Institute of Mathematics of PAN - Mathematical Physics and Differential Geometry

Department of Mathematical Physics and Differential Geometry

Head:

Janusz Grabowski (Professor)
email
pok. 112, tel.: 022 5228 112

Staff:

Witold Kondracki (Associate Professor)
email
pok. 507, tel.: 022 5228 175
Andrzej Królak (Professor)
email
pok. 507, tel.: 022 5228 175
Daniele Malafarina (Assistant Professor) (May.2008 - Oct.2008)
Danuta Przeworska-Rolewicz (Professor)
email
Stefan Rolewicz (Professor)
email
pok. 115, tel.: 022 5228 115
Piotr Wojdyłło (Assistant Professor)
email
pok. 418, tel.: 022 5228 239

Research activities:

Principal structures of geometry and physics (Janusz Grabowski)
Canonical algebraic and geometric structures that play fundamental role in description of various systems in physics are studied in the language of differential geometry and super-geometry. Among them are Poisson and Jacobi structures (e.g. symplectic and contact), Lie and Courant algebroids, Dirac structures, generalized (e.g. complex) geometries, Nijenhuis tensors and the corresponding contractions, principles of variational calculus – all this with applications to Theoretical Mechanics, especially to frame-independent description of mechanical systems, foundations of Quantum Mechanics and the geometry of quantum states, quantum information and description of entanglement.
1. J. Grabowski, K. Grabowska: Variational calculus with constraints on general algebroids, J. Phys A: Math. Theor. 41 (2008), to appear.
2. J. Grabowski, G. Marmo and M. Kuś: On the relation between states and maps in infinite dimensions, Open Syst. Inf. Dyn. 14 (2007), 355-370
Mathematical methods of physics (Witold Kondracki, Andrzej Królak)
Main activity are applications of methods of statistics and the theory of stochastic processes to the problem of detection of gravitational waves in the noise of the detector. In particuar in the paper [1] optimal statistic to search for modulated periodic gravitational waves from rotating neutron stars is derived. This optimal statistic is now commonly used in the gravitational wave data analysis in particular in the Einstein@Home project.

In paper [2] it is shown that the response of a detector to the superposition of tens of millions of gravitational wave signals from binary white dwarf systems is a cyclo-stationary random process. Using these methods real data from resonant bar detectors EXPLORER and NAUTILUS are analyzed. Also several problems of the Lorentzian geometry in the large, in particular properties of Cauchy horizons have been studied.
1. P. Jaranowski, A. Królak, and B. F. Schutz: Data analysis of gravitational-wave signals from pulsars. I. The signal and its detection, Phys. Rev. D58 (1998) 063001.
2. J. Edlund, M. Tinto, A. Królak, and G. Nelemans: The White Dwarf - White Dwarf galactic background in the LISA data, Phys. Rev. D71 (2005) 122003.
The main areas of research are geometrical structures of nonlinear field theories, in particular of the Einstein theory of gravitation and Yang-Mills gauge theories, and statistical methods of detection and estimation of signals of gravitational and electromagnetic origin.

In particular we carried out research on the long-standing hypothesis of Roger Penrose - the cosmic censorship hypothesis, which asserts that the final state of gravitational collapse of a star of sufficiently large mass is always a black hole. To approach this problem we have used methods of geometry and differential topology. We work on theoretical and practical aspects of analysis of data from gravitational-wave detectors. Ground-based detectors of gravitational waves are currently working in Germany, France, Italy, Japan, and the USA. NASA and the European Space Agency are planning to put a gravitational wave detector (LISA project) in an orbit around the Sun. Detection of gravitational waves will be a final confirmation of Einstein’s theory of gravity and will open a new window on the Universe. Our main interest is detection of very weak, quasi-periodic signals in large parameter spaces. Using our theoretical methods and algorithms we currently analyse data from Italian NAUTILUS resonant bar detector. Analysis is performed on a large network of computers. We study theoretical methods and we develop data analysis tools to analyse the gravitation wave signal originating from superposition of many signals from binary systems in our Galaxy. This is the dominant gravitational wave signal that will be present in the data of a space-borne detector LISA.

