Volume 2009, Issue 636

We construct a model complete and o-minimal expansion of the field of real numbers such that, for any planar analytic vector field ξ and any isolated, non-resonant hyperbolic singularity p of ξ , a transition map for ξ at p is definable in . This expansion also defines all convergent generalized power series with natural support and is polynomially bounded.

We provide a proof for an inequality between volume and L 2 -Betti numbers of aspherical manifolds for which Gromov outlined a strategy based on general ideas of Connes. The implementation of that strategy involves measured equivalence relations, Gaboriau's theory of L 2 -Betti numbers of ℛ-simplicial complexes, and other themes of measurable group theory. Further, we prove new vanishing theorems for L 2 -Betti numbers that generalize a classical result of Cheeger and Gromov. As one of the corollaries, we obtain a gap theorem which implies vanishing of L 2 -Betti numbers of an aspherical manifold when its minimal volume is sufficiently small.

We introduce the notion of ℒ-optimal transportation, and use it to construct a natural monotonic quantity for Ricci flow which includes a selection of other monotonicity results, including some key discoveries of Perelman [The entropy formula for the Ricci flow and its geometric applications, 2002] (both related to entropy and to ℒ-length) and a recent result of McCann and the author [Topping, Amer. J. Math.].

This paper is the follow-up of [Merkulov and Vallette, J. reine angew. Math. 634: 51–106, 2009].

In this note we prove that the Bass stable rank of is two. This establishes the validity of a conjecture by S. Treil. We accomplish this in two different ways, one by giving a direct proof, and the other, by first showing that the topological stable rank of is two. We apply these results to give new proofs of results by R. Rupp and A. Sasane stating that the Bass stable rank of is two and the topological stable rank of is two, settling a conjecture by the second author. We also present a -free proof of the second author's characterization of the reducible pairs in .

We propose a three dimensional generalization of the geometric McKay correspondence described by Gonzales-Sprinberg and Verdier in dimension two. We work it out in detail in abelian case. More precisely, we show that the Bridgeland-King-Reid derived category equivalence induces a natural geometric correspondence between irreducible representations of G and subschemes of the exceptional set of G -Hilb. This correspondence appears to be related to Reid's recipe.