a) Conformal nets of local algebras

• Topological sectors associated with conformal theories that are orbifolds with respect to cyclic permutations will be studied to yield a quantum index theorem for such sectors (with the Jones index). Applications to constructing sectors with infinite statistics and to strong additivity for rational theories are foreseen. (Longo, Xu)
• Twisted sectors associated with conformal theories that are orbifolds with respect to the full permutation group will be studied and their fusion rules calculated. There will be applications to the theory of Lie algebras in infinite dimensions (Kac-Moody algebras). (Longo, Xu)
• The algebraic approach to conformal theories with boundary will be tackled in relation to conformal theories on two dimensional Minkowski space. (Longo, Rehren)
• An intrinsic entropy associated with a conformal net of von Neumann algebras will be introduced and its relation to the entropy of a black hole studied. (Longo)
• We plan to study positive energy sectors with infinite statistics for conformal field theory on the two dimensional spacetim,e and the possible extension of the local picture of superselection structure as well as the extension of results previously obtained to the case of central charge larger than one (Carpi)

b) Quantum fields on curved spacetime

• The theory of sectors on a globally hyperbolic spacetime with compact Cauchy surface will be studied with particular reference to the existence of a conjugate. (J. Roberts, G. Ruzzi)
• Physics on the forward light cone will be compared with that on the whole spacetime with particular regard to theory of sectors. (D. Buchholz, J. Roberts)

c) Algebraic formulation of the renormalization group

• The analysis of the ultraviolet limit of the superselection structure will be studied in particular models of quantum field theory defined in terms of local C*-algebras, both in the case of localized and of topological sectors, with the aim of testing and illustrating general results previously attained in this field. (C. D'Antoni, G. Morsella)
• The formulation of generally covariant algebraic theories will be studied further and extended to the case of non-globally hyperbolic spacetime using the generally covariant formulation of the renormalization group a la Buchholz and Verch both in the perturbative and the non-perturbative case. (Brunetti)
• The relations between the theory of renormalization in algebraic quantum field theory and the non-commutative generalization of Gromov's notion of tangent cone will be studied. (Guido, Isola, Verch)

d) Field theory on noncommutative manifolds

• Models of quantum field theories on non-commutative manifolds of DFR type and their generalizations will be studied. (Morsella )
• We plan to study deformations of quantum spacetime where the commutator of the coordinates is not central and, in connection with the new mechanisms for ultraviolet regularization associated to spacetime noncommutativity, to investigate the adiabatic and large scale limits, as well as the roots the breakdown of Lorentz invariance.

e) C*-tensor categories - Quantum groups

• The results recently obtained on the characterization of the actions of the regular representation of quantum groups on C*algebras suggest the problem of reconstructing such an action from a rigid braided tensor C*category of endomorphisms generated by its irreducible objects. (Pinzari, Roberts)
• We plan to face this problem first for actions of the dual of quantum deformations of unitary groups at real values of the parameter. We also plan to study the possibility of representing a rigid braided tensor C* category as a tensor category of of Hilbert bimodules in situations where there is no representation in the category of Hilbert spaces. (Pinzari, Roberts)
• We plan to study asymptotically abelian tensor categories and mechanisms for the emergence of braidings or symmetries, and application to the problem of existence of an intrinsic notion of statistics in quantum field theories with massless particles. (Pinzari, Roberts)
• Conjugation on locally compact quantum groups will be studied further and extended to the case of quantum group frames following a recent article of van Daele and Maes. (J. Roberts, D. Parashar)

f) Noncommutative geometry

• Measures and tangential dimensions for classical and non-commutative fractals (Guido, Isola)
• Measures and geometric dimensions for spectral triples in non-commutative geometry. (Guido, Isola)
• Novikov-Shubin invariants and asymptotic dimension for graphs and complexes associated with fractals. (Cipriani, Guido, Isola)
• Noncommutative metric spaces in the sense of M. Rieffel and noncommutative Gromov-Hausdorff convergence. (Guido, Isola)

g) Free probability and factors of type II₁

• The relation between the conjecture of Connes, a non-commutative version of Hilbert's 17th problem and matrix integration will be studied. (Radulescu)
• The moments of products of traces of words in the unitary group U(N) with respect to Haar measure will be computed and applied to embed the group algebra of a group generated by a single relation into the ultrafilter algebra. (Radulescu)

h) Probability and quantum statistics

• Stochastic monotony of the scalar curvature of the BKM metric. (Gibilisco, Isola)
• Non-commutative exponential manifolds. (Gibilisco, Isola)
• Markov fields and Markov chains will be studied further, as well as the natural applications to the statistical mechanics of multidimensional spin models. (Fidaleo)

i) Noncommutative dynamical systems

• We intend to study abstract C*-algebraic versions of Perron-Frobenius-Ruelle theorem (Pinzari-Renault) and the occurence of a variational principle for pressure in asymptotically abelian, non finitely abelian dynamical systems (Kerr-Pinzari).

l) Statistical mechanics of disordered systems

• The structure of temperature states of disordered quantum systems (such as spin glasses) will be studied further, using standard techniques of operator algebras and trying to clarify the connections with "symmetry breaking by replicas." (Fidaleo)