Results 1  10
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21
ON THE IDEALS OF SECANT VARIETIES OF SEGRE VARIETIES
, 2003
"... 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 settheoretically for an arbitrary n ..."
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Cited by 39 (9 self)
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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 settheoretically 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.
AN OVERVIEW OF MATHEMATICAL ISSUES ARISING IN THE GEOMETRIC COMPLEXITY THEORY APPROACH TO VP != VNP
"... 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 t ..."
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Cited by 15 (6 self)
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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.
On the ideals and singularities of secant varieties of segre varieties
 math.AG/0601452, Bull. London Math. Soc
"... 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 rationa ..."
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Cited by 14 (4 self)
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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 CohenMacaulay in the second two. 1.
Geometry and the complexity of matrix multiplication
, 2007
"... 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, ..."
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Cited by 14 (2 self)
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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.
ON THE RANKS AND BORDER RANKS OF SYMMETRIC TENSORS
"... 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 ..."
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Cited by 9 (0 self)
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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.
On tangential varieties of rational homogeneous varieties
 Jour. Lond. Math. Soc
"... 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 boun ..."
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Cited by 5 (1 self)
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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.
ON SECANT VARIETIES OF COMPACT HERMITIAN SYMMETRIC SPACES
"... 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 21dimensional spinor v ..."
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Cited by 5 (2 self)
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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 21dimensional 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.
FImodules: a new approach to stability for Snrepresentations
, 2012
"... In this paper we introduce and develop the theory of FImodules. We apply this theory to obtain new theorems about: • the cohomology of the configuration space of ordered ntuples on an arbitrary manifold • the diagonal coinvariant algebra on r sets of n variables • the cohomology and tautological r ..."
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Cited by 5 (1 self)
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In this paper we introduce and develop the theory of FImodules. We apply this theory to obtain new theorems about: • the cohomology of the configuration space of ordered ntuples on an arbitrary manifold • the diagonal coinvariant algebra on r sets of n variables • the cohomology and tautological ring of the moduli space of npointed curves • the space of polynomials on rank varieties of n × n matrices • the subalgebra of the cohomology of the genus n Torelli group generated by H 1 and more. The symmetric group Sn acts on each of these vector spaces. In most cases almost nothing is known about the characters of these representations, or even their dimensions. We prove that in each fixed degree the character is given, for n large enough, by a polynomial in the cyclecounting functions that is independent of n. In particular, the dimension is eventually a polynomial in n. FImodules are a refinement of Church–Farb’s theory of representation stability for Snrepresentations. In this framework, a complicated sequence of Snrepresentations becomes a
Report on “Geometry and representation theory of tensors for computer science, statistics and other areas
, 2008
"... This workshop was sponsored by AIM and the NSF and it brought in participants from the US, Canada and the European Union to Palo Alto, CA to work to translate questions from quantum computing, complexity theory, statistical learning theory, signal processing, and data analysis to problems in geometr ..."
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Cited by 3 (0 self)
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This workshop was sponsored by AIM and the NSF and it brought in participants from the US, Canada and the European Union to Palo Alto, CA to work to translate questions from quantum computing, complexity theory, statistical learning theory, signal processing, and data analysis to problems in geometry and representation theory.