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Computational Homology
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I am interested in developing efficient algorithms for computing
homology and homology maps. My original interest in the subject was
motivated by my work in rigorous computer assisted proofs in dynamics.
However, these tools are proving to be useful in the study of geometric
properties of either numerically or experimentally generated data sets.
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Book
- T.
Kaczynski, K. Mischaikow, and M.
Mrozek, Computational Homology
Applied Mathematical Sciences 157 Springer-Verlag, 2004.
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Theory
- K. Mischaikow, M.
Mrozek , P.
Pilarczyk , Graph
Approach to the Computation of the Homology of Continuous Maps, The
final
version can be found in Foundations
of Computational Mathematics 5 (2005) 199-229.
- T.
Kaczynski, K. Mischaikow and M. Mrozek , Computing
Homology (The final version can be found at Homology,
Homotopy and Applications 5 (233-256)
- W. Kalies,
K.
Mischaikow, and G.
Watson, Cubical
Approximation and Computation of Homology (The final
version can be found in Conley Index Theory, Banach
Center Publications, 47, 1999.)
- K. Mischaikow and T. Wanner, Probabilistic validation of
homology computations for nodal domains, Annals of Applied Probability 17 (2007) 980-1018.
- S. Day, W. Kalies, K. Mischaikow and T. Wanner,
Probabilistic and numerical validation of homology computations for
nodal domains, Electronic Research
Announcements, 13
(2007) 60-73.
- K. Mischaikow and T. Wanner, Topology-guided sampling of
nonhomogeneous random fields, (2008)
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Applications
- K. Krishan, H. Kurtuldu, M. F. Schatz, M.
Gameiro, K. Mischaikow, and M. Madruga, Homological and symmetry
breaking in Rayleigh-Benard convection: Experiments and simulations, Physics of Fluids, 19 (2007).
- K. Krishan, M.
Gameiro, K. Mischaikow, and M. F. Schatz, Homological
characterizations of spiral defect chaos in Rayleigh-Benard convection
- M.
Gameiro,
K. Mischaikow and T. Wanner,
Evolution
of Pattern Complexity in the Cahn-Hilliard Theory of Phase Separation,
(The rfinal version
appears in Acta Materialia).
- M.
Gameiro, W. D.
Kalies, and K. Mischaikow, Topological
Characterization of Spatial Temporal Chaos, (The final version can
be found in Phys. Rev. E.
- M. Niethammer, A. N. Stein, W. D. Kalies, P. Pilarczyk, K.
Mischaikow, A.
Tannenbaum, Analysis
of Blood Vessel Topology by Cubical Homology
- M. Allili, K. Mischaikow, A.
Tannenbaum, Cubical homology and the
topological
classification of 2D and 3D imagery, IEEE International Conference on Image
Processing, v 2, 2001, p 173-176
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