projects in noncommutative geometry

Tomorrow I’ll start with the course Projects in non-commutative geometry in our masterclass. The idea of this course (and its companion Projects in non-commutative algebra run by Fred Van Oystaeyen) is that students should make a small (original if possible) work, that may eventually lead to a publication.
At this moment the students have seen the following : definition and examples of quasi-free algebras (aka formally smooth algebras, non-commutative curves), their representation varieties, their connected component semigroup and the Euler-form on it. Last week, Markus Reineke used all this in his mini-course Rational points of varieties associated to quasi-free algebras. In it, Markus gave a method to compute (at least in principle) the number of points of the non-commutative Hilbert scheme and the varieties of simple representations over a finite field. Here, in principle means that Markus demands a lot of knowledge in advance : the number of points of all connected components of all representation schemes of the algebra as well as of its scalar extensions over finite field extensions, together with the action of the Galois group on them … Sadly, I do not know too many examples were all this information is known (apart from path algebras of quivers).
Therefore, it seems like a good idea to run through Markus’ calculations in some specific examples were I think one can get all this : free products of semi-simple algebras. The motivating examples being the groupalgebra of the (projective) modular group PSL(2,Z) = Z(2) * Z(3) and the free matrix-products $M(n,Fq) * M(m,Fq)$. I will explain how one begins to compute things in these examples and will also explain how to get the One quiver to rule them all in these cases. It would be interesting to compare the calculations we will find with those corresponding to the path algebra of this one quiver.
As Markus set the good example of writing out his notes and posting them, I will try to do the same for my previous two sessions on quasi-free algebras over the next couple of weeks.

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