Mathematical Physics and Differential Geometry

Faculty

Mathematical methods of physics (Witold Kondracki, Andrzej Królak) 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 the statistical theory of detection and estimation of parameters of signals of gravitational and electromagnetic origin.

In particular, Roger Penrose's cosmic censorship hypothesis is studied, asserting that the final state of gravitational collapse of a star of sufficiently large mass is always a black hole. To approach this problem, the methods of geometry and differential topology are used. A. Królak works on theoretical analysis of data from the large-scale gravitational-wave detectors in Germany, France, Italy, Japan, and the USA and recently from the radiotelescope located near Toruń (Poland). The main interest is detection of very weak quasi-periodic signals with a large number of parameters and in a large parameter space.

Algebraic analysis (Danuta Przeworska-Rolewicz) The main idea of Algebraic Analysis is investigating properties of right invertible operators in various spaces. Main advantages of Algebraic Analysis are: (i) simplification of proofs due to an algebraic description of problems under consideration; (ii) algorithms for solving "similar" problems, although these similarities could be only very formal, and (iii) several new unexpected results even in the classical case of the operator d/dt. There are numerous applications to ordinary and partial differential equations with scalar and variable coefficients, functional-differential equations and their discrete analogues, for instance, difference equations (see Encyclopaedia of Mathematics, Supplement, Vol. I, Kluwer, 1997).

Mathematical optimization (Krzysztof Maciej Przyłuski and Stefan Rolewicz) K. M. Przyłuski investigates linear systems. He considers problems of controllability, stability and stabilizability, and observability of such systems. His work is concentrated in two directions. The first one is investigation of infinite dimensional non-stationary systems described in the standard form. They have the form

in the continuous time case and

in the discrete time case, where x(t), x_k belong to a Banach space X called the space of states and u(t), u_k belong to a Banach space U called the space of controls.

The second direction are implicit finite dimensional stationary systems. They have the form

in the continuous case and

in the discrete case. In the case when E is an invertible square matrix they can be reduced to systems in the standard form, but several engineering and economical models require implicit systems.

S. Rolewicz is working in abstract convex analysis and nondifferentiable optimization. The optimization problems are considered in a very general setting, where convexity is defined in spaces without linear structure (see the book [1]).

In the last three years two books were published:

1. D. Pallaschke and S. Rolewicz, Foundations of Mathematical Optimization. Convex Analysis without Linearity, Math. Appl. 388, Kluwer, 1997.
2. D. Przeworska-Rolewicz, Logarithms and Antilogarithms. An Algebraic Analysis Approach, With Appendix by Zbigniew Binderman, Math. Appl. 437, Kluwer, 1998.

In the same period more than 40 other publications were prepared.

In 1997 Song Wen obtained his Ph.D. at the Institute based on his paper on vector optimization and duality in infinite dimensional spaces. His thesis advisor was S. Rolewicz. Song Wen had a graduate fellowship from the Institute.

W. Kondracki and A. Królak together with P. Chruściel organized the Banach Center semester "Mathematical Aspects of Theories of Gravitation", held in 1996. The proceedings were published in Banach Center Publications, vol. 41 (2 parts).

W. Kondracki was among the organizers of the Banach Center symposium "Differential Geometry and Mathematical Physics" and the Banach Center semester "Quantum Groups and Quantum Spaces". The proceedings of the two events, co-edited by W. Kondracki, were published as Banach Center Publications, vols. 39 and 40.

In 1996 a special issue of Integral Transforms and Special Functions, vol. 4, 1-2, was published by Gordon and Breach; it contains the proceedings of the conference "Different Aspects of Differentiability II" (Warsaw, 1995), edited by Danuta Przeworska-Rolewicz.