We have taken part in organization of several conferences in Banach International Mathematical Center:
- Symplectic Singularities and Geometry of Gauge Fields
- Quantum Groups and Quantum Spaces
- Mathematics of Gravitation
- Mathematics of Gravitation II
The team has strong collabor
1. W. Kondracki and J. Rogulski, Metody geometryczne w mechanice klasycznej i klasycznej teorii pola, Wydawnictwa Uniwersytetu Warszawskiego, 1978.
2. C. J. S. Clarke and A. Królak, Conditions for the occurrence of strong curvature singularities, Journal of Geometry and Physics, Vol. 12 (1985), 127.
3. W. Kondracki and J. Rogulski, On the stratification of the orbit space for the action of automorphisms on connections, Dissertationes Mathematicae, Vol. 250 (1986).
4. A. Królak and B. F. Schutz, Coalescing binaries - probe to the Universe, General Relativity and Gravitation, Vol. 19 (1987), 1163. (Second Prize by Gravity Research Foundation, Ma, USA).
5. A. Królak, Towards the proof of the cosmic censorship hypothesis , Classical and Quantum Gravity Vol.3 (1986), 267.
6. S. Klimek and W. Kondracki, A construction of two-dimensional quantum chromodynamics, Commun. Math. Phys. Vol. 113 (1987), 389.
7. P. Jaranowski and A. Królak, Optimal solution of the inverse problem for the gravitational wave signal of a coalescing compact binary, Phys. Rev. D49 (1994), 1723.
8. P. Jaranowski, A. Królak, and B. F. Schutz, Data analysis of gravitational-wave signals from pulsars. I. The signal and its detection, Phys. Rev. D58 (1998), 063001.
9. R. Budzyński, W. Kondracki, and A. Królak, New properties of Cauchy and event horizons, Nonlinear Analysis Vol. 47 (2001), 2983.
10. A. Królak, Cosmic Censorship Hypothesis, Contemporary Mathematics, Vol.359 (2004), 51.
11. P. Jaranowski and A. Królak, Gravitational Wave Data Analysis: Formalism and Applications, Living Reviews, lrr-2005-3, 2005
Algebraic Analysis, algebras with logarithms (Danuta Przeworska-Rolewicz)
New approach to logarithmic mappings as invertible selectors of multifunctions is proposed together with some applications of logarithmic and antilogarithmic mappings to linear and nonlinear equations in algebras of matrices. Also problems of algebraic analysis of algebraic structures with right-invertible operators.
1. D. Przeworska-Rolewicz: Logarithms and Antilogarithms. An Algebraic Analysis Approach, Kluwer Academic Publishers, Dordrecht, 1998.
2. [2] D. Przeworska-Rolewicz: Algebraic Analysis in structures with Kaplansky-Jacobson property, Studia Math. 168(2005), 165-186.
Applications of Functional Analysis and Theory of Metric Spaces to Optimization. Paraconvex analysis. (Stefan Rolewicz)
Problems concerning strongly paraconvex functions on an open convex subset in a Banach space X are studied, e.g. their differentiability [1], as well as an extension of the notion ofstrongly paraconvex (uniformly approximate convex) functions on differentiable manifolds and Φ-convexity in metric spaces [2].
1. S. Rolewicz: On differentiability of strongly α(.)-paraconvex functions in non-separable Asplund spaces, Studia Math. 167, (2005), 235 – 244.
2. S. Rolewicz: Φ-convex functions de¯ned on metric spaces, Journal of Mathematical Sciences (ed. Plenum) 115 (2003), 2631 - 2652.
Wavelets and Data Analysis (Piotr Wojdyłło)
Wilson systems play an essential role in the construction of orthonormal bases from the Gabor tight frames with certain redundancy. The research shows that there is a strong connection between the Wilson system construction and the automorphisms of the underlying system of operators, especially as they are unitarily implemented.

The wavelet analysis of time series is investigated, showing the so called Long-Range Dependence. This behaviour is observed, e.g. in packet load in web traffic or in economical processes. To this group of stochastic processes belong, in particular, 1/f noises.
1. P.Wojdyłło:Characterization of Wilson Systems for General Lattices, International Journal of Wavelets, Multiresolution and Information Processing, 6/2(2008), 305-314.
2. P.Wojdyłło: Modified Wilson Orthonormal Bases, Sampling Theory in Signal and Image Processing, 6 (2007), 223-235.
Symplectic geometry (Aleksy Tralle, Bogusław Hajduk)
Explicit constructions of symplectic manifolds with prescribed properties are performed. The most important problem in this area is the existence problem for symplectic structures on closed manifolds. It is generally believed that closed symplectic manifolds do not possess any special topological properties, however, the main problem in this area (Thurston's conjecture) still is not solved. Several questions of the same type were aked about some other homotopic properties. For example, it was asked if there are relations between hard Lefschetz property, formality and Betti numbers of closed symplectic manifolds.
1. B. Hajduk, A. Tralle, Diffeomorphisms and almost complex structures on tori, Annals Global Anal. Geom.28 (2005), 337-349.
2. B. Hajduk, A. Tralle, Exotic smooth structures and symplectic forms on closed manifolds, Geom. Dedicata 132 (2008), 31-42.
Bihamiltonian Systems and Integrability (Andriy Panasyuk)
Integrability, and Poisson pensils are studied in the context of bihamiltonian systems.
1. A. Panasyuk:Reduction by stages and the Raïs-type formula for the index of a Lie algebra with an ideal, Ann. Global Anal. Geom. 33 (2008), 1-10.
2. A. Panasyuk:Algebraic Nijenhuis operators and Kronecker Poisson pencils, Differential Geom. Appl. 24 (2006), 482-491.

Annual Scientific Report

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