Mathematics and Cosmology

Main Article Content

Michał Heller

Abstract

The mathematical and cosmological works of a group associated with the Copernicus Center for Interdisciplinary Studies in Cracow are summarized. The group consists mainly of M. Heller, L. Pysiak, W. Sasin, Z. Odrzygóźdź and J. Gruszczak. The first paper by members of the group was published in 1988, and research has been continued to the present day. The main mathematical tool used in the first part of the group’s activity was the theory of differential spaces and, in the second, methods of noncommutative geometry. Among the main topics investigated have been classical singularities in relativistic cosmology and the unification of general relativity with quantum mechanics.

Article Details

Section
Anniversary articles

References

Connes, A., Noncommutative Geometry, Academic Press, New York, 1994.

Demaret, J., Heller, M., Sasin, W., Noncommutative Unification of General Relativity with Quantum Mechanics and Canonical Gravity Quantization, Revue des Questions Scientifiques, 172 (4), 2001, 357-370.

Geroch, R., Einstein Algebras, Communications in Mathematical Physics 26, 1972, 271-275.

Gruszczak, J,. Heller, M., Multarzyński, P. A Generalization of Manifolds as Space-Time Models, J. Math. Phys. 29, 1988, 2576-2580.

Gruszczak, J., Heller, M., Differential Structure of Space-Time and Its Prolongations to Singular Boundaries, International Journal of Theoretical Physics 32, 1993, 625-648.

Gruszczak, J., Heller, M., Pogoda, Z., Cauchy Boundary and b-Incompletness of Space-Time, International Journal of Theoretical Physics 30, 1991, 555-565.

Gruszczak, J., Heller, M., Sasin, W., Quasiregular Singularity of a Cosmic String, Acta Cosmologica 18, 1992, 45-55.

Heller, M., A Noncommutative Friedman Cosmological Model, Annalen der Physik 19, 2010, 196-201.

Heller, M., Algebraic Self-Duality as the "Ultimate Explanation", Foundations of Science, 9, 2004, 369-385.

Heller, M., Commutative and Non-Commutative Einstein Algebras, Acta Cosmologica 21, 1995, 111-130.

Heller, M., Einstein Algebras and General Relativity, International Journal of Theoretical Physics 31, 1992, 277-278.

Heller, M., Multarzyński, P., Sasin, W., Żekanowski, Z., Local Differential Dimension of Space-Time, Acta Cosmologica 17, 1991, 19-26.

Heller, M., Multarzyński, P., Sasin, W., Żekanowski, Z., On Some Generalizations of the Manifold Concept, Acta Cosmologica 18, 1992, 31-44.

Heller, M., Odrzygóźdź, Z., Pysiak, L., Sasin, W., Anatomy of Malicious Singularities, Journal of Mathematical Physics 48, 2007, 092504-092511.

Heller, M., Odrzygóźdź, Z., Pysiak, L., Sasin, W., Gravitational Aharonov-Bohm Effect, International Journal of Theoretical Physics 47, 2008, 2566-2575.

Heller, M., Odrzygóźdź, Z., Pysiak, L., Sasin, W., Hopf-Sikorski Algebras, Demonstatio Mathematica, vol. 44, 2011, 213-221.

Heller, M., Odrzygóźdź, Z., Pysiak, L., Sasin, W., Noncommutative Unification of General Relativity and Quantum Mechanics. A Finite Model, General Relativity and Gravitation 36, 2004, 111-126.

Heller, M., Odrzygóźdź, Z., Pysiak, L., Sasin, W., Quantum Groupoids of the Final Type and Quantization on Orbit Spaces, Demonstratio Mathematica, 37, 2004, 671-678.

Heller, M., Odrzygóźdź, Z., Pysiak, L., Sasin, W., Structure of Malicious Singularities, International Journal of Theoretical Physics 42, 2003, 427-441.

Heller, M., Pysiak, L., Sasin, W., Conceptual Unification of Gravity and Quanta, International Journal of Theoretical Physics 46, 2007, 2492-2512.

Heller, M., Pysiak, L., Sasin, W., Fundamental Problems in the Unification of Physic, Foundations of Physics 41, 2011, 905-918.

Heller, M., Pysiak, L., Sasin, W., Geometry of Non-Hausdorff Spaces and Its Significance for Physics, Journal of Mathematical Physics 52, 2011, 043506.

Heller, M., Pysiak, L., Sasin, W., Golda, Z., Noncommutative Closed Friedman Universe, General Relativity and Gravitation 41, 2009, 1625-1637.

Heller, M., Pysiak, L., Sasin, W., Inner Geometry of Random Operators, Demonstratio Mathematica 39 (nr 4), 2006, 971-978.

Heller, M., Pysiak, L., Sasin, W., Noncommutative Dynamics of Random Operators, International Journal of Theoretical Physics 44, 2005, 619-628.

Heller, M., Pysiak, L., Sasin, W., Noncommutative Unification of General Relativity and Quantum Mechanics, Journal of Mathematical Physics 46, 2005, 122501-16.

Heller, M., Sasin, M., Anatomy of the Elementary Quasi-Regular Singularity, Acta Cosmologica, 21, 1995, 47-60.

