Discrete Morse Theory for Computing Cellular Sheaf Cohomology

@article{Curry2016DiscreteMT,
  title={Discrete Morse Theory for Computing Cellular Sheaf Cohomology},
  author={Justin Curry and Robert Ghrist and Vidit Nanda},
  journal={Foundations of Computational Mathematics},
  year={2016},
  volume={16},
  pages={875-897}
}
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. 
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References

SHOWING 1-10 OF 92 REFERENCES
Discrete Morse Theoretic Algorithms for Computing Homology of Complexes and Maps
TLDR
A new Morse theoretic preprocessing framework for deriving chain maps from set-valued maps is introduced, and hence an effective scheme for computing the morphism induced on homology by the approximated continuous function is provided.
The Nyquist theorem for cellular sheaves
We develop a unified sampling theory based on sheaves and show that the Shannon-Nyquist theorem is a cohomological consequence of an exact sequence of sheaves. Our theory indicates that there are
Morse Theory for Filtrations and Efficient Computation of Persistent Homology
TLDR
An efficient preprocessing algorithm is introduced to reduce the number of cells in a filtered cell complex while preserving its persistent homology groups through an extension of combinatorial Morse theory from complexes to filtrations.
Morse theory from an algebraic viewpoint
Forman's discrete Morse theory is studied from an algebraic viewpoint, and we show how this theory can be extended to chain complexes of modules over arbitrary rings. As applications we compute the
Elements of algebraic topology
Elements of Algebraic Topology provides the most concrete approach to the subject. With coverage of homology and cohomology theory, universal coefficient theorems, Kunneth theorem, duality in
Discrete Morse Theory for free chain complexes
An introduction to homological algebra
Preface 1. Generalities concerning modules 2. Tensor products and groups of homomorphisms 3. Categories and functors 4. Homology functors 5. Projective and injective modules 6. Derived functors 7.
On discrete Morse functions and combinatorial decompositions
Computing persistent homology
TLDR
The analysis establishes the existence of a simple description of persistent homology groups over arbitrary fields and derives an algorithm for computing individual persistent homological groups over an arbitrary principal ideal domain in any dimension.
Zigzag persistent homology and real-valued functions
TLDR
The algorithmic results provide a way to compute zigzag persistence for any sequence of homology groups, but combined with the structural results give a novel algorithm for computing extended persistence that is easily parallelizable and uses (asymptotically) less memory.
...
...