Seminars:
Computational mathematics Thu 16:15-17:45
room 1016
Image and Data Processing Tue 11:30-13:00
room 1146
Differential equations and related issues Fri 12:15-13:45
room 1016
Partial differential equations Tue 12:15-13:45
room 1016
Marian Mrozek, PhD (chair)
room:2064
address:Łojasiewicza 6, PL-30-348 Kraków, Poland
phone:(+48) 012 664 6665
e-mail:Marian.Mrozek At ii.uj.edu.pl
www:http://ww2.ii.uj.edu.pl/~mrozek

Research interests:
  • topological dynamics
  • rigorous numerical analysis
  • copmuter assisted proofs
  • computational topology

CV light:
1999M.S. in matchematics Jagiellonian University, Kraków, Poland
1982PhD in mathematics Jagiellonian University, Kraków, Poland
1990habilitation Jagiellonian University, Kraków, Poland
1991-1993Senior Visiting ResearcherGeorgia Institute of Technology, Atlanta, USA
1996-1997Visiting Associate ProfessorGeorgia Institute of Technology, Atlanta, USA
1997-presentProfessor Graduate School of Business, Nowy Sącz, Poland
1999Professor of Mathematical SciencesPresident of Republic of Poland
2004-presentFull ProfessorJagiellonian University, Kraków, Poland

Awards and scholarships:
1983award for PhD thesisPolish Ministry of Science and Higher Education
1991award for habilitationRector of Jagiellonian University
1996 the paper on chaos in Lorenz system among the 4 best results in mathematics in 1995 Encyclopedia Britannica
1997Wacław Sierpiński awardPolish Mathematical Society
2004 award for outstanding scientific and educational achievements Rector of Jagiellonian University
2008 award for outstanding scientific achievements in the development of science Polish Ministry of Science and Higher Education
2011 Laur Jagielloński Rector of Jagiellonian University

Research grants:
1994-1997Computer assisted proofs in dynamicsPolish Scientific Research Commitee
1997-2000 Topological methods in computer assisted proofs in dynamics Polish Scientific Research Commitee
2000-2003 Topological methods in rigorous computational dynamics Polish Scientific Research Commitee
2003-2006Computational topology and dynamicsPolish Scientific Research Commitee
2006-2009Algorithmic methods in topology and dynamicsPolish Scientific Research Commitee
2010-2013 Algorithmic methods in computational topology and computational dynamics National Science Centre
2011-2012Homology for massive data analysisGoogle Awards
2010-2015 Geometry and Topology in Physical Models Foundation for Polish Science
2011-2015 Applied and Computational Algebraic Topology European Science Foundation Networking Programme
2012-2015 TOPOSYS - Topological Complex Systems EU 7th Framework Programme FET proactive
2015-2020 MAESTRO: Computational topological dynamics National Science Centre

PhD students:
  • (1996) Piotr Zgliczyński Metody topologiczne w chaotycznych układach dynamicznych
  • (2001) Paweł Pilarczyk Topologiczny alorytm ścisłej weryfikacji istnienia trajektorii okresowych
  • (2002) Marcin Żelawski Ścisła numeryczna weryfikacja istnienia podwójnie heteroklinicznej bifurkacji w równaniach Kuramoto-Sivashinsky'ego
  • (2003) Daniel Wilczak Metoda relacji nakrywających w ścisłej analizie numerycznej równań różniczkowych
  • (2003) Kinga Stolot Homotopijny indeks Conleya dla wielowartościowych dyskretnych układów dynamicznych
