The new technique arises from an insight that instead of looking at just one moduli space at a time it is important to look at an associated family of moduli spaces and that this family of moduli spaces is then birational to the finite dimensional representations of a (highly) noncommutative ring. To make this clearer I need to say how this family of moduli spaces is chosen and what it means informally for this family to be birational to the finite dimensional representations of a ring.
The objects we are trying to classify by the points of the moduli space are in all cases objects in some abelian category and they have both continuous and discrete invariants. Thus vector bundles over a smooth projective curve have a rank and a degree which are discrete invariants; they also have a determinant line bundle which is a continuous invariant. In all cases considered the natural thing to do is to consider the moduli space of objects whose discrete part of the Grothendieck class is fixed. This fixes all discrete invariants. In the case of vector bundles over smooth projective curves it fixes precisely the rank and degree. Now instead of considering simply this moduli space we should consider the moduli spaces of all objects whose discrete part of the Grothendieck class is a rational multiple of the one we are interested in. In the case of vector bundles over a smooth projective curve this amounts to considering vector bundles such that the ratio of the rank and degree is fixed. In all cases considered there is a smallest Grothendieck class β of which all the others are an integer multiple. Thus the set of moduli spaces we are interested in is parametrised by the natural numbers. We'll refer to this as the β family of moduli spaces.
The moduli spaces of finite dimensional representations of a ring are also parametrised by the natural numbers since the dimension of the representations parametrised by a connected variety will be constant. Now it is clear what we mean when we say that the β family of moduli spaces should be birational to the finite dimensional representations of a ring; the moduli space of objects of class nβ should be birational to the moduli space of representations of dimension n.
From this point of view, there is an obvious notion of noncommutative rationality for such a family; we'll say that a β family is noncommutative rational if it is birational to the finite dimensional representations of a finitely generated free algebra. There are two cases where I showed such a result. If we consider vector bundles over the projective plane, then in this case the Grothendieck group is free abelian of rank 3 and if we fix a Grothendieck class β which is not an integer multiple of any other class (an indivisible class) and for which there are vector bundles of this class then the corresponding β family is noncommutative rational. Again if we consider representations of a quiver with n vertices, then the Grothendieck group is free abelian of rank n and if we fix an indivisible class β for which there are representations with trivial endomorphism ring then the β family is noncommutative rational.
This noncommutative rationality result implies all the previously studied cases where rationality had been proved because of what is known about the rationality problem for moduli spaces of finite dimensional representations of the free algebra. At present, all we know about this is that for n=1,2,3 and 4, this moduli space is rational, for n dividing 420 it is stably rational, and for n squarefree or twice a squarefree it is retract rational.
In the case of vector bundles over a smooth projective curve, we have a rather different answer. Let C be a smooth projective curve of genus g>1. Let R be the coordinate ring of an affine open piece of the curve; consider the ring T such that Mg(T) is isomorphic to the algebra freely generated by Mg(k) and R. The discrete part of the Grothendieck group is isomorphic to the free abelian group of rank 2. Fixing the discrete Grothendieck class simply prescribes the rank and degree of a vector bundle. If β is an indivisible class we are simply asserting that the rank and degree are coprime and the β family classifies the semistable vector bundles whose slope is fixed. In this case, I can show that this β family is birational to the finite dimensional representations of a ring U which is noncommutatively stably isomorphic to T.
Here noncommutatively stably isomorphic means that there exist free algebras F1 and F2 such that the algebra freely generated by U and F1 is isomorphic to the algebra freely generated by T and F2. In the circumstances we are interested in it implies that the moduli spaces of representations of dimension n are stably birational.