90 citations · 333 across the 51 of their papers we have counts for
31 papers · 1 filter
On Linear spaces of of matrices bounded rank
Hang Huang, J. M. Landsberg
Motivated by questions in theoretical computer science and quantum information theory, we study the classical problem of determining linear spaces of matrices of bounded rank. Spac…
Equivariant spaces of matrices of constant rank
J. M. Landsberg, L. Manivel
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 symple…
Secant varieties and the complexity of matrix multiplication
J. M. Landsberg
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…
Algebraic Geometry and Representation theory in the study of matrix multiplication complexity and other problems in theoretical computer science
J. M. Landsberg
Many fundamental questions in theoretical computer science are naturally expressed as special cases of the following problem: Let be a complex reductive group, let be a …
New lower bounds for matrix multiplication and the 3x3 determinant
Austin Conner, Alicia Harper, J. M. Landsberg
Let denote the matrix multiplication tensor (and write ) and let $det_3\in ( C^9)^{\o…
Towards a Geometric Approach to Strassen's Asymptotic Rank Conjecture
Austin Conner, Fulvio Gesmundo, Joseph M. Landsberg +2
We make a first geometric study of three varieties in (for each ), including the Zariski closure of the set of tight ten…
The method of shifted partial derivatives cannot separate the permanent from the determinant
Klim Efremenko, J. M. Landsberg, Hal Schenck +1
The method of shifted partial derivatives was used to prove a super-polynomial lower bound on the size of depth four circuits needed to compute the permanent. We show that this met…
An explicit description of the irreducible components of the set of matrix pencils with bounded normal rank
Fernando De Terán, Froilán M. Dopico, J. M. Landsberg
The set of mxn singular matrix pencils with normal rank at most r is an algebraic set with r+1 irreducible components. These components are the closure of the orbits (under strict…
Geometric Complexity Theory: an introduction for geometers
J. M. Landsberg
This article is a survey of recent developments in, and a tutorial on, the approach to P v. NP and related questions called Geometric Complexity Theory (GCT). It is written to be a…
Equations for secant varieties of Veronese and other varieties
J. M. Landsberg, Giorgio Ottaviani
New classes of modules of equations for secant varieties of Veronese varieties are defined using representation theory and geometry. Some old modules of equations (catalecticant mi…
On the third secant variety
Jarosław Buczyński, J. M. Landsberg
We determine normal forms and ranks of tensors of border rank at most three. We present a differential-geometric analysis of limits of secant planes in a more general context. In p…
Determinantal equations for secant varieties and the Eisenbud-Koh-Stillman conjecture
Jarosław Buczyński, Adam Ginensky, J. M. Landsberg
We address special cases of a question of Eisenbud on the ideals of secant varieties of Veronese re-embeddings of arbitrary varieties. Eisenbud's question generalizes a conjecture…
Ranks of tensors and a generalization of secant varieties
Jarosław Buczyński, J. M. Landsberg
We introduce subspace rank as a tool for studying ranks of tensors and X-rank more generally. We derive a new upper bound for the rank of a tensor and determine the ranks of partia…
Kruskal's theorem
J. M. Landsberg
This is just a short proof of Kruskal's theorem regarding uniqueness of expressions for tensors, phrased in geometric language.
On the ranks and border ranks of symmetric tensors
J. M. Landsberg, Zach Teitler
Motivated by questions arising in signal processing, computational complexity, and other areas, we study the ranks and border ranks of symmetric tensors using geometric methods. We…
On the Debarre-de Jong and Beheshti-Starr conjectures on hypersurfaces with too many lines
J. M. Landsberg, Orsola Tommasi
We show that the Debarre-de Jong conjecture that the Fano scheme of lines on a smooth hypersurface of degree at most n in n-dimensional projective space must have its expected dime…
Lines on hypersurfaces
J. M. Landsberg, C. Robles
This is a detailed study of the infinitesimal variation of the variety of lines through a point of a low degree hypersurface in pro jective space. The motion is governed by a syste…
Exterior differential systems, Lie algebra cohomology, and the rigidity of homogenous varieties
J. M. Landsberg
These are expository notes from the 2008 Srni Winter School. They have two purposes: (1) to give a quick introduction to exterior differential systems (EDS), which is a collection…
On the ideals and singularities of secant varieties of Segre varieties
J. M. Landsberg, Jerzy Weyman
We find generators for the ideals of secant varieties of Segre varieties in several the cases including the Garcia-Stillmann-Ssturmfels conjecture for four factors and prove result…
Fubini's Theorem in codimension two
J. M. Landsberg, Colleen Robles
We classify codimension two analytic submanifolds X of projective space having the property that any line through a general point p having contact to order two with X at p automati…
Legendrian varieties
J. M. Landsberg, L. Manivel
We investigate the geometry of Legendrian complex projective manifolds $X\subset\PP V$. By definition, this means is a complex vector space of dimension , endowed with a…
Series of nilpotent orbits
J. M. Landsberg, L. Manivel, B. W. Westbury
We organize the nilpotent orbits in the exceptional complex Lie algebras into series using the triality model and show that within each series the dimension of the orbit is a linea…
Griffiths-Harris rigidity of compact Hermitian symmetric spaces
J. M. Landsberg
I show that any complex manifold that resembles a rank two compact Hermitian symmetric space (other than a quadric hypersurface) to order two at a general point must be an open sub…
Series of Lie Groups
J. M. Landsberg, L. Manivel
For various series of complex semi-simple Lie algebras $\fg (t)$ equipped with irreducible representations , we decompose the tensor powers of into irreducible factors…