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Dustin Cartwright

  1. The Number of Eigenvalues of a Tensor.

    Чт, 04/29/2010 - 15:01 — SomNambul
    Authors: Dustin Cartwright, Bernd Sturmfels
    Subjects: Numerical Analysis
    Abstract

    Eigenvectors of tensors, as studied recently in numerical multilinear
    algebra, correspond to fixed points of self-maps of a projective space. We
    determine the number of eigenvectors and eigenvalues of a generic tensor, and
    we show that the number of normalized eigenvalues of a symmetric tensor is
    always finite. We also examine the characteristic polynomial and how its
    coefficients are related to discriminants and resultants.

  2. Mustafin Varieties.

    Втр, 02/09/2010 - 16:01 — SomNambul
    Authors: Dustin Cartwright, Bernd Sturmfels, Mathias Häbich, Annette Werner
    Subjects: Algebraic Geometry
    Abstract

    A Mustafin variety is a degeneration of projective space induced by a point
    configuration in a Bruhat-Tits building. The special fiber is reduced and
    Cohen-Macaulay, and its irreducible components form interesting combinatorial
    patterns. For configurations that lie in one apartment, these patterns are
    regular mixed subdivisions of scaled simplices, and the Mustafin variety is a
    twisted Veronese variety built from such a subdivision. This connects our study
    to tropical and toric geometry.

  3. Three notions of tropical rank for symmetric matrices.

    Ср, 12/09/2009 - 16:01 — SomNambul
    Authors: Dustin Cartwright, Melody Chan
    Subjects: Combinatorics
    Abstract

    We introduce and study three different notions of tropical rank for symmetric
    and dissimilarity matrices in terms of minimal decompositions into rank 1
    symmetric matrices, star tree matrices, and tree matrices. Our results provide
    a close study of the tropical secant sets of certain nice tropical varieties,
    including the tropical Grassmannian. In particular, we determine the dimension
    of each secant set, the convex hull of the variety, and in most cases, the
    smallest secant set which is equal to the convex hull.

  4. The Hilbert scheme of the diagonal in a product of projective spaces.

    Пт, 08/28/2009 - 23:56 — SomNambul
    Authors: Dustin Cartwright, Bernd Sturmfels
    Subjects: Algebraic Geometry
    Abstract

    The diagonal in a product of projective spaces is cut out by the ideal of
    2x2-minors of a matrix of unknowns. The multigraded Hilbert scheme which
    classifies its degenerations has a unique Borel-fixed ideal. This Hilbert
    scheme is generally reducible, and its main component is a compactification of
    PGL(d)^n/PGL(d). For n=2 we recover the manifold of complete collineations. For
    projective lines we obtain a space of trees that is irreducible but singular.
    All ideals in our Hilbert scheme are radical. We also explore connections to
    affine buildings and Deligne schemes.

  5. The Hilbert scheme of the diagonal in a product of projective spaces.

    Пт, 08/28/2009 - 23:56 — SomNambul
    Authors: Dustin Cartwright, Bernd Sturmfels
    Subjects: Algebraic Geometry
    Abstract

    The diagonal in a product of projective spaces is cut out by the ideal of
    2x2-minors of a matrix of unknowns. The multigraded Hilbert scheme which
    classifies its degenerations has a unique Borel-fixed ideal. This Hilbert
    scheme is generally reducible, and its main component is a compactification of
    PGL(d)^n/PGL(d). For n=2 we recover the manifold of complete collineations. For
    projective lines we obtain a space of trees that is irreducible but singular.
    All ideals in our Hilbert scheme are radical. We also explore connections to
    affine buildings and Deligne schemes.

Math-arch

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