Heller, M., Sasin, M., Sheaves of Einstein Algebras, International Journal of Theoretical Physics, 34, 1995, 387-398.

Heller, M., Sasin, M., Structured Spaces and Their Application to Relativistic Physics, Journal of Mathematical Physics 36, 1995, 3644-3662.

Heller, M., Sasin, M., Superstructured Spaces, Acta Cosmologica, 21, 1995, 61-70.

Heller, M., Sasin, W. Regular Singularities in Space-Time, Acta Cosmologica 17 1991, 7-18.

Heller, M., Sasin, W., Differential Groupoids and Their Application to the Theory of Spacetime Singularities, International Journal of Theoretical Physics 41, 2002, 919-937.

Heller, M., Sasin, W., Generalized Friedman's Equation and Its Singularities, Acta Cosmologica, 19, 1993, 23-33.

Heller, M., Sasin, W., Lambert, D., Groupoid Approach to Noncommutative Quantization of Gravity, Journal of Mathematical Physics 38, 1997, 5840-5853.

Heller, M., Sasin, W., Non-Commutative Differential Geometry, Acta Cosmologica 21, 1955, 235-245.

Heller, M., Sasin, W., Noncommutative Structure of Singularities in General Relativity, Journal of Mathematical Physics, 37, 1996, 5665-5671.

Heller, M., Sasin, W., Noncommutative Unification of General Relativity and Quantum Mechanics, International Journal of Theoretical Physics 38, 1999, 1619-1642.

Heller, M., Sasin, W., Odrzygóźdź, Z., Noncommutative Quantum Dynamics, Gravitation and Cosmology 7, 2001, 135-139.

Heller, M., Sasin, W., Odrzygóźdź, Z., State Vector Reduction as a Shadow of Noncommutative Dynamics, Journal of Mathematical Physics 41, 2000, 5168-5179.

Heller, M., Sasin, W., Origin of Classical Singularities, General Relativity and Gravitation 31, 1999, 555-570.

Heller, M., Sasin, W., Rigorous Model of Classical Spacetime Foam, International Journal of Theoretical Physics 36, 1997, 1441-1455.

Heller, M., Sasin, W., The Closed Friedman World Model with the Initial and Final Singularities as Non-commutative Space, Banach Center Publications, vol. 41: Mathematics of Gravitation, part I: Lorentzian Geometry and Einstein Equations, red.: P.T. Chruściel, Warszawa 1997, pp. 153-162.

Heller, M., Sasin, W., The Structure of the b-Completion of Space-Time, General Relativity and Gravitation, 26, 1994, 797-811.

Heller, M., Sasin, W., Trafny, A., Żekanowski, Z., Differential Spaces and New Aspects of Schmidt's b-Boundary of Space-Time, Acta Cosmologica 18, 1992, 57-75.

Heller, M., Some Conceptual Problems of the Groupoid Approach to Noncommutative Quantization of Gravity, Acta Cosmologica 24, 1998, 71-85.

Heller, M., Staruszkiewicz, A., Fuzzy Space-Time, Zeitschrift für Naturforschung 36a, 1981, 609-610.

Heller, M., Time and Physics - A Noncommutative Revolution, w: A Collection of Polish Works on Philosophical Problems of Time and Spacetime, Synthese Library, vol. 309, red. H. Eilstein, Kluwer Academic Publishers, Dordrecht-Boston-London 2002, ss. 47-56.

Heller, M., Time of the Universe, w: The Far-Future Universe – Eschatology from a Cosmic Perspective, red.: G.F.R. Ellis, Templeton Foundation Press, Philadelphia-London, 2002, ss. 63-64.

Heller, M., Where Physics Meets Metaphysics, in: A. Connes, M. Heller, Sh. Majid, R. Penrose, J. Polkinghorne, A. Tylor, On Space and Time, red.: Shahn Majid, Cambridge University Press, Cambridge 2008, pp. 238-277.

Multarzyński, P., Heller, M., The Differential and Cone Structures of Spacetime, Foundations of Physics 20, 1990, 1005-1015.

Pysiak, L., Eckstein, M., Heller, M., Sasin, W., Semidirect Product of Groupoids, Its Representations and Random Operators, arXiv:1107.1775 [math-ph]

Pysiak, L., Heller, M., Odrzygóźdź, Z., Sasin, W., Observables in a Noncommutative Approach to the Unification of Quanta and Gravity: A Finite Model, General Relativity and Gravitation 37 (3), 2005, 541-555.

Raszewki, P. K., Geometria Riemanna i analiza tensorowa, PWN, Warszawa 1958.

Sasin, W., Heller, M., Space-Time with Boundary as a Generalized Differntial Space, Acta Cosmologica, 19, 1993, 35-44.

Schmidt, B.G., A New Definition of Singular Points in General Relativity, General Relativity and Gravitation 1, 1971, 269–280.

Seminar on Differential Spaces, Demonstratio Mathematica [numer specjalny] 24, no 3-4, 1991, 347-348.

Sikorski, R., Geometria różniczkowa, PWN, Warszawa 1972.