| Research Programme in relation to the state-of-the-art:
In 1998, Connes and Moscovici discovered a new type of cyclic cohomology. Now our aim is to develop general cyclic theory with coefficients, to discover its geometrical meaning, and to apply it in the thus far unreachable examples (Hajac, Maszczyk, Zielinski).
The other main strand of our research is related to topological quantum groups. These have been initiated by the pioneering work of Woronowicz in late 1980s, reached maturity around 2000, and are now the subject of intensive investigation. In our group, the main focus is put on the study of actions of quantum groups on C*-algebras (Baum, Hajac, Rudnik, Soltan) and deep connections to classical and quantum probability (Skalski). In particular, Soltan and Skalski intend to exploit their complementary research experience and expertise to construct and analyse non-compact quantum symmetry groups. On the purely algebraic side, the categorical understanding of Hopf algebras seems to unify Hopf-Galois theory, theory of invariants, faithfully flat descend of Grothendieck and algebraic geometry. This path of investigations towards an appropriate framework for noncommutative geometry is pursued by Maszczyk.
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