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There is a longstanding conjecture, due to Gregory Cherlin and Boris Zilber, that all simple groups of finite Morley rank are simple algebraic groups. Towards this end, the development of the theory of groups of finite Morley rank has achieved a good theory of Sylow 2-subgroups. It is now common practice to divide the Cherlin-Zilber conjecture into… (More)

- Boris Zilber
- Ann. Pure Appl. Logic
- 2005

- Boris Zilber
- 2008

Model theoretically one can interprete the sequence as a structure in various ways. The simplest algebraic structure on the sequence which bears an interesting algebro-geometric information is the one with the additive group structure in the middle and with the full algebraic geometry on C. The latter is equivalent to treating C as C \ {0} with the full… (More)

- Boris Zilber
- 2005

1 This paper grew out of an observation that some new stable structures discovered in the 1990' as counterexamples to well-known conjectures in pure model theory might be related to non-commutative geometry. The general meaning of the conjectures was that " very good " , or more technically, very stable structures must be in a certain way reducible to… (More)

- Boris Zilber
- 2014

1.1 This paper is a part of a much broader project which aims to establish, for a typical ’co-ordinate algebra’ A in the sense of non-commutative geometry, or a scheme, a geometric counterpart VA. Here “geometric” should also be understood in some broad but well-defined sense. We believe this is possible in the setting of model theory, where a crucial new… (More)

We describe definable relations in the real field augmented by a binary relation which is an arbitrary multiplicative group of complex numbers contained in the divisible hull of a finitely generated subgroup of the unit circle. We give a complete axiom system for this structure which admits quantifier elimination down to Boolean combinations od existential… (More)

We describe the structure QHO = QHON (dependent on the positive integer number N) on the universe L which is a finite cover, of order N, of the projective line P = P(F), F an algebraically closed field of characteristic 0. We prove that QHO is a complete irreducible Zariski geometry of dimension 1. We also prove that QHO is not classical in the sense that… (More)

- Boris Zilber
- 2007

We undertake a case study of two series of nonclassical Zariski geometries. We show that these geometries can be realised as representations of certain noncommutative C∗-algebras and introduce a natural limit construction which for each of the two series produces a classical U(1)-gauge field over a 2-dimensional Riemann surface.

- Tomer Kotek, Johann A. Makowsky, Boris Zilber
- CSL
- 2008

The notion of graph polynomials definable in Monadic Second Order Logic, MSOL, was introduced in [Mak04]. It was shown that the Tutte polynomial and its generalization, as well as the matching polynomial, the cover polynomial and the various interlace polynomials fall into this category. In this paper we present a framework of graph polynomials based on… (More)

- Boris Zilber
- 2000

One of the questions frequently asked nowadays about model theory is whether it is still logic. The reason for asking the question is mainly that more and more of model theoretic research focuses on concrete mathematical fields, uses extensively their tools and attacks their inner problems. Nevertheless the logical roots in the case of model theoretic… (More)