While the classification project for the simple groups of finite Morley rank is unlikely to produce a classification of the simple groups of finite Morley rank, the enterprise has already arrived at a considerably closer approximation to that ideal goal than could have been realistically anticipated, with a mix of results of several flavors, some classificatory and others more structural, which can be combined when the stars are suitably aligned to produce results at a level of generality which, in parallel areas of group theory, would normally require either some additional geometric structure, or an explicit classification. And Bruno Poizat is generally awesome, though sometimes he goes too far.

In this short note we present a result of Perelman with detailed proof. The result states that if $g(t)$ is the Kähler Ricci flow on a compact, Kähler manifold $M$ with $c_1(M)>0$, the scalar curvature and diameter of $(M,g(t))$ stay uniformly bounded along the flow, for $t\in[0,\infty)$. We learned about this result and its proof from Grigori Perelman when he was visiting MIT in the spring of 2003. This may be helpful to people studying the Kähler Ricci flow.

We study model-theoretic and stability-theoretic properties of the non-abelian free group in the light of Sela's recent result on stability and results announced by Bestvina and Feighn on ‘negligible subsets' of free groups. We point out analogies between the free group and so-called bad groups of finite Morley rank, and prove ‘non-CM-triviality' of the free group.

We establish an identification result of the projective special linear group of dimension 2 among a certain class of groups the Morley rank of which is finite.

The Poisson and Martin boundaries for invariant random walks on the dual of the orthogonal quantum groups $A_{\mathrm{o}}(F)$ are identified with higher-dimensional Podleś spheres that we describe in terms of generators and relations. This provides the first such identification for random walks on non-amenable discrete quantum groups.

We introduce a new topology, weaker than the gap topology, on the space of self-adjoint operators affiliated to a semifinite von Neumann algebra. We define the real-valued spectral flow for a continuous path of self-adjoint Breuer–Fredholm operators in terms of a generalization of the winding number. We compare our definition with Phillips's analytical definition and derive integral formulae for the spectral flow for certain paths of unbounded operators with common domain, generalizing those of Carey and Phillips. Furthermore, we prove the homotopy invariance of the real-valued index. As an example we consider invariant symmetric elliptic differential operators on Galois coverings.

The notion of an angular function has been introduced by Zilber as one possible way of connecting non-commutative geometry with two ‘counterexamples’ from model theory: the non-classical Zariski curves of Hrushovski and Zilber, and Poizat's field with green points. This article discusses some questions of Zilber relating to existentially closed structures in the class of algebraically closed fields with an angular function.

We consider a new subgroup In(G) in any group G of finite Morley rank. This definably characteristic subgroup is the smallest normal subgroup of G from which we can hope to build a geometry over the quotient group G/ In(G). We say that G is a geometric group if In(G) is trivial.

This paper is a discussion of a conjecture which states that every geometric group G of finite Morley rank is definably linear over a ring K1 ⊕…⊕ Kn where K1,…,Kn are some interpretable fields. This linearity conjecture seems to generalize the Cherlin–Zil'ber conjecture in a very large class of groups of finite Morley rank.

We show that, if this linearity conjecture is true, then there is a Rosenlicht theorem for groups of finite Morley rank, in the sense that the quotient group of any connected group of finite Morley rank by its hypercentre is definably linear.

In order to construct a counterexample to Zilber's conjecture—that a strongly minimal set has a degenerate, affine or field-like geometry—Ehud Hrushovski invented an amalgamation technique which has yielded all the exotic geometries so far. We shall present a framework for this construction in the language of standard geometric stability and show how some of the recent constructions fit into this setting. We also ask some fundamental questions concerning this method.

We construct the free fusion of two geometric thories over a common ω-categorical strongly minimal reduct. If the two theories are supersimple of rank 1 (and satisfy an additional hypothesis true in particular for stable theories or trivial reduct), the completions of the free fusion are supersimple of rank at most ω.

The classes in Valiant's theory are classes of polynomials defined by arithmetic circuits. We characterize them by different notions of tensor calculus, in the vein of Damm, Holzer and McKenzie. This characterization underlines in particular the role played by properties of parallelization in these classes. We also give a first natural complete sequence for the class VPnb, the analogue of the class P in this context.

We prove that if M0 is a model of a simple theory, and p(x) is a complete type of Cantor–Bendixon rank 1 over M0, then p is stationary and regular. As a consequence we obtain another proof that any countable model M0 of a countable complete simple theory T has infinitely many countable elementary extensions up to M0-isomorphism. The latter extends earlier results of the author in the stable case, and is a special case of a recent result of Tanovic.