# Quantum Groups and Noncommutative Geometry

## Research topics

- Non-commutative geometry and quantum field theory.
- Dirac operators on quantum homogeneous spaces and construction of spectral triples.
- ADHM construction of non-commutative instatons in SUq(2) gauge theories. Non commutative moduli spaces.
- Representations of SUq(2) at roots of unity and extensions of the spin covering with applications to exotic statistics.

## Research Plan

The research plans for next few years concentrate on two main lines.

- The construction of instantons and their moduli spaces on quantum spaces, and their applications. A first task is to complete the study of quantum moduli spaces for instantons on quantum S4. In the longer perspective this serves to valuate the possibility to set up the stage for a generic quantum 4-manifold, in view of non-commutative generalizations of the Donaldson theorem.
- The study of the differential structure (Riemannian metric and spin) on quantum spaces, and their applications. Firstly, we intend to proceed with the study of the relations between the quantum group deformations and the Connes spectral triples.

## Prin

Noncommutative geometry of quantum groups and their homogeneous spaces