Noncommutative geometry entered the research programme of IMPAN in 1999. Five years later, with the help of the Warsaw University transfer-of-knowledge grant Quantum Geometry, this branch of IMPAN's mathematics gained an international dimension. Since 2004, there are about 10-20 visitors per year who contribute their research experience and give talks at the weekly Noncommutative Geometry Seminar held in the Institute. Among our invitees were Alain Connes and Maxim Kontsevitch, and the seminar talks are announced to about 200 mathematicians worldwide.
The aforementioned scientific activity helped to cristalize a local research team consisting of Piotr M. Hajac, Ulrich Krähmer, Tomasz Maszczyk and Bartosz Zieliński. Ulrich Krähmer was a Marie Curie fellow in the years 2005-2007. In January 2008, the Noncommutative Geometry Research Unit was formally created by the Institute Scientific Council. In October 2008, the team was enlarged by Emily Burgunder, who chose IMPAN for her European Postdoctoral Institute fellowship.
Another chapter of Noncommutative Geometry at IMPAN opened in 2009 with the EU- project Geometry and Symmetry of Quantum Spaces. Co-sponsored by the Polish Government, this 4-year international research staff exchange programme established a transcontinental network of 12 nodes with IMPAN as the co-ordinating node. In particular, we welcomed in our group Paul F. Baum who joined us as a Visiting Professor working at IMPAN a month each year. Our mathematical environment was further enriched by Adam Skalski who came as another Marie Curie Postdoctoral Fellow for the years 2010-2012. Together with Paweł Kasprzak, Andrzej Sitarz, and Piotr M. Sołtan employed on short-term position, our Research Unit got top expertise in topological quantum groups and spectral geometry. On the other hand, a Ph.D.-student Jan Rudnik started his collaboration with Baum and Hajac on computing the K-theory of triple-pullback C*-algebras.
The key words characterizing IMPAN's research in noncommutative geometry are: K-theory of operator algebras and free actions of compact quantum groups on unital C*-algebras, multi-pullback C*-algebras and free distributive lattices of ideals, index theory of Fredholm modules and spectral geometry of Dirac operators, locally compact quantum groups and universal (free) quantum groups, Hopf-cyclic homology with coefficients and Chern-Galois character, corings and monoidal categories. The assumed research strategy is to explore the feedback between solving concrete difficult problems and developing new mathematical structures. The proposed approach is to unite rather than separate different fields of mathematics by taking advantage of complemetary tools that they offer. To this end, a large scale and intensive international collaboration is currently sustained and planned for the future.
A more systematic and detailed description of the aforementioned research profile is as follows:
1. K-theory of operator algebras. Computing K-theory of C*-algebras of quantum projective spaces of Toeplitz-type. Noncommutative version of the Borsuk-Ulam for a family quantum spheres. Non-existence of Z/2-equivariant homomorphisms from a unital C*-algebra A into its unital suspension SA. Computing the K-theory of noncommutative Bieberbach manifolds. An example of a published paper in this area of research is: P. F. Baum, P. M. Hajac, R. Matthes, W. Szymański, The K-theory of Heegaard-type quantum 3-spheres, K-Theory (2005) 35:159-186.
2. Hopf-cyclic homology and Chern-Galois character. Applying new coefficients of Hopf-cyclic homology and cohomology. Studying relationships between De Rham cohomology with coefficients in flat vector bundles and cyclic homology with coefficients. Version of the local index formula of Connes-Moscovici for twisted cyclic cohomology. Extending cyclic (co)homology with coefficients from algebras to monoidal functors. Among pivotal papers for this area of research are: P. M. Hajac, M. Khalkhali, B. Rangipour, Y. Sommerhäuser, Hopf-cyclic homology and cohomology with coefficients, C. R. Acad. Sci. Paris, Ser. I 338 (2004) 667-672; T. Brzeziński, P. M. Hajac, The Chern-Galois character, C. R. Acad. Sci. Paris, Ser. I 338 (2004) 113-116.
3. Spectral geometry and index theorem. Proving an analogue of the 2-dimensional Gauss-Bonnet for spectral triples. Stability of spectral triples and regular spectral geometries. Constructing spectral triples on cross-products of C*-algebras and studying their topological properties. Studying contact structures by means of noncommutative geometry. An example of a published paper related to this area of research is: P. M. Hajac, R. Matthes, W. Szymański, Noncommutative index theory of mirror quantum spheres, C. R. Acad. Sci. Paris, Ser. I 343 (2006) 731-736.
4. Quantum group actions. Equivalence of principality of actions by compact quantum groups on unital C*-algebras and the Hopf-Galois condition for induced coactions of corresponding polynomial Hopf algebras. Applying non-contractibility of compact quantum groups to prove non-triviality of noncommutative principal bundles obtained by means of the join construction of compact quantum groups. Proving principality of piecewise principal actions. Constructing examples of quantum spaces without group structure and actions of non-compact locally compact quantum groups. Quotienting of locally compact quantum groups by their closed subgroups. Studying quantum symmetry groups of group C*-algebras, especially focused on their representation theory. In particular, classification of compact group actions on the C*-algebra of 2 by 2 matrices. Finding non-classical quantum permutations on two elements. Extending Hopf-Galois theory to monoidal categories. An example of a published paper in this area of research is: P. M. Sołtan: Examples of non-compact quantum group actions, J. Math. Anal. Appl. 372 (2010), 224-236.
5. Quantum spaces, sets, and cohomology. Noncommutative deformations of complex projective spaces glued from Toeplitz cubes and other multi-pullback constructions of algebras. Constructing of a category of quantum sets (or quantum algebraic sets) and extenting of the contravariant adjunction between the category of sets and the category of commutative algebras to noncommutative sets and associative algebras. Generalizing the notion of a discrete (or algebraic) group to the corresponding notion making sense for quantum sets (or quantum algebraic sets). Constructing a recursive algorithm for computing genus zero Gromov-Witten invariants of some Fano varieties, generalizing the formula of Kontsevich. An example of a published paper in this area is: P. M. Hajac, A. Kaygun, Bartosz Zieliński, Quantum complex projective spaces from Toeplitz cubes, J. Noncommutative Geometry, 6: 3 (2012) 603-621.