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147
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.
Heterotic compactification, an algorithmic approach
 arXiv:hepth/0702210. – “Complete Intersections, Monads and Heterotic Compactification
"... We approach string phenomenology from the perspective of computational algebraic geometry, by providing new and efficient techniques for proving stability and calculating particle spectra in heterotic compactifications. This is done in the context of complete intersection CalabiYau manifolds in a s ..."
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Cited by 14 (8 self)
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We approach string phenomenology from the perspective of computational algebraic geometry, by providing new and efficient techniques for proving stability and calculating particle spectra in heterotic compactifications. This is done in the context of complete intersection CalabiYau manifolds in a single projective space where we classify positive monad bundles. Using a combination of analytic methods and computer algebra we prove stability for all such bundles and compute the complete particle spectrum, including gauge singlets. In particular, we find that the number of antigenerations vanishes
Extraspecial 2groups and images of braid group representations
 J. Knot Theory Ramifications
"... Abstract. We investigate a family of (reducible) representations of the braid groups Bn corresponding to a specific solution to the YangBaxter equation. The images of Bn under these representations are finite groups, and we identify them precisely as extensions of extraspecial 2groups. The decompo ..."
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Cited by 13 (6 self)
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Abstract. We investigate a family of (reducible) representations of the braid groups Bn corresponding to a specific solution to the YangBaxter equation. The images of Bn under these representations are finite groups, and we identify them precisely as extensions of extraspecial 2groups. The decompositions of the representations into their irreducible constituents are determined, which allows us to relate them to the wellknown Jones representations of Bn factoring over TemperleyLieb algebras and the corresponding link invariants. 1.
Combinatorial representation theory
 in New Perspectives in Algebraic Combinatorics (Berkeley, CA, 1996–97), MSRI Publ. 38
, 1999
"... Abstract. We survey the field of combinatorial representation theory, describe the main results and main questions and give an update of its current status. Answers to the main questions are given in Part I for the fundamental structures, Sn and GL(n, �), and later for certain generalizations, when ..."
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Cited by 13 (0 self)
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Abstract. We survey the field of combinatorial representation theory, describe the main results and main questions and give an update of its current status. Answers to the main questions are given in Part I for the fundamental structures, Sn and GL(n, �), and later for certain generalizations, when known. Background material and more specialized results are given in a series of appendices. We give a personal view of the field while remaining aware that there is much important and beautiful work that we have been unable to mention.
The generalized triangle inequalities for rank 3 symmetric spaces of noncompact type
 Contemp. Math
"... We compute the generalized triangle inequalities explicitly for all rank 3 symmetric spaces of noncompact type. For SL(4, C) there are 50 inequalities none of them redundant by [KTW]. For both Sp(6, C) and Spin(7, C) there are 135 inequalities of which 24 are trivially redundant in the sense that th ..."
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Cited by 13 (11 self)
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We compute the generalized triangle inequalities explicitly for all rank 3 symmetric spaces of noncompact type. For SL(4, C) there are 50 inequalities none of them redundant by [KTW]. For both Sp(6, C) and Spin(7, C) there are 135 inequalities of which 24 are trivially redundant in the sense that they follow from the inequalities defining the Weyl chamber ∆. There are 9 more redundant inequalities for each of these two groups. One interesting feature is that these inequalities do not occur for the other system (and consequently must be redundant because the two polyhedral cones are the same by Theorem 1.8). The two equal polyhedral cones D3(B3) = D3(C3) have precisely 102 facets and 51 generators (edges). 1
THE MODULARITY CONJECTURE FOR RIGID CALABI–YAU Threefolds Over Q
, 2000
"... We formulate the modularity conjecture for rigid Calabi–Yau threefolds defined over the field Q of rational numbers. We establish the modularity for the rigid Calabi–Yau threefold arising from the root lattice A3. Our proof is based on geometric analysis. ..."
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Cited by 12 (1 self)
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We formulate the modularity conjecture for rigid Calabi–Yau threefolds defined over the field Q of rational numbers. We establish the modularity for the rigid Calabi–Yau threefold arising from the root lattice A3. Our proof is based on geometric analysis.
EXISTENCE OF GLOBAL INVARIANT JET DIFFERENTIALS ON PROJECTIVE HYPERSURFACES OF High Degree
, 2008
"... Let X ⊂ P n+1 be a smooth complex projective hypersurface. In this paper we show that, if the degree of X is large enough, then there exist global sections of the bundle of invariant jet differentials of order n on X, vanishing on an ample divisor. We also prove a logarithmic version, effective in ..."
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Cited by 12 (2 self)
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Let X ⊂ P n+1 be a smooth complex projective hypersurface. In this paper we show that, if the degree of X is large enough, then there exist global sections of the bundle of invariant jet differentials of order n on X, vanishing on an ample divisor. We also prove a logarithmic version, effective in low dimension, for the logpair (P n, D), where D is a smooth irreducible divisor of high degree. Moreover, these result are sharp, i.e. one cannot have such jet differentials of order less than n.
Differential equations on complex projective hypersurfaces of low dimension
 Compos. Math
"... Abstract. In this paper we prove that every holomorphic entire curve in a smooth projective hypersurface of degree d≥329 in P 5 must satisfy an algebraic differential equation of order 4. Moreover we show that there is no such algebraic differential equations of order less than n for a smooth hypers ..."
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Cited by 12 (5 self)
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Abstract. In this paper we prove that every holomorphic entire curve in a smooth projective hypersurface of degree d≥329 in P 5 must satisfy an algebraic differential equation of order 4. Moreover we show that there is no such algebraic differential equations of order less than n for a smooth hypersurface in P n+1. 1.
Vertices of GelfandTsetlin polytopes
 Discrete Comput. Geom
"... This paper is dedicated to Louis Billera on the occasion of his sixtieth birthday. Abstract: This paper is a study of the polyhedral geometry of GelfandTsetlin patterns arising in the representation theory gl nC and algebraic combinatorics. We present a combinatorial characterization of the vertice ..."
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Cited by 12 (4 self)
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This paper is dedicated to Louis Billera on the occasion of his sixtieth birthday. Abstract: This paper is a study of the polyhedral geometry of GelfandTsetlin patterns arising in the representation theory gl nC and algebraic combinatorics. We present a combinatorial characterization of the vertices and a method to calculate the dimension of the lowestdimensional face containing a given GelfandTsetlin pattern. As an application, we disprove a conjecture of Berenstein and Kirillov [1] about the integrality of all vertices of the GelfandTsetlin polytopes. We can construct for each n ≥ 5 a counterexample, with arbitrarily increasing denominators as n grows, of a nonintegral vertex. This is the first infinite family of nonintegral polyhedra for which the Ehrhart counting function is still a polynomial. We also derive a bound on the denominators for the nonintegral vertices when n is fixed. 1
Preprojective algebras and cluster algebras, Trends in representation theory of algebras and related topics, EMS
, 2008
"... Abstract. We use the representation theory of preprojective algebras to construct and study certain cluster algebras related to semisimple algebraic groups. ..."
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Cited by 11 (0 self)
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Abstract. We use the representation theory of preprojective algebras to construct and study certain cluster algebras related to semisimple algebraic groups.