CHomP

CHomP, Computational homology project. Much of the fascination and challenge of studying nonlinear systems arises from the complicated spatial, temporal and even spatial-temporal behavior they exhibit. On the level of mathematics this complicated behavior can occur at all scales, both in the state space and in parameter space. Somewhat paradoxically, this points to the need for a coherent set of mathematical techniques that is capable of extracting coarse but robust information about the structure of these systems. Furthermore, most of our understanding of specific systems comes from experimental observation or numerical simulations and thus it is important that these techniques be computationally efficient. Algebraic Topology is the classical mathematical tool for the global analysis of nonlinear spaces and functions, within which homology is perhaps the most computable subset. In particular, it provides a well understood framework through which the information hidden in large datasets can be reduced to compact algebraic expressions that provide insight into underlying geometric structures and properties. The material described through these web pages represents our ongoing effort to develop and apply efficient and effective topologically based methods to the analysis of nonlinear systems.


References in zbMATH (referenced in 30 articles )

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  1. Adiprasito, Karim A.; Benedetti, Bruno; Lutz, Frank H.: Extremal examples of collapsible complexes and random discrete Morse theory (2017)
  2. Jaquette, Jonathan; Kramár, Miroslav: On $\varepsilon $ approximations of persistence diagrams (2017)
  3. Toth, Csaba D. (ed.); Goodman, Jacob E. (ed.); O’Rourke, Joseph (ed.): Handbook of discrete and computational geometry (2017)
  4. Bush, Justin; Cowan, Wes; Harker, Shaun; Mischaikow, Konstantin: Conley-Morse databases for the angular dynamics of Newton’s method on the plane (2016)
  5. de Floriani, Leila; Fugacci, Ulderico; Iuricich, Federico: Homological shape analysis through discrete Morse theory (2016)
  6. Harker, Shaun; Kokubu, Hiroshi; Mischaikow, Konstantin; Pilarczyk, Paweł: Inducing a map on homology from a correspondence (2016)
  7. Haro, Àlex; Canadell, Marta; Figueras, Jordi-Lluís; Luque, Alejandro; Mondelo, Josep-Maria: The parameterization method for invariant manifolds. From rigorous results to effective computations (2016)
  8. Krčál, Marek; Pilarczyk, Paweł: Computation of cubical Steenrod squares (2016)
  9. Alexander, Zachary; Bradley, Elizabeth; Meiss, James D.; Sanderson, Nicole F.: Simplicial multivalued maps and the witness complex for dynamical analysis of time series (2015)
  10. Gameiro, Marcio; Hiraoka, Yasuaki; Izumi, Shunsuke; Kramar, Miroslav; Mischaikow, Konstantin; Nanda, Vidit: A topological measurement of protein compressibility (2015)
  11. Giusti, Chad; Pastalkova, Eva; Curto, Carina; Itskov, Vladimir: Clique topology reveals intrinsic geometric structure in neural correlations (2015)
  12. Pilarczyk, Paweł; Real, Pedro: Computation of cubical homology, cohomology, and (co)homological operations via chain contraction (2015)
  13. Benedetti, Bruno; Lutz, Frank H.: Random discrete Morse theory and a new library of triangulations (2014)
  14. Berwald, Jesse; Gidea, Marian: Critical transitions in a model of a genetic regulatory system (2014)
  15. Joswig, Michael; Lutz, Frank H.; Tsuruga, Mimi: Heuristics for sphere recognition (2014)
  16. Cochran, Gregory S.; Wanner, Thomas; Dłotko, Paweł: A randomized subdivision algorithm for determining the topology of nodal sets (2013)
  17. Kaczynski, Tomasz; Mrozek, Marian: The cubical cohomology ring: an algorithmic approach (2013)
  18. Pellikka, M.; Suuriniemi, S.; Kettunen, L.; Geuzaine, C.: Homology and cohomology computation in finite element modeling (2013)
  19. van den Berg, Jan Bouwe; Day, Sarah; Vandervorst, Robert: Braided connecting orbits in parabolic equations via computational homology (2013)
  20. Berciano, A.; Molina-Abril, H.; Real, P.: Searching high order invariants in computer imagery (2012)

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