  • (2010) Natalia Żelazna Algorytmy homologii zbiorów kostkowych metodą podzbioru acyklicznego
  • (2012) Paweł Dłotko Algorytmy redukcji w obliczeniowej teorii homologii i kohomologii z zastosowaniami
  • (2013) Mateusz Juda Teoretyczne i praktyczne aspekty liczenia homologii metodami geometrycznych koredukcji
  • (2014) Hubert Wagner Efficient computation of persistent homology
  • (2015) Grzegorz Jabłoński Persistent homology of a self-map
  • (2016) Frank Weilandt Rigorous Numerical Computation of the Conley Index for Flows

Books:
  • M. Erdmann, J. Erickson, R. Ghrist, M. Mrozek, A. Zomorodian (Ed.),
    Advances in Applied and Computational Topology: AMS Short Course Computational Topology
    Applied Mathematical Sciences 157, American Mathematical Society, 2011
  • T. Kaczynski, K. Mischaikow, M. Mrozek,
    Computational Homology
    Applied Mathematical Sciences 157, Springer-Verlag, 2004
  • K. Mischaikow, M. Mrozek,
    The Conley Index
    Handbook of Dynamical Systems II: Towards Applications, (B. Fiedler, ed.) North-Holland , 2002
  • K. Mischaikow, M. Mrozek, P. Zgliczyński ,
    Conley Index Theory
    Banach Center Publications, Polish Academy of Sciences , 1999

Publications:
  1. Tamal Dey, Mateusz Juda, Tomasz Kapela, Jacek Kubica, Michał Lipiński, Marian Mrozek,
    Persistent Homology of Morse Decompositions in Combinatorial Dynamics
    SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS, vol. 18-1, 510-530 (2019)
  2. Marian Mrozek,
    Conley-Morse-Forman theory for combinatorial multivector fields on Lefschetz complexes
    FOUNDATIONS OF COMPUTATIONAL MATHEMATICS, vol. 17, 1585-1633 (2017)
  3. Marc Ethier, Grzegorz Jabłoński, Marian Mrozek,
    Finding Eigenvalues of Self-maps with the Kronecker Canonical Form
    Springer, vol. 198, 119-136 (2017)
  4. Bogdan Batko, Marian Mrozek,
    Weak index pairs and the Conley index for discrete multivalued dynamical systems
    SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS, vol. 15.2, 1143-1162 (2016)
  5. K. Mischaikow, M. Mrozek, F. Weilandt,
    Discretization strategies for computing Conley indices and Morse decompositions of flows
    Journal of Computational Dynamics , (2016)
  6. Tomasz Kaczynski, Marian Mrozek, Thomas Wanner,
    Towards a formal tie between combinatorial and classical vector field dynamics
    Journal of Computational Dynamics, vol. 3 (1), 17-50 (2016)
  7. Andrzej Gryglewski, Brandon Michael Henry, Marian Mrozek, Krzysztof Piech, Krzysztof Tomaszewski, Marcin Żelawski,
    Sensitivity and Specificity of Fractal Analysis to Distinguish Between Healthy and Pathologic Rectal Mucosa Microvasculature Seen During Colonoscopy
    SURGICAL LAPAROSCOPY ENDOSCOPY & PERCUTANEOUS TECHNIQUES, vol. 26(5), 358–363 (2016)
  8. K. Mischaikow, Marian Mrozek, Frank Weilandt,
    Discretization strategies for computing Conley indices and Morse decompositions of flows
    Journal of Computational Dynamics, vol. 3, 1-16 (2016)
  9. Piotr Brendel, Paweł Dłotko, Graham Ellis, Mateusz Juda, Marian Mrozek,
    Computing fundamental groups from point clouds
    APPLICABLE ALGEBRA IN ENGINEERING COMMUNICATION AND COMPUTING, vol. 26, 27-48 (2015)
  10. Herbert Edelsbrunner, Grzegorz Jabłoński, Marian Mrozek,
    The Persistent Homology of a Self-Map
    FOUNDATIONS OF COMPUTATIONAL MATHEMATICS, vol. 15, 1213-1244 (2015)
