3,936 research outputs found
Discrete Morse theory for computing cellular sheaf cohomology
Sheaves and sheaf cohomology are powerful tools in computational topology,
greatly generalizing persistent homology. We develop an algorithm for
simplifying the computation of cellular sheaf cohomology via (discrete)
Morse-theoretic techniques. As a consequence, we derive efficient techniques
for distributed computation of (ordinary) cohomology of a cell complex.Comment: 19 pages, 1 Figure. Added Section 5.
Geometric Quantization of real polarizations via sheaves
In this article we develop tools to compute the Geometric Quantization of a
symplectic manifold with respect to a regular Lagrangian foliation via sheaf
cohomology and obtain important new applications in the case of real
polarizations. The starting point is the definition of representation spaces
due to Kostant. Besides the classical examples of Gelfand-Cetlin systems due to
Guillemin and Sternberg very few examples of explicit computations of real
polarizations are known. The computation of Geometric Quantization for
Gelfand-Cetlin systems is based on a theorem due to \'Sniatycki for fibrations
which identifies the representation space with the set of Bohr-Sommerfeld
leaves determined by the integral action coordinates.
In this article we check that the associated sheaf cohomology apparatus of
Geometric Quantization satisfies Mayer-Vietoris and K\"unneth formulae. As a
consequence, a new short proof of this classical result for fibrations due to
\'Sniatycki is obtained. We also compute Geometric Quantization with respect to
any generic regular Lagrangian foliation on a 2-torus and the case of the
irrational flow. In the way, we recover some classical results in the
computation of foliated cohomology of these polarizations.Comment: 35 pages, 4 figures, minor change
(0,2) Target Space Duality, CICYs and Reflexive Sheaves
It is shown that the recently proposed target space duality for (0,2) models
is not limited to models admitting a Landau-Ginzburg description. By studying
some generic examples it is established for the broader class of vector bundles
over complete intersections in toric varieties. Instead of sharing a common
Landau-Ginzburg locus, a pair of dual models agrees in more general
non-geometric phases. The mathematical tools for treating reflexive sheaves are
provided, as well.Comment: 20 pages, TeX, harvma
Cohomology of Line Bundles: Applications
Massless modes of both heterotic and Type II string compactifications on
compact manifolds are determined by vector bundle valued cohomology classes.
Various applications of our recent algorithm for the computation of line bundle
valued cohomology classes over toric varieties are presented. For the heterotic
string, the prime examples are so-called monad constructions on Calabi-Yau
manifolds. In the context of Type II orientifolds, one often needs to compute
equivariant cohomology for line bundles, necessitating us to generalize our
algorithm to this case. Moreover, we exemplify that the different terms in
Batyrev's formula and its generalizations can be given a one-to-one
cohomological interpretation.
This paper is considered the third in the row of arXiv:1003.5217 and
arXiv:1006.2392.Comment: 56 pages, 8 tables, cohomCalg incl. Koszul extension available at
http://wwwth.mppmu.mpg.de/members/blumenha/cohomcalg
Positive Alexander Duality for Pursuit and Evasion
Considered is a class of pursuit-evasion games, in which an evader tries to
avoid detection. Such games can be formulated as the search for sections to the
complement of a coverage region in a Euclidean space over a timeline. Prior
results give homological criteria for evasion in the general case that are not
necessary and sufficient. This paper provides a necessary and sufficient
positive cohomological criterion for evasion in a general case. The principal
tools are (1) a refinement of the Cech cohomology of a coverage region with a
positive cone encoding spatial orientation, (2) a refinement of the Borel-Moore
homology of the coverage gaps with a positive cone encoding time orientation,
and (3) a positive variant of Alexander Duality. Positive cohomology decomposes
as the global sections of a sheaf of local positive cohomology over the time
axis; we show how this decomposition makes positive cohomology computable as a
linear program.Comment: 19 pages, 6 figures; improvements made throughout: e.g. positive
(co)homology generalized to arbitrary degrees; Positive Alexander Duality
generalized from homological degrees 0,1; Morse and smoothness conditions
generalized; illustrations of positive homology added. minor corrections in
proofs, notation, organization, and language made throughout. variant of
Borel-Moore homology now use
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