Katedra Algebry i Geometrii / Division of Algebra and Geometry

Słoneczna 54, 10-710 Olsztyn
Tel.: (4889) 524 60 34
e-mail: marzena.smiech@matman.uwm.edu.pl

Last update: 20/12/2011

Research at the
DIVISION OF ALGEBRA AND GEOMETRY


At present members of the Division are primarily occupied with the following research topics:

Symplectic geometry and topology (A. Matraś, P. Rusznicki, A. Tralle)
Geometric discretization of integrable systems (A. Doliwa)
Foundations of geometry (J. Jakóbowski, A. Matraś)
Universal algebra (B. Staruch)
Mathematical logic (A. Orlicki).

PARTICULAR RESEARCH PROJECTS REALIZED IN THE DIVISION
(years 2003-2005)

Hamiltonian Lie group actions on closed symplectic manifolds (project coordinator A. Tralle)

This is a project financed by the University. It lies in the framework of a general topic of studying closed symplectic manifolds equipped with a compact Lie group acting by symplectomorphisms. It is known that the existence of such structures imposes some topological restrictions on manifolds (and, in particular, on their equivariant cohomology). The project is aimed at studying the nature of these restrictions, and, in a broader perspective, homotopy properties of symplectic reduction.

Geometric methods of discretization of integrable systems (project coordinator A. Doliwa)

The project is aimed at developing differential-geometric methods of finding discrete analogues of some differential equations of physical interest (e.g. solitons), as well as geometric interpretations of these objects. An important feature of this work is developing of methods which can be translated in case of finite geometries. This, in turn, has an application to the theory of cellular automata.

Symplectic structures on symplectically aspherical manifolds (project coordinator B. Hajduk, University of Wroclaw, project financed by Polish Research Committeee.

It is known that symplectically aspherical manifolds constitute a natural toolkit in symplectic topology. They appear in many contexts, for example, the Arnold conjecture, in its original formulation, is completely proved for such manifolds. The project is aimed at looking for explicit constructions of symplectically aspherical manifolds and studying their topological properties. The natural broader context for such questions is the studying of groups of symplectomorphisms and exotic symplectic structures on such manifolds.

Topology of symplectic manifolds and related problems of mechanics (project coordinators J. Rychlewski and Manuel de Leon (CSIC, Madrid, Spain).

This is a project realized under the agreement between Polish Academy of Sciences and Consejo Superior de Investigaciones Cientificas, with individuals from the Division involved.

The project is primarily occupied with explicit constructions of closed symplectic manifolds with prescribed (topological) properties.

Problems of teaching of mathematics and physics in the newly-reformed school (project coordinator Sz. Brym)

This is an educational project related with problems of integrated teaching in gymnasium, where mathematics and physics are taught in the context of natural sciences. One of the aims of the project is to study the consequences of the school reform and introducing the concept of integrated teaching in the Warmia and Mazury region.

INTERNATIONAL COOPERATION

The research of our division is developed in a cooperation with persons and Universities all over the world, icluding Germany (Technical University of Munich, Professor Hans-Joachim Kroll), Italy (University "La Sapienza", Rome, Professor Santini), USA (Prof. Yuli Rudyak, University of Florida), Spain (Universities in Bilbao and Zaragoza, Professors Raul Ibanez and Luis Ugarte, CSIC, Madrid).

Aleksy Tralle is a member of Group Action Forum and a member of Organizing Committee of the workshop "Symplectic Topology" in the Banach Center in 2004, together with Professors Bogusław Hajduk (University of Wroclaw), and Leonid Polterovich (Tel-Aviv University) and dr Jaroslaw Kedra (Universities of Szczecin and Munich).