  11. Marian Mrozek, Roman Srzednicki, Frank Weilandt,
    A topological approach to the algorithmic computation of the Conley index for Poincaré maps
    SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS, vol. 14, 1348-1386 (2015)
  12. Marian Mrozek,
    A Topological Method for Finding Invariant Sets of Continuous Systems
    Springer Verlag (połaczony z Kluwer Academic Publishing), vol. Lecture Notes in Computer Science, 63-75 (2015)
  13. Mateusz Juda, Marian Mrozek,
    CAPD::RedHom v2 - Homology Software Based on Reduction Algorithms
    LECTURE NOTES IN COMPUTER SCIENCE, vol. 8592, 160-166 (2014)
  14. Tomasz Kaczynski, Marian Mrozek,
    Cubical Cohomology Ring: Algorithmic Approach
    FOUNDATIONS OF COMPUTATIONAL MATHEMATICS, vol. 13, 789-818 (2013)
  15. Paweł Dłotko, Robert Ghrist, Mateusz Juda, Marian Mrozek,
    Distributed computation of coverage in sensor networks by homological methods
    APPLICABLE ALGEBRA IN ENGINEERING COMMUNICATION AND COMPUTING, vol. 23, Issue 1-2, 29-58 (2012)
  16. Andrzej Gryglewski, Andrzej Krajniak, Marian Mrozek, Han Sejin, Marcin Żelawski,
    Homological methods for extraction and analysis of linear features in multidimensional images
    PATTERN RECOGNITION, vol. 45(1), 285-298 (2012)
  17. Piotr Brendel, Paweł Dłotko, Marian Mrozek, Natalia Żelazna,
    Homology Computations via Acyclic Subspace
    LECTURE NOTES IN COMPUTER SCIENCE, vol. 7309, 117-127 (2012)
  18. Marian Mrozek,
    Topological Dynamics: Rigorous Numerics via Cubical Homology
    American Mathematical Society (AMS), vol. 70, 41-73 (2012)
  19. Mateusz Juda, Marian Mrozek,
    Z_2-homology of weak (p-2)-faceless p-pseudomanifolds may be computed in O(n) time
    Topological Methods in Nonlinear Analysis, vol. 40, 137-159 (2012)
  20. Paweł Dłotko, Marian Mrozek, Hubert Wagner,
    Computational Topology in Text Mining
    LECTURE NOTES IN COMPUTER SCIENCE, vol. 7309, 68 - 79 (2012)
  21. Tomasz Baron, Marcin Cwynar, Tomasz Grodzicki, Barbara Gryglewska, Marian Mrozek, Mirosław Nęcki, Marcin Żelawski,
    Fractal dimensions of skin microcirculation flow in subjects with familial predisposition or newly diagnosed hypertension
    CARDIOLOGY JOURNAL, FORMERLY FOLIA CARDIOLOGICA, vol. 18, No. 1, 26-32 (2011)
  22. Paweł Dłotko, Tomasz Kaczynski, Marian Mrozek, Thomas Wanner,
    Coreduction Homology Algorithm for Regular CW-Complexes
    DISCRETE & COMPUTATIONAL GEOMETRY, vol. 46, 361-388 (2011)
  23. Marian Mrozek,
    Homology of representable sets
    Annales Polonici Mathematici, vol. 97.3, 243–252 (2010)
  24. Marian Mrozek, Roman Srzednicki,
    Topological approach to rigorous numerics of chaotic dynamical systems with strong expansion of error bounds
    FOUNDATIONS OF COMPUTATIONAL MATHEMATICS, 191-220 (2010)
  25. Marian Mrozek,
    ˇCech Type Approach to Computing Homology of Maps
    DISCRETE & COMPUTATIONAL GEOMETRY, vol. 44.3, 546–576 (2010)
  26. Marian Mrozek, Thomas Wanner,
    Coreduction homology algorithm for inclusions and persistent homology
    COMPUTERS & MATHEMATICS WITH APPLICATIONS, vol. 60.10, 2812-2833 (2010)
  27. Paweł Dłotko, Tomasz Kaczynski, Marian Mrozek,
    Computing the cubical cohomology ring
    Image A, Proceedings of the 3rd International Workshop on computational Topology in Image Context, 137-142 (2010)
