J. M. Landsberg

On the Ranks and Border Ranks of Symmetric Tensors

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 provide improved lower bounds for the rank of a symmetric tensor (i.e., a homogeneous polynomial) obtained by considering the singularities of the hypersurface defined by the polynomial. We obtain normal forms for polynomials of border rank up to five, and compute or bound the ranks of several classes of polynomials, including monomials, the determinant, and the permanent.

On the Ideals of Secant Varieties of Segre Varieties

Abstract We establish basic techniques for determining the ideals of secant varieties of Segre varieties.We solve a conjecture of Garcia, Stillman, and Sturmfels on the generators of the ideal of the first secant variety in the case of three factors and solve the conjecture set-theoretically for an arbitrary number of factors. We determine the low degree components of the ideals of secant varieties of small dimension in a few cases.

Geometry and the complexity of matrix multiplication

It has been known for some time that algebraic geometry and representation theory are useful for proving lower bounds on the complexity of the matrix multiplication operator. In this talk I will explain how geometry can be used to prove both upper and lower bounds. After a discussion of the problem and its history, I will present very recent work with M. Michalek on border rank algorithms. Faculty Host: Eric Allender

Determinantal equations for secant varieties and the Eisenbud–Koh–Stillman conjecture

We address special cases of a question of Eisenbud on the ideals of secant varieties of Veronese re-embeddings of arbitrary varieties. Eisenbuds question generalizes a conjecture of Eisenbud, Koh and Stillman (EKS) for curves. We prove that set-theoretic equations of small secant varieties to a high degree Veronese re-embedding of a smooth variety are determined by equations of the ambient Veronese variety and linear equations. However this is false for singular varieties, and we give explicit counter-examples to the EKS conjecture for singular curves. The techniques we use also allow us to prove a gap and uniqueness theorem for symmetric tensor rank. We put Eisenbuds question in a more general context about the behaviour of border rank under specialisation to a linear subspace, and provide an overview of conjectures coming from signal processing and complexity theory in this context.

Ranks of tensors and a generalization of secant varieties

Abstract 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 partially symmetric tensors in C 2 ⊗ C b ⊗ C b . We review the literature from a geometric perspective.

New lower bounds for the border rank of matrix multiplication

The border rank of the matrix multiplication operator for n by n matrices is a standard measure of its complexity. Using techniques from algebraic geometry and representation theory, we show the border rank is at least 2n^2-n. Our bounds are better than the previous lower bound (due to Lickteig in 1985) of 3/2 n^2+ n/2 -1 for all n>2. The bounds are obtained by finding new equations that bilinear maps of small border rank must satisfy, i.e., new equations for secant varieties of triple Segre products, that matrix multiplication fails to satisfy.

Generalizations of Strassen's Equations for Secant Varieties of Segre Varieties

We define many new examples of modules of equations for secant varieties of Segre varieties that generalize Strassens commutation equations (Strassen, 1988). Our modules of equations are obtained by constructing subspaces of matrices from tensors that satisfy various commutation properties.

The border rank of the multiplication of 2×2 matrices is seven

One of the leading problems of algebraic complexity theory is matrix multiplication. The naive multiplication of two n× n matrices uses n multiplications. In 1969, Strassen [20] presented an explicit algorithm for multiplying 2 × 2 matrices using seven multiplications. In the opposite direction, Hopcroft and Kerr [12] and Winograd [22] proved independently that there is no algorithm for multiplying 2×2 matrices using only six multiplications. The precise number of multiplications needed to execute matrix multiplication (or any given bilinear map) is called the rank of the bilinear map. A related problem is to determine the border rank of matrix multiplication (or any given bilinear map), first introduced in [6, 5]. Roughly speaking, some bilinear maps may be approximated with arbitrary precision by less complicated bilinear maps and the border rank of a bilinear map is the complexity of arbitrarily small “good” perturbations of the map. These perturbed maps can give rise to fast exact algorithms for matrix multiplication; see [7]. The border rank made appearances in the literature in the 1980s and early 1990s (see, e.g., [6, 5, 19, 8, 15, 2, 10, 11, 4, 3, 17, 16, 18, 1, 9, 21]), but to our knowledge there has not been much progress on the question since then. More precisely, for any complex projective variety X ⊂ CP = PV and point p ∈ PV , define the X-rank of p to be the smallest number r such that p is in the linear span of r points of X. Define σr(X), the r-th secant variety of X, to be the Zariski closure of the set of points of X-rank r, and define the X-border rank of p to be the smallest r such that p ∈ σr(X). The terminology is motivated by the case X = Seg(Pa−1 × Pb−1) ⊂ P(C ⊗ C), the Segre variety of rank one matrices. Then the X-rank of a matrix is just its usual rank. Let A∗, B∗, C be vector spaces and let f : A∗ ×B∗ → C be a bilinear map, i.e., an element of A⊗B⊗C. Let X = Seg(PA×PB×PC) ⊂ P(A⊗B⊗C) denote the Segre variety of decomposable tensors in A⊗B⊗C. The border rank of a bilinear map is its X-border rank. While for the Segre product of two projective spaces, border rank coincides with rank, here they can be quite different. In this paper we prove the theorem stated in the title. Let MMult ∈ C ⊗C ⊗ C denote the matrix multiplication operator for 2 × 2 matrices. Strassen [18] showed that MMult / ∈ σ5(P × P × P). Our method of proof is to decompose

Representation Theory and Projective Geometry

This article consists of three parts that are largely independent of one another. The first part deals with the projective geometry of homogeneous varieties, in particular their secant and tangential varieties. It culminates with an elementary construction of the compact Hermitian symmetric spaces and the closed orbits in the projectivization of the adjoint representation of a simple Lie algebra. The second part discusses division algebras, triality, Jordan agebras and the Freudenthal magic square. The third part describes work of Deligne and Vogel inspired by knot theory and several perspectives for understanding this work.

Hypersurfaces with degenerate duals and the Geometric Complexity Theory Program

We determine set-theoretic defining equations for the variety of hypersurfaces of degree d in an N-dimensional complex vector space that have dual variety of dimension at most k. We apply these equations to the Mulmuley-Sohoni variety, the GL_{n^2} orbit closure of the determinant, showing it is an irreducible component of the variety of hypersurfaces of degree