MSRI- Geometry and Representation Theory of Tensors for Computer Science, Statistics, and other areas
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Title: Phylogenetic algebraic geometry
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Title: Constructibility of the set of tensors of a given rank
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Title: § 4.6,7,8 Highest weight vectors, bases of highest weight space. Ideals of Segre, Veronese varieties and homogeneous varieties in general, decomposing S^d(A_1\otimes \cdots \otimes A_n), characters.
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Title: Toric varieties, toric ideals, moment map, exponential families.
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Title: Conditioning, computations, applications
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Title: § 4.3,4,5 - Representations of the symmetric group, Young diagrams, Young symmetrizers and wiring diagrams. Using these tools to decompose V^{\otimes d} as a GL(V) module. Schur-Weyl Duality.
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Title: Finish Chap 3 - Terracini's lemma cont'd and applications to computing the dimension of secant varieties. The geometric definition of border rank, projective second fundamental form.
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Title: What is quantum information theory?
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Title: Notions of tensor ranks: rank, border rank, multilinear rank, nonnegative rank
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Title: § 3.3,4,5,6 Tangent spaces to varieties, joins, cones, secant varieties, their dimension, Terracini's lemma.
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Title: Finish Ch. 2: skew-symmetric tensors, equations for rank at most r linear mappings, border rank, decomposing V^{\ot 3}., G-modules, isotypic components. § 4.1,2 Representations, Schur's Lemma, G-modules and decomposing spaces of tensors
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Title: Tensor approximations
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Title: Algebraic varieties § 3.1, 3.2. Basic definitions from algebraic geometry: projective space, variety, ideal, Zariski topology. Segre, Veronese, and other examples of varieties. Graphical models and motivating examples in statistics and information t
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Title: Complexity of matrix multiplication, an overview of Ch. 2 including tensors, rank of tensors, and wiring diagrams.
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