  28. Paweł Dłotko, Shaun Harker, Mateusz Juda, Konstantin Mischaikov, Marian Mrozek, Vidit Nanda, Hubert Wagner,
    The efficiency of a homology algorithm based on discrete Morse Theory and Coreductions
    Image A Proceedings of the 3rd International Workshop on computational Topology in Image Context, 41-47 (2010)
  29. Andrzej Gryglewski, Andrzej Krajniak, Marian Mrozek, Han Sejin, Marcin Żelawski,
    Homological Methods in Feature Extraction of Multidimensional Images
    Proceedings of the 2009 2nd International Congress on Image and Signal Processing, IEEE, 1061-1066 (2009)
  30. Marian Mrozek, Roman Srzednicki,
    Conley index of Poincare maps in isolating segments
    NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, 2123—2131 (2009)
  31. Bogdan Batko, Marian Mrozek,
    Coreduction homology algorithm
    DISCRETE & COMPUTATIONAL GEOMETRY, vol. 41, 96-118 (2009)
  32. Marian Mrozek, Paweł Pilarczyk, Natalia Żelazna,
    Homology algorithm based on acyclic subspace
    COMPUTERS & MATHEMATICS WITH APPLICATIONS, vol. 55, 2395-2412 (2008)
  33. Tomasz Kaczynski, M. Mrozek, Anik Trahan,
    Ideas from Zariski topology in the study of cubical sets, cubical maps, and their homology
    Canadian J. Math., 1008-1028 (2007)
  34. Bogdan Batko, Marian Mrozek,
    The Euler-Poincare characteristic of index maps
    TOPOLOGY AND ITS APPLICATIONS, vol. 154.4, 859-866 (2007)
  35. M. Mrozek,
    Index Pairs Algorithms
    Foundations of Computational Mathematics, 457-493 (2006)
  36. M. Mrozek,
    The Method of Topological Sections in the Rigorous Numerics of Dynamical Systems
    Canadian Applied Mathematics Quarterly, 209-222 (2006)
  37. Marian Mrozek, Klaudiusz Wójcik,
    Discrete version of a geometric method for detecting chaotic dynamics
    TOPOLOGY AND ITS APPLICATIONS, vol. 152, 70-82 (2005)
  38. K. Mischaikow, M. Mrozek, P. Pilarczyk,
    Graph Approach to the Computation of the Homology of Continuous Maps
    Foundations of Computational Mathematics, 199-229 (2005)
  39. T. Kaczynski, K. Mischaikow, M. Mrozek,
    Computing Homology
    Homotopy, Homology and Applications, 233-256 (2003)
  40. K. Mischaikow, M. Mrozek,
    The Conley Index Theory
    Handbook of Dynamical Systems III: Towards Applications , 393-460 (2002)
  41. M. Mrozek, P. Pilarczyk,
    The Conley Index and Rigorous Numerics of Attracting Periodic Orbits
    Variational and Topological Methods in the Study of Nonlinear Phenomena, V. Benci et al. Editors, Progress in Nonlinear Differential Equations and Their Applications, Vol. 49 , (2002)
  42. K. Mischaikow, M. Mrozek, A. Szymczak,
    Chaos in the Lorenz equations: a computer assisted proof. Part III: the classical parameter values
    J. Diff. Equ., 17-56 (2001)
  43. M. Mrozek, P. Zgliczyński ,
    Set arithmetic and the enclosing problem in dynamics
    Annales Polonici Mathematici, 237-259 (2000)
  44. M. Mrozek, J. Reineck, R. Srzednicki,
    The Conley index over a circle
    J. Dynamics and Differential Equations,, 385--409 (2000)
  45. Marian Mrozek, Roman Srzednicki,
    The Conley index over the circle
    , (2000)
  46. Marian Mrozek, James F. Reineck, Roman Srzednicki,
    The Conley index over a base
    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, vol. 352, 4171-4194 (2000)
