The Differential Geometry and Topology Team was founded in 1991. It is headed by Assistant Prof. Jan Kubarski, Professor of the Technical University of ŁódĽ.

The subject matter of the research work includes the application of Lie algebroids to many fields of differential geometry, mainly principal fibre bundles, Poisson manifolds, or some regular transversally complete foliations. Recently, there have been started the geometrical studies of Lie-Rinehart algebras, which are an algebraic equivalent of Lie algebroids. These algebras have been applied to the theory of Poisson algebras and some problems of mathematical physics. To the above studies the material of differential topology, including the theory of homotopy, homology, and characteristic classes, has been applied. Another field of research is foliations on differential spaces; some studies have been devoted to Ahlfors operator, and Riemannian manifolds with magnetic tensor.

Under grant No. 173/P03/97/13, awarded by the Scientific Research Board, there have been realised the following subjects:

• fibre integral in the cohomology of Lie algebroids,
• Poincaré duality and its interpretation for geometrical objects determining Lie algebroid infinitesimally (e.g., principal fibre bundles, Lie groupoids, TC-foliations, Poisson manifolds),
• Gysin class and sequence,
• index of flat connection in an isolated singularity and the sum of the indexes (Euler-Poincaré-Hopf Theorem).

There have also been studied the following subjects:

• Chern-Weil characteristic classes of pairs of Lie algebroids,
• the secondary homomorphism characteristic for regular Lie algebroids,
• homotopic classification of endomorphisms of Lie algebroids
• Jacobi equation and Lapunov index for magnetic streams,
• boundary issues for Ahlfors operator.

• Lie-Rinehart nonunital algebras (including tensor product of modules over nonunital algebras and Picard group),
• foliations in commutative algebras,
• Lefschetz class of Lie transitive algebroids,
• K-C-F unification of three theories of secondary characteristic classes of Lie algebroids (Kubarski-Crainic-Fernandes),
• the secondary characteristic homomorphism of Lie-Rinehart algebras,
• foliations in differential spaces.

Since 1998 the Team has been organising a cycle of international conferences “Geometry and Topology of Manifolds”, with the aim to make a survey of the research in geometry, topology, and analysis on manifolds in Poland and Europe (mainly Central and Eastern), in particular that which can be applied to the theory of Lie algebroids and, generally, to these fields of differential geometry, where Lie functor appears in the category of algebroids, cf., http://im0.p.lodz.pl/konferencje

The Team co-operates with some universities from Poland (the University of ŁódĽ, the University of Warsaw, the Jagiellonian University, and the University of Mining and Metallurgy) and abroad (the Moscow University – Prof. Alexander Mishchenko, the University of Toulouse – Prof. Jean Pradines, the University of Haifa – Prof. Izu Vaisman). There has also been started some co-operation with Departamento de Matematica Instituto Superior Technico (Lisbon, Portugal), and with the National Technical University of Ukraine (Kiev Polytechnic).

Members of the Team are reviewers of Mathematical Reviews, and Zentralblatt für Mathematik. They have also numerously acted as reviewers for some Polish journals (e.g., Demonstratio Mathematicae, and Banach Center Publications), as well as some foreign ones (e.g., Acta Sci Math. Szeged, Journal Acta Math. U.C.). Some books published by WNT (Scientific and Technical Publishers) or university publishers (the Technical University of Koszalin), as well as several doctoral and habilitation theses have also been reviewed in the Team.

In the Team one habilitation and two doctoral theses have been written. The Head of the Team tutors four research workers; two of them have been working on their doctoral theses. Members of the Team have also published a course of lectures for the students of the Technical University of ŁódĽ.

The Head of the Team was awarded a two-year grant by the State Committee for Scientific Research (VII-97-VI-99). Within its scope he has prepared six papers for publication and delivered two lectures at international conferences.

The results of the Team’s scientific research from 1990s have been published in nine papers in renowned journals, like: Transactions of the AMS, Publicationes du Dept. de Math. Univ. C.Bernard LYON 1, Revista Matematica Universidad Complutense Madrid, Bull. Soc. Math. France, Topology and Applicationes, Publ.Math.Debrecen, Rendiconti del Circolo Matematico di Palermo. Moreover, members of the Team have published twelve original papers in materials from big international conferences; some of them have the status of “high level” conferences (e.g., conferences in Tokyo, Santiago, Budapest, Vigo, and some conferences at the Stefan Banach International Mathematical Centre). In total, thirty lectures have been delivered at international conferences (in Japan, Spain, the Czech Republic, Slovakia, France, Russia, Hungary, Brazil, Ukraine), including three at the Congresses of European Mathematics (Spain, Hungary, France).