arXiv subjects
Publications and source records attributed to Boris Zilber.
We study analytic Zariski structures from the point of view of non-elementary model theory. We show how to associate an abstract elementary class with a one-dimensional analytic Zariski structure and prove that the class is stable, quasi-minimal and homogeneous over models. We also demonstrate how Hrushovski's predimension arises in this general context as a natural geometric notion and use it as one of our main tools.
We construct a new non-desarguesian projective plane from a complex analytic structure. At the same time the construction can be explained in terms of so called Hrushovski's construction. This supports the hypothesis that in general structures produced by Hrushovski's construction have "prototypes" in complex geometry.
We describe a structure over the complex numbers associated with the non-commutative algebra Aq called quantum 2-tori. These turn out to have uncountably categorical L_omega1,omega-theory, and are similar to other pseudo-analytic structures considered by the second author. The first-order theory of a quantum torus for generic q interprets arithmetic and so is unstable and undecidable. But certain interesting reduct of the structure, a quantum line bundle, is superstable.
We use the language and tools available in model theory to redefine and clarify the rather involved notion of a {\em special subvariety} known from the theory of Shimura varieties (mixed and pure).
We consider the theory of algebraically closed fields of characteristic zero with multivalued operations $x\mapsto x^r$ (raising to powers). It is in fact the theory of equations in exponential sums. In an earlier paper we have described complete first-order theories of such structures, conditional on a Diophantine conjecture that generalises the Mordell-Lang conjecture (CIT). Here we get this result unconditionally. The field of complex numbers with raising to real powers satisfies the corresponding theory if Schanuel's conjecture holds. In particular, we have proved that a (weaker) version of Schanuel's conjecture implies that every well-defined system of exponential sums with real exponents has a solution in the complex numbers. Recent result by Bays, Kirby and Wilkie states that the required version of Schanuel's conjecture holds for almost every choice of exponents. It follows that for the corresponding choice of real exponents we have an unconditional description of the first order theory of the complex numbers with raising to these powers.
In this paper we will present an ongoing project which aims to use model theory as a suitable mathematical setting for studying the formalism of quantum mechanics. We will argue that this approach provides a geometric semantics for such formalism by means of establishing a (non-commutative) duality between certain algebraic and geometric objects.
We give an axiomatization of the class ECF of exponentially closed fields, which includes the pseudo-exponential fields previously introduced by the second author, and show that it is superstable over its interpretation of arithmetic. Furthermore, ECF is exactly the elementary class of the pseudo-exponential fields if and only if the diophantine conjecture CIT on atypical intersections of tori with subvarieties is true.
We explicitly present expansions of the complex field which are models of the theories of green points in the multiplicative group case and in the case of an elliptic curve without complex multiplication defined over $\mathbb{R}$. In fact, in both cases we give families of structures depending on parameters and prove that they are all models of the theories, provided certain instances of Schanuel's conjecture or an analogous conjecture for the exponential map of the elliptic curve hold. In the multiplicative group case, however, the results are unconditional for generic choices of the parameters.
We carry out a model-theoretic analysis of the Heisenberg algebra. To this end, a geometric structure is associated to the Heisenberg algebra and is shown to be a Zariski geometry. Furthermore, this Zariski geometry is shown to be non-classical, in the sense that it is not interpretable in an algebraically closed field. On assuming self-adjointness of the position and momentum operators, one obtains a discrete substructure of which the original Zariski geometry is seen as the complexification.
We show that algebraic analogues of universal group covers, surjective group homomorphisms from a $\mathbb{Q}$-vector space to $F^{\times}$ with "standard kernel", are determined up to isomorphism of the algebraic structure by the characteristic and transcendence degree of $F$ and, in positive characteristic, the restriction of the cover to finite fields. This extends the main result of "Covers of the Multiplicative Group of an Algebraically Closed Field of Characteristic Zero" (B. Zilber, JLMS 2007), and our proof fills a hole in the proof given there.
We study the structure of an algebraically closed field with extra function resembling the classical exponentiation on complex numbers.
We characterize the Zariski topologies over an algebraically closed field in terms of general dimension-theoretic properties. Some applications are given to complex manifold and to strongly minimal sets.