  47. K. Mischaikow, M. Mrozek, J. Reineck,
    Singular index pairs
    J. Dyn. Diff. Equ., 399-426 (1999)
  48. K. Mischaikow, M. Mrozek, J. Reiss, A. Szymczak,
    Construction of Symbolic Dynamics from Experimental Time Series
    Physical Review Letters, 1144 (1999)
  49. M. Mrozek,
    Construction and properties of the Conley index
    Conley index theory, Banach Center Publications, Vol. 47 , 29-40 (1999)
  50. M. Mrozek,
    An algorithmic approach to the Conley index theory
    J. Dyn. Diff. Equ., 711-734 (1999)
  51. T. Kaczynski, M. Mrozek, M. Slusarek,
    Homology Computation by Reduction of Chain Complexes,
    Computers and Mathematics with Appl., 59--70 (1998)
  52. K. Mischaikow, M. Mrozek,
    Chaos in the Lorenz equations: a computer assisted proof. Part II: details
    Mathematics of Computation, 1023-1046 (1998)
  53. K. Mischaikow, M. Mrozek, J. Reineck,
    Singular Index Pairs
    Univ. Iag. Acta Math., 223-226 (1998)
  54. M. Mrozek,
    Algorithms and the Conley index theory,
    Univ. Iag. Acta Math.,, 227-230 (1998)
  55. T. Kaczynski, M. Mrozek,
    Connected simple systems and the Conley functor
    Topol. Methods Nonlin. Anal., 183-193 (1997)
  56. T. Kaczynski, M. Mrozek,
    Stable index pairs for discrete dynamical systems
    Bulletin Canadian Math. Soc., 448-455 (1997)
  57. M. Mrozek, J. Urbaniec,
    Evolution of Mathematical Proof
    Foundations of Science, 77-85 (1997)
  58. M. Mrozek, M. Żelawski,
    Heteroclinic Connections in the Kuramoto-Sivashinsky Equation: a Computer Assisted Proof
    Reliable Computing, 277-285 (1997)
  59. M. Mrozek,
    Topological invariants, multivalued maps and computer assisted proofs
    Computers and Mathematics with Appl., 83-104 (1996)
  60. M. Mrozek,
    Rigorous error analysis of numerical algorithms via symbolic computations
    J. Symb. Comp., 435-458 (1996)
  61. M. Mrozek,
    The Conley Index and Rigorous Numerics,
    Nonlinear Analysis and Boundary Value Problems for Ordinary Differential Equations, CISM Courses ans Lectures, Vol. 371 , 175-196 (1996)
  62. M. Mrozek,
    Inheritable properties and computer assisted proofs in dynamics
    Scientific Computing and validated numerics, G. Alefeld, A. Frommer and B. Lang editors, Akademie Verlag, Berlin , 245-253 (1996)
  63. Ch. McCord, K. Mischaikow, M. Mrozek,
    Zeta Functions, Periodic Trajectories and the Conley Index
    J. Diff. Equ, 258-292 (1995)
  64. T. Kaczynski, M. Mrozek,
    Conley index for discrete multivalued dynamical systems
    Topology & its Appl., 83-96 (1995)
  65. K. Mischaikow, M. Mrozek,
    Isolating neighborhoods and chaos
    Jap. J. Ind. and Appl. Math., 205-236 (1995)
  66. K. Mischaikow, M. Mrozek,
    Chaos in Lorenz equations: a computer assisted proof
    Bull. Amer. Math. Soc. (N.S.), 66-72 (1995)
  67. M. Mrozek,
    Rigorous numerics of chaotic dynamical suystems
    Lect. Notes in Physic, 283-296 (1995)
  68. M. Mrozek,
    From the Theorem of Ważewski to computer assisted proofs in dynamics
    Panoramas of Mathematics, Banach Center Publications, Vol. 34 , 105-120 (1995)
  69. M. Mrozek,
    Shape index and other indices of Conley type for continuous maps in locally compact metric spaces
    Fundamenta Mathematicae, 15-37 (1994)
  70. M. Degiovanni, M. Mrozek,
