Showing papers by "Joseph M. Landsberg published in 2022"

Concise tensors of minimal border rank

Abstract: We determine defining equations for the set of concise tensors of minimal border rank in $${\mathbb {C}}^m{\mathord { \otimes } }{\mathbb {C}}^m{\mathord { \otimes } }{\mathbb {C}}^m$$ C m ⊗ C m ⊗ C m when $$m=5$$ m = 5 and the set of concise minimal border rank $$1_*$$ 1 ∗ -generic tensors when $$m=5,6$$ m = 5 , 6 . We solve the classical problem in algebraic complexity theory of classifying minimal border rank tensors in the special case $$m=5$$ m = 5 . Our proofs utilize two recent developments: the 111-equations defined by Buczyńska–Buczyński and results of Jelisiejew–Šivic on the variety of commuting matrices. We introduce a new algebraic invariant of a concise tensor, its 111-algebra, and exploit it to give a strengthening of Friedland’s normal form for 1-degenerate tensors satisfying Strassen’s equations. We use the 111-algebra to characterize wild minimal border rank tensors and classify them in $${\mathbb {C}}^5{\mathord { \otimes } }{\mathbb {C}}^5{\mathord { \otimes } }{\mathbb {C}}^5$$ C 5 ⊗ C 5 ⊗ C 5 .

Secant varieties and the complexity of matrix multiplication

Abstract: . This is a survey primarily about determining the border rank of tensors, especially those relevant for the study of the complexity of matrix multiplication. This is a subject that on the one hand is of great significance in theoretical computer science, and on the other hand touches on many beautiful topics in algebraic geometry such as classical and recent results on equations for secant varieties (e.g., via vector bundle and representation-theoretic methods) and the geometry and deformation theory of zero dimensional schemes.

Equivariant spaces of matrices of constant rank

Abstract: . We use representation theory to construct spaces of matrices of constant rank . These spaces are parametrized by the natural representation of the general linear group or the symplectic group. We present variants of this idea, with more complicated representations, and others with the orthogonal group. Our spaces of matrices correspond to vector bundles which are homogeneous but sometimes admit deformations to non-homogeneous vector bundles, showing that these spaces of matrices sometimes admit large families of deformations.