We establish basic techniques for studying 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.

Abstract. We survey results in algebraic complexity theory, focusing on matrix multiplication. Our goals are (i) to show how open questions in algebraic complexity theory are naturally posed as questions in geometry and representation theory, (ii) to motivate researchers to work on these questions, and (iii) to point out relations with more general problems in geometry. The key geometric objects for our study are the secant varieties of Segre varieties. We explain how these varieties are also useful for algebraic statistics, the study of phylogenetic invariants, and quantum computing.

In a remarkable paper Mats Boij and Jonas Söderberg [2006] have conjectured that the Betti table of a Cohen-Macaulay module over a polynomial ring can be decomposed in a certain way as a positive linear combination of Betti tables of modules with pure resolutions. We prove, in characteristic zero, a strengthened form of their conjecture. Applications include a proof (in characteristic zero) of the Multiplicity Conjecture of Huneke and Srinivasan, a proof of the convexity of a fan naturally associated to the Young lattice, and bounds on the possible cohomology modules of vector bundles on projective spaces.

Abstract. We find minimal generators for the ideals of secant varieties of Segre varieties in the cases of σk(P 1 × P n × P m) for all k, n, m, σ2(P n × P m × P p × P r) for all n, m, p, r (GSS conjecture for four factors), and σ3(P n ×P m ×P p) for all n, m, p and prove they are normal with rational singularities in the first case and arithmetically Cohen-Macaulay in the second two. 1.

Abstract. 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. 1.

Abstract. We determine which tangential varieties of homogeneously embedded rational homogeneous varieties are spherical. We determine the homogeneous coordinate rings and rings of covariants of the tangential varieties of homogenously embedded compact Hermitian symmetric spaces (CHSS). We give bounds on the degrees of generators of the ideals of tangential varieties of CHSS and obtain more explicit infomation about the ideals in certain cases. 1.

Abstract. We show that the secant varieties of rank three compact Hermitian symmetric spaces in their minimal homogeneous embeddings are normal, with rational singularities. We show that their ideals are generated in degree three- with one exception, the secant variety of the 21-dimensional spinor variety in P 63, whose ideal is generated in degree four. We also discuss the coordinate ring of secant varieties of compact Hermitian symmetric spaces. 1.

It is well-known that the intersection multiplicities of Schubert classes in the Grassmanian are Littlewood-Richardson coefficients. For a partition λ inside a r × (n − r) rectangle, let Yλ be the Schubert variety inside the Grassmanian Grass(r, n) corresponding to λ and let [Yλ] be its cohomology class. We have

Abstract. We discuss the geometry of orbit closures and the asymptotic behavior of Kronecker coefficients in the context of the Geometric Complexity Theory program to prove a variant of Valiant’s algebraic analog of the P ̸ = NP conjecture. We also describe the precise separation of complexity classes that their program proposes to demonstrate.

Abstract. We investigate differences between X-rank and X-border rank, focusing on the cases of tensors and partially symmetric tensors. As an aid to our study, and as an object of interest in its own right, we define notions of X-rank and border rank for a linear subspace. Results include determining and bounding the maximum X-rank of points in several cases of interest. 1.