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32
Combinatorial secant varieties
 Quart. J. Pure Applied Math
"... Abstract. The construction of joins and secant varieties is studied in the combinatorial context of monomial ideals. For ideals generated by quadratic monomials, the generators of the secant ideals are obstructions to graph colorings, and this leads to a commutative algebra version of the Strong Per ..."
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Cited by 16 (1 self)
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Abstract. The construction of joins and secant varieties is studied in the combinatorial context of monomial ideals. For ideals generated by quadratic monomials, the generators of the secant ideals are obstructions to graph colorings, and this leads to a commutative algebra version of the Strong Perfect Graph Theorem. Given any projective variety and any term order, we explore whether the initial ideal of the secant ideal coincides with the secant ideal of the initial ideal. For toric varieties, this leads to the notion of delightful triangulations of convex polytopes. 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 15 (1 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.
Performance of a New Invariants Method on Homogeneous and Nonhomogeneous Quartet Trees
, 2006
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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.
The strand symmetric model
, 2005
"... This chapter is devoted to the study of strand symmetric Markov models on trees from the standpoint of algebraic statistics. By a strand symmetric Markov model, we mean one whose mutation structure reflects the symmetry induced by the doublestranded structure of DNA. In particular, a strand ..."
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Cited by 8 (4 self)
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This chapter is devoted to the study of strand symmetric Markov models on trees from the standpoint of algebraic statistics. By a strand symmetric Markov model, we mean one whose mutation structure reflects the symmetry induced by the doublestranded structure of DNA. In particular, a strand
Toric fiber products
"... Abstract. We introduce and study the toric fiber product of two ideals in polynomial rings that are homogeneous with respect to the same multigrading. Under the assumption that the set of degrees of the variables form a linearly independent set, we can explicitly describe generating sets and Gröbner ..."
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Cited by 7 (1 self)
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Abstract. We introduce and study the toric fiber product of two ideals in polynomial rings that are homogeneous with respect to the same multigrading. Under the assumption that the set of degrees of the variables form a linearly independent set, we can explicitly describe generating sets and Gröbner bases for these ideals. This allows us to unify and generalize some results in algebraic statistics. 1.
Fourier transform inequalities for phylogenetic trees
, 2008
"... Phylogenetic invariants are not the only constraints on sitepattern frequency vectors for phylogenetic trees. A mutation matrix, by its definition, is the exponential of a matrix with nonnegative offdiagonal entries; this nonnegativity requirement implies nontrivial constraints on the sitepatt ..."
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Cited by 5 (0 self)
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Phylogenetic invariants are not the only constraints on sitepattern frequency vectors for phylogenetic trees. A mutation matrix, by its definition, is the exponential of a matrix with nonnegative offdiagonal entries; this nonnegativity requirement implies nontrivial constraints on the sitepattern frequency vectors. We call these additional constraints “edgeparameter inequalities. ” In this paper, we first motivate the edgeparameter inequalities by considering a pathological sitepattern frequency vector corresponding to a quartet tree with a negative internal edge. This sitepattern frequency vector nevertheless satisfies all of the constraints described up to now in the literature. We next describe two complete sets of edgeparameter inequalities for the groupbased models; these constraints are squarefree monomial inequalities in the Fourier transformed coordinates. These inequalities, along with the phylogenetic invariants, form a complete description of the set of sitepattern frequency vectors corresponding to bona fide trees. Said in mathematical language, this paper explicitly presents two finite lists of inequalities in Fourier coordinates of the form “monomial ≤ 1, ” each list characterizing the phylogenetically relevant semialgebraic subsets of the phylogenetic varieties.
Most tensor problems are NP hard
 CORR
, 2009
"... The idea that one might extend numerical linear algebra, the collection of matrix computational methods that form the workhorse of scientific and engineering computing, to numerical multilinear algebra, an analogous collection of tools involving hypermatrices/tensors, appears very promising and has ..."
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The idea that one might extend numerical linear algebra, the collection of matrix computational methods that form the workhorse of scientific and engineering computing, to numerical multilinear algebra, an analogous collection of tools involving hypermatrices/tensors, appears very promising and has attracted a lot of attention recently. We examine here the computational tractability of some core problems in numerical multilinear algebra. We show that tensor analogues of several standard problems that are readily computable in the matrix (i.e. 2tensor) case are NP hard. Our list here includes: determining the feasibility of a system of bilinear equations, determining an eigenvalue, a singular value, or the spectral norm of a 3tensor, determining a best rank1 approximation to a 3tensor, determining the rank of a 3tensor over R or C. Hence making tensor computations feasible is likely to be a challenge.
Open problems in algebraic statistics
"... Abstract. Algebraic statistics is concerned with the study of probabilistic models and techniques for statistical inference using methods from algebra and geometry. This article presents a list of open mathematical problems in this emerging field, with main emphasis on graphical models with hidden v ..."
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Cited by 4 (1 self)
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Abstract. Algebraic statistics is concerned with the study of probabilistic models and techniques for statistical inference using methods from algebra and geometry. This article presents a list of open mathematical problems in this emerging field, with main emphasis on graphical models with hidden variables, maximum likelihood estimation, and multivariate Gaussian distributions. These are notes from a lecture presented at the IMA in Minneapolis during the 2006/07 program on Applications of Algebraic Geometry. Key words. Algebraic statistics, contingency tables, hidden variables, Schur modules,
Geometry of the Kimura 3parameter model
 Adv. Appl. Math
"... Abstract. The Kimura 3parameter model on a tree of n leaves is one of the most used in phylogenetics. The affine algebraic variety W associated to it is a toric variety. We study its geometry and we prove that it is isomorphic to a geometric quotient of the affine space by a finite group acting on ..."
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Abstract. The Kimura 3parameter model on a tree of n leaves is one of the most used in phylogenetics. The affine algebraic variety W associated to it is a toric variety. We study its geometry and we prove that it is isomorphic to a geometric quotient of the affine space by a finite group acting on it. As a consequence, we are able to study the singularities of W and prove that the biologically meaningful points are smooth points. Then we give an algorithm for constructing a set of minimal generators of the localized ideal at these points, for an arbitrary number of leaves n. This leads to a major improvement of phylogenetic reconstruction methods based on algebraic geometry. 1.