# 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.

## 40 Citations

Matroid Filtrations and Computational Persistent Homology

- Computer Science
- 2016

A novel approach to efficient computation in homological algebra over fields, with particular emphasis on computing the persistent homology of a filtered topological cell complex, based on a novel relationship between discrete Morse theory, matroid theory, and classical matrix factorizations.

Graph de Rham Cohomology and the Automorphsim Group

- Mathematics
- 2020

We introduce a graph-theoretical interpretation of an induced action of Aut$(\Gamma)$ in the discrete de Rham cohomology of a finite graph $\Gamma$. This action produces a splitting of Aut$(\Gamma)$…

Distributing Persistent Homology via Spectral Sequences

- Mathematics
- 2019

We set up the theory for a distributive algorithm for computing persistent homology. For this purpose we develop linear algebra of persistence modules. We present bases of persistence modules, and…

Local Cohomology and Stratification

- MathematicsFound. Comput. Math.
- 2020

An algorithm to recover the canonical stratification of a given finite-dimensional regular CW complex into cohomology manifolds, each of which is a union of cells, with the property that two cells are isomorphic in the last category if and only if they lie in the same canonical stratum.

Weighted sheaves and homology of Artin groups

- MathematicsAlgebraic & Geometric Topology
- 2018

In this paper we expand the theory of weighted sheaves over posets, and use it to study the local homology of Artin groups. First, we use such theory to relate the homology of classical braid groups…

ALGEBRAIC TOPOLOGY FOR DATA ANALYSIS

- Mathematics
- 2015

I develop algebraic-topological theories, algorithms and software for the analysis of nonlinear data and complex systems arising in various scientific contexts. In particular, I employ discrete…

Canonical Stratifications Along Bisheaves

- MathematicsTopological Data Analysis
- 2020

A theory of bisheaves has been recently introduced to measure the homological stability of fibers of maps to manifolds. A bisheaf over a topological space is a triple consisting of a sheaf, a…

Topological Methods in Data Analysis

- Mathematics
- 2017

I develop algebraic-topological theories, algorithms and software for the analysis of nonlinear data and complex systems arising in various scientific contexts. In particular, I employ discrete…

Discrete Morse Theory for Computing Zigzag Persistence

- MathematicsWADS
- 2019

An algorithm to compute the zigzag persistence of a filtration that depends mostly on the number of critical cells of the complexes, and it is shown experimentally that it performs better in practice.

## References

SHOWING 1-10 OF 92 REFERENCES

Discrete Morse Theoretic Algorithms for Computing Homology of Complexes and Maps

- Mathematics, Computer ScienceFound. Comput. Math.
- 2014

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

- Mathematics
- 2013

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

- MathematicsDiscret. Comput. Geom.
- 2013

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

- Mathematics
- 2005

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

- Mathematics
- 1984

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…

An introduction to homological algebra

- Mathematics
- 1960

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

- Mathematics, Computer ScienceDiscret. Math.
- 2000

Computing persistent homology

- Mathematics, Computer ScienceSCG '04
- 2004

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

- MathematicsSCG '09
- 2009

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.