Noncommutative geometry and quantum symmetry

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AbstractMoscovici The goal of this project is to refine and further develop the new insights and tools for the treatment of quantum symmetryin noncommutative geometry which emerged from the recent joint work of A. Connes and the PI, in order to expand their range of applications to transverse differential geometry, the algebraic and the analytic theory of quantum groups, as well as to quantum field theory. The main new insight, that symmetry in Diff-equivariantgeometry comes organized in the form of specific Hopf algebras,will be investigated for all geometrically interesting classes oftransverse geometries and then employed to settle the correspondingtransverse index problem. The main new tool, provided by the adaptation of cyclic cohomology to Hopf algebras, will be appliedto obtain new cohomological invariants of quantum groups and of theiractions. The proposed work is motivated by the necessity of developingappropriate mathematical tools for handling a wealth of examplesof non-classical spaces, arising in many areas of mathematics and physics, whose common feature is that their measurable local parameters behavelike infinite-dimensional matrices rather than numbers. This requires aconsiderable stretching and rethinking of the existing geometric techniques and led to the development of noncomutative geometry. Theemphasis of the present project is on the the treatment of suchnoncommutative spaces in the presence of non-classical symmetries.