Today the last day of the workshop, with a morning full of seminar talks and afternoon with final discussion and closing remarks.

The seminar of Mairie Sakellariadou contained an overview of cosmological consequences of the noncommutative spectral model, and vice versa, how cosmological observations put restrictions on some of the parameters in that model. An interesting point raisen was on topological defects that might appear in the course of spontaneous symmetry breaking. As the Pati-Salam model seems to be suggested by noncommutative geometry, it is interesting to see whether breaking this to the Standard Model fits with cosmology.

Next, John Barrett presented an interesting list of matrix geometries, building Dirac operators using Clifford and matrix algebras. In particular, fuzzy circles and fuzzy spheres fit in this scheme.

The long-awaited almost-associative geometries were presented in the last two seminars of the workshop by Latham Boyle and Shane Farnsworth. Though the general theory of non-associative noncommutative geometry is still to be developed, there are interesting models available with potential applications to unified theories. Intriguing similarities were displayed with the inner perturbations in the abscence of the first-order condition.

After lunch, we had concluding remarks from the study groups.The Lorentzian NCG group made some progress in analyzing a distance formula for Lorentzian manifolds, working towards a translation of causal conditions in terms of spectral data. The Higgs vacuum stability group concluded that in order to say something about the cosmological consequences, a better understanding is needed about the physical meaning of the spectral action. Summarizing, this left the participants of the workshop with the following crucial, but unanswered question:

If the spectral action

that yields the Standard Model Lagrangian is understood as an effective theory at scaleTr f(D/Λ)Λ, how can one supress the higher-order terms proportional towhen the bosonic fields have momenta of that same order?1/Λ

A possible answer was suggested earlier this week by Alain Connes, by taking the function ** f **to be a cutoff function so that the asymptotic expansion actually only contains the nonnegative powers of

*. However*

**Λ***,*this raised the question of what the physical meaning is of such a choice of cutoff function.

Nevertheless, I consider the clarity with which we arrived at this open problem as one of the great successes of this week! Thanks to all the participants!