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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 35 (5 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 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 14 (3 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 11 (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.
Noncommutative desingularization of determinantal varieties
 I.” Inventiones Mathematicae
, 2010
"... In our paper “Noncommutative desingularization of determinantal varieties I”, we constructed and studied noncommutative resolutions of determinantal varieties defined by maximal minors. At the end of the introduction, we asserted that the results could be generalized to determinantal varieties de ..."
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Cited by 7 (1 self)
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In our paper “Noncommutative desingularization of determinantal varieties I”, we constructed and studied noncommutative resolutions of determinantal varieties defined by maximal minors. At the end of the introduction, we asserted that the results could be generalized to determinantal varieties defined by nonmaximal minors, at least in characteristic zero. In this paper, we prove the existence of noncommutative resolutions in the general case in a manner which is still characteristic free, and carry out the explicit description by generators and relations in characteristic zero. As an application of our results, we prove that there is a fully faithful embedding between the bounded derived categories of the two canonical (commutative) resolutions of a determinantal variety, confirming a wellknown conjecture of Bondal and Orlov in this special case. 1
ON THE HOLONOMIC RANK PROBLEM
, 1302
"... Abstract. A tautological system, introduced in [15][16], arises as a regular holonomic system of partial differential equations that govern the period integrals of a family of complete intersections in a complex manifold X, equipped with a suitable Lie group action. In this article, we introduce two ..."
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Cited by 2 (1 self)
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Abstract. A tautological system, introduced in [15][16], arises as a regular holonomic system of partial differential equations that govern the period integrals of a family of complete intersections in a complex manifold X, equipped with a suitable Lie group action. In this article, we introduce two formulas – one purely algebraic, the other geometric – to compute the rank of the solution sheaf of such a system for CY hypersurfaces in a generalized flag variety. The algebraic version gives the local solution space as a Lie algebra homology group, while the geometric one as the middle de Rham cohomology of the complement of a hyperplane section in X. We use both formulas to find certain degenerate points for which the rank of the solution sheaf becomes 1. These rank 1 points appear to be good candidates for the socalled large complex structure limits in mirror symmetry. The formulas are also used to prove a conjecture of Hosono, Lian and Yau on the completeness of the extended GKZ system when X is Pn. Contents
Workshop on Syzygies in Algebraic Geometry, with an exploration of a connection with String Theory
, 2012
"... Free resolutions are often naturally attached to geometric objects. A question of prime interest has been to understand what constraints the geometry of a variety imposes on the corresponding Betti numbers and structure of the resolution. In recent years, free resolutions have been also impressively ..."
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Free resolutions are often naturally attached to geometric objects. A question of prime interest has been to understand what constraints the geometry of a variety imposes on the corresponding Betti numbers and structure of the resolution. In recent years, free resolutions have been also impressively used to study the birational geometry of moduli spaces of curves. One of the most exciting and challenging longstanding open conjectures on free resolutions is the Regularity EisenbudGoto Conjecture that the regularity of a prime ideal is bounded above by its multiplicity. The conjecture has roots in Castelnuovo’s work, and is known to hold in only a few cases: the CohenMacaulay case, for curves, and for smooth surfaces. In the 80’s, Mark Green [11], [12] conjectured that the the minimal resolution of the canonical ring a nonhyperelliptic curve X of genus g satisfies the Np property if and only if the Clifford index of X is greater than P. Recall that N1 says that the homogenous ideal of X in P g−1 is generated by quadrics. For p ≥ 2, Np means that the first p − 1 steps of the minimal resolution of the homogenous ideal of X are given by matrices of linear forms. Using geometry of the Hilbert schemes of K3 surfaces, Voisin [20], [21] showed that the Green conjecture holds for generic curves. In the last few years, by the work of Schreyer, Texidor, Aprodu and Farkas [17]. [19], [1], [2], we know that Green’s conjecture holds also in many other
GEXCEPTIONAL VECTOR BUNDLES ON P2 AND REPRESENTATIONS OF QUIVERS
"... Abstract. It is known that the category of homogeneous bundles on P2 is equivalent to the category of representations of a quiver with relation. In this paper we make use of this equivalence to describe a family of Gexceptional bundles on P2 and to prove that they are stable. We also study the Gex ..."
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Abstract. It is known that the category of homogeneous bundles on P2 is equivalent to the category of representations of a quiver with relation. In this paper we make use of this equivalence to describe a family of Gexceptional bundles on P2 and to prove that they are stable. We also study the Gexceptionality of Fibonacci bundles on P2. Contents