Showing papers by "Joseph M. Landsberg published in 2021"
TL;DR: In this paper, a geometric framework for Strassen's asymptotic rank conjecture that the rank of any tight tensor is minimal was developed. And the dimension of the set of tight tensors with continuous regular symmetry was shown to be the same as that of oblique tensors.
Abstract: We make a first geometric study of three varieties in $$\mathbb {C}^m\otimes \mathbb {C}^m\otimes \mathbb {C}^m$$
(for each m), including the Zariski closure of the set of tight tensors, the tensors with continuous regular symmetry. Our motivation is to develop a geometric framework for Strassen’s asymptotic rank conjecture that the asymptotic rank of any tight tensor is minimal. In particular, we determine the dimension of the set of tight tensors. We prove that this dimension equals the dimension of the set of oblique tensors, a less restrictive class introduced by Strassen.
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TL;DR: In this article, the complexity of matrix multiplication has been studied as a special case of the problem of determining if a matrix is in the $G$-orbit closure of a matrix.
Abstract: Many fundamental questions in theoretical computer science are naturally expressed as special cases of the following problem: Let $G$ be a complex reductive group, let $V$ be a $G$-module, and let $v,w$ be elements of $V$. Determine if $w$ is in the $G$-orbit closure of $v$. I explain the computer science problems, the questions in representation theory and algebraic geometry that they give rise to, and the new perspectives on old areas such as invariant theory that have arisen in light of these questions. I focus primarily on the complexity of matrix multiplication.