83 research outputs found
A Hopf bundle over a quantum four-sphere from the symplectic group
We construct a quantum version of the SU(2) Hopf bundle . The
quantum sphere arises from the symplectic group and a quantum
4-sphere is obtained via a suitable self-adjoint idempotent whose
entries generate the algebra of polynomial functions over it. This
projection determines a deformation of an (anti-)instanton bundle over the
classical sphere . We compute the fundamental -homology class of
and pair it with the class of in the -theory getting the value
-1 for the topological charge. There is a right coaction of on
such that the algebra is a non trivial quantum principal
bundle over with structure quantum group .Comment: 27 pages. Latex. v2 several substantial changes and improvements; to
appear in CM
Changing Views on Curves and Surfaces
Visual events in computer vision are studied from the perspective of
algebraic geometry. Given a sufficiently general curve or surface in 3-space,
we consider the image or contour curve that arises by projecting from a
viewpoint. Qualitative changes in that curve occur when the viewpoint crosses
the visual event surface. We examine the components of this ruled surface, and
observe that these coincide with the iterated singular loci of the coisotropic
hypersurfaces associated with the original curve or surface. We derive
formulas, due to Salmon and Petitjean, for the degrees of these surfaces, and
show how to compute exact representations for all visual event surfaces using
algebraic methods.Comment: 31 page
Bases in the solution space of the Mellin system
Local holomorphic solutions z=z(a) to a univariate sparse polynomial equation
p(z) =0, in terms of its vector of complex coefficients a, are classically
known to satisfy holonomic systems of linear partial differential equations
with polynomial coefficients. In this paper we investigate one of such systems
of differential equations which was introduced by Mellin. We compute the
holonomic rank of the Mellin system as well as the dimension of the space of
its algebraic solutions. Moreover, we construct explicit bases of solutions in
terms of the roots of p and their logarithms. We show that the monodromy of the
Mellin system is always reducible and give some factorization results in the
univariate case
New examples of four dimensional AS-regular algebras
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2005.Includes bibliographical references (p. 48-49).This thesis deals with AS-regular algebras, first defined by Michael Artin and William Schelter in Graded Algebras of Global Dimension 3. All such algebras of dimension three have been classified, but the corresponding problem in higher dimensions remains open. We construct new examples of four dimensional AS-regular algebras, and provide some information about their module structure. Results are provided for proving the regularity of such algebras. In addition we classify the AS-regular algebras of dimension four satisfying certain conditions.by Ian Caines.Ph.D
Coordinate-wise Powers of Algebraic Varieties
We introduce and study coordinate-wise powers of subvarieties of
, i.e. varieties arising from raising all points in a given
subvariety of to the -th power, coordinate by coordinate.
This corresponds to studying the image of a subvariety of under
the quotient of by the action of the finite group
. We determine the degree of coordinate-wise powers and
study their defining equations, particularly for hypersurfaces and linear
spaces. Applying these results, we compute the degree of the variety of
orthostochastic matrices and determine iterated dual and reciprocal varieties
of power sum hypersurfaces. We also establish a link between coordinate-wise
squares of linear spaces and the study of real symmetric matrices with a
degenerate eigenspectrum.Comment: 26 page
F-theory and linear sigma models
We present an explicit method for translating between the linear sigma model
and the spectral cover description of SU(r) stable bundles over an elliptically
fibered Calabi-Yau manifold. We use this to investigate the 4-dimensional
duality between (0,2) heterotic and F-theory compactifications. We indirectly
find that much interesting heterotic information must be contained in the
`spectral bundle' and in its dual description as a gauge theory on multiple
F-theory 7-branes.
A by-product of these efforts is a method for analyzing semistability and the
splitting type of vector bundles over an elliptic curve given as the sheaf
cohomology of a monad.Comment: 40 pages, no figures; minor cosmetic reorganization of section 4;
reference [6] update
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