high-dimensional tensor space. In this talk, I will introduce two variants: homogeneous matrix product states and uniform matrix product states. Studying the linear spans of these varieties leads to a natural connection with invariant theory of matrices. For homogeneous matrix product states, a classical result on polynomial identities of matrices leads to a formula for the dimension of the linear span, in the case of 2x2 matrices. I will explain this connection, and the difficulties that arise when generalizing it to the case of uniform matrix product states. This talk is partially based on joint work with Claudia De Lazzari and Harshit Motwani.