Conley
Index
Computational Homology
Computational Dynamics
Mathematical
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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.

Book
 T.
Kaczynski, K. Mischaikow, and M.
Mrozek, Computational
Homology
Applied Mathematical Sciences 157 SpringerVerlag,
2004.

Theory
 S. Harker, K. Mischaikow, M. Mrozek, and V. Nanda,
Discrete Morse Theoretic Algorithms for Computing
Homology of Complexes and Maps, (2012) preprint.
 K. Mischaikow and V. Nanda, Reconstructing the
Induced Map on Homology from Images of Random
Samples, (2012) preprint.
 K. Mischaikow and V. Nanda, Morse Theory for
Filtrations and Efficient Computation of Persistent
Homology, (2011) preprint.
 S. Harker, K. Mischaikow, M. Mrozek, V. Nanda, H.
Wagner, M. Juda, P. Dlotko, The Eciency of a
Homology Algorithm based on Discrete Morse Theory
and Coreductions, Proceedings of the 3rd
International Workshop on Computational Topology in
Image Context, Chipiona, Spain, November 2010,
(Rocio Gonzalez Diaz Pedro Real Jurado (Eds.)),
Image A Vol. 1(2010), 4147.
 K. Mischaikow and T. Wanner, TopologyGuided
Sampling of Nonhomogeneous Random Processes, Annals
of Applied Probability, 20 10681097 (2010).
 S. Day, W. Kalies, K. Mischaikow, and T. Wanner,
Probabilistic and Numerical Validation of Homology
Computations for Nodal Domains, Electronic Research
Announcements, 13 (2007) 6073.
 K. Mischaikow and T. Wanner, Probabilistic
validation of homology computations for nodal
domains, Annals of Applied Probability 17 (2007)
9801018.
 K. Mischaikow, M. Mrozek and Pawel Pilarczyk, Graph Approach to the
Computation of the Homology of Continuous Maps,
Foundations of Computational Mathematics 5 (2005)
199229.
 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.)

Applications  Condensed Matter
 M. Kramar, A. Goullet, L. Kondic, K. Mischaikow,
Persistence of Force Networks in Compressed Granular
Media, (2012).
 L. Kondic, A. Goullet, C. S. O'Hern, M. Kramar, K.
Mischaikow, and R. P. Behringer, Topology of force
networks in compressed granular media, EPL, 97 (5)
54001 (2012).
 Huseyin Kurtuldu, Konstantin Mischaikow, and
Michael F. Schatz, Measuring the departures from the
Boussinesq approximation in RayleighBenard
convection experiments, J. Fluid Mechanics, (2011)
682: 543557.
 Huseyin Kurtuldu, Konstantin Mischaikow, and
Michael F. Schatz, Extensive scaling from
computational homology and KarhunenLoeve
decomposition analysis of RayleighBenard
convection experiments, Physical Review Letters, 107
034503 (2011).
 James R.Wilson, Marcio Gameiro, Konstantin
Mischaikow, William Kalies, Peter W. Voorhees, and
Scott A. Barnett, ThreeDimensional Analysis of
Solid Oxide Fuel Cell NiYSZ Anode
Interconnectivity, Microscopy and Microanalysis 15
(2009) 7177.
 K. Krishan, H. Kurtuldu, M. F. Schatz, M. Gameiro,
K. Mischaikow, and S. Madruga, Homological and
symmetry breaking in RayleighBenard convection:
Experiments and simulations, Physics of Fluids, 19
117105 (2007).
 M.
Gameiro,
K. Mischaikow and T. Wanner,
Evolution
of Pattern Complexity in the CahnHilliard Theory
of Phase Separation,
(A revised version
will appear 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.

Applications  Biology

Applications  Image Processing
 A. Szymczak, A. Stillman, A. Tannenbaum and K.
Mischaikow, Coronary vessel cores from 3D imagery: a
topological approach, Medical Image Analysis, 10
(2006) 548559.
 M. Niethammer, W. Kalies, K. Mischaikow and A.
Tannenbaum, On the detection of simple points in
higher dimensions using cubical homology, IEEE
Transactions on Image Processing, 15 (2006)
24622469.
 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 173176