    The Conley index for maps in absence of compactness
    Proc. Royal Soc. Edinburgh, 75-94 (1993)
  71. Marian Mrozek, Roman Srzednicki,
    On time-duality of the Conley index
    , (1993)
  72. M. Mrozek, K.P. Rybakowski,
    Discretized ordinary differential equations and the Conley index
    J. of Dynamics and Differential Equations, 57-63 (1992)
  73. M. Mrozek,
    Normal Functors and Retractors in Categories of Endomorphisms
    Univ. Iag. Acta Math., 181-198 (1992)
  74. M. Mrozek, K.P. Rybakowski,
    Nontrivial full bounded solutions of time-periodic semilinear PDEs
    Proc. Roy. Soc. Edinburgh, 305-315 (1991)
  75. M. Mrozek, K.P. Rybakowski,
    A cohomological Conley index for maps on metric spaces
    Journal of Diff. Equations, 143-171 (1991)
  76. M. Mrozek,
    Some Remarks on the Garay's Conjecture
    Acta Math. Hungarica, 53-59 (1991)
  77. M. Mrozek,
    The Morse Equation in Conley's Index Theory for Homeomorphisms
    Topology and its Applications, 45-60 (1991)
  78. M. Mrozek,
    Leray Functor and the Cohomological Conley Index for Discrete Time Dynamical Systems
    Transactions of the Amer. Math. Soc., 149-178 (1990)
  79. M. Mrozek,
    A cohomological index of Conley type for multivalued admissible flows
    J. Diff. Equations, 15-51 (1990)
  80. M. Mrozek,
    Open index pairs, the fixed point index and rationality of zeta functions
    Erg. Th. and Dyn. Syst., 555-564 (1990)
  81. M. Mrozek,
    The Conley index on compact ANR's is of finite type
    Results in Mathematics, 306-313 (1990)
  82. M. Mrozek,
    Index pairs and the Fixed Point Index for Semidynamical Systems with Discrete Time
    Fund. Mathematicae, 177-192 (1989)
  83. M. Mrozek,
    Transitively Reduced and Transitively Closed Event Networks
    Networks, 47-72 (1989)
  84. Marian Mrozek, Roman Srzednicki,
    The continuity axiom in multi-valued dynamical systems
    , (1988)
  85. M. Mrozek,
    Periodic and Stationary Trajectories of Flows and Ordinary Differential Equations
    Univ. Iag. Acta Math., 29-37 (1988)
  86. M. Mrozek,
    The Fixed Point Index of a Translation Operator of a Semiflow
    Univ. Iag. Acta Math., 13-22 (1988)
  87. M. Mrozek,
    Strongly Isolating Invariant Sets and Strongly Isolating Blocks
    Bull. Acad. Pol. Sc. Ser. Sci. Math., 1-2 (1987)
  88. Marian Mrozek, Marek Zaionc,
    A computer model of evolution and ecological processes
    Modeling, Simulation & Control C, 59-63 (1984)
  89. M. Mrozek,
    A Note on Minimum-Dummy-Activities PERT Networks
    RAIRO Rech. Oper., 415-421 (1984)
  90. M. Mrozek, M. Zaionc,
    Simulation model of evolution in a simplified ecological system
    Wiad. Ekol., 49-59 (1981)
  91. M. Mrozek,
    Generalized Inverse of a Finite Graph
    Fund. Inform., 499-509 (1981)
  92. M. Mrozek,
    Inverse Limits of Torus-Groups and Minimal Sets of some Almost Periodic Continuous Flows
    Zesz. Nauk. UJ, Pr. Mat., 29-50 (1979)
  93. K. Mischaikow, Marian Mrozek, S. Harker, V. Nanda,
    Discrete Morse Theoretic Algorithms for Computing Homology of Complexes and Maps
    FOUNDATIONS OF COMPUTATIONAL MATHEMATICS, ()
  94. Marian Mrozek, Roman Srzednicki,
    On a generalization of the Conley index
    , ()
Division of Computational Mathematics, Institute of Computer Science, Łojasiewicza 6, 30-348 Kraków, Poland