Boris Zilber

Exponential Sums Equations and the Schanuel Conjecture

A uniform version of the Schanuel conjecture is discussed that has some model-theoretical motivation. This conjecture is assumed, and it is proved that any ‘non-obviously-contradictory’ system of equations in the form of exponential sums with real exponents has a solution.

Pseudo-exponentiation on algebraically closed fields of characteristic zero

Abstract We construct and study structures imitating the field of complex numbers with exponentiation. We give a natural, albeit non first-order, axiomatisation for the corresponding class of structures and prove that the class has a unique model in every uncountable cardinality. This gives grounds to conjecture that the unique model of cardinality continuum is isomorphic to the field of complex numbers with exponentiation.

Covers of the Multiplicative Group of an Algebraically Closed Field of Characteristic Zero

We study the universal cover of the complex one-dimensional torus as a model-theoretic structure in a natural language. We consider also abstract covers of one-dimensional tori over algebraically closed fields of characteristic zero. The main result states that the structures can be described uniquely up to isomorphism by a simple (non-first order) sentence, given a fixed uncountable cardinality of the underlying field. The proof reduces to the Kummer theory of cyclic Galois extensions over some, not necessarily finitely generated fields.

Covers of multiplicative groups of algebraically closed fields of arbitrary characteristic

An automatic degaussing circuit for a color picture tube (CPT) capable of automatically degaussing the CPT when a power conservation mode is released in a monitor having a power conservation function. The degaussing circuit includes a degaussing coil for eliminating electromagnetic waves produced from the CPT, a power conservation mode detecting section for detecting the release of the power conservation mode in accordance with a power conservation mode signal of the CPT inputted thereto and providing a degaussing control signal for controlling the operation of the degaussing coil if the release of the power conservation mode is detected, and a degaussing coil driving section for driving the degaussing coil for a predetermined time in accordance with the degaussing control signal when the power conservation mode is released.

Zariski geometries : geometry from the logician's point of view

This book presents methods and results from the theory of Zariski structures and discusses their applications in geometry as well as various other mathematical fields. Its logical approach helps us understand why algebraic geometry is so fundamental throughout mathematics and why the extension to noncommutative geometry, which has been forced by recent developments in quantum physics, is both natural and necessary. Beginning with a crash course in model theory, this book will suit not only model theorists but also readers with a more classical geometric background.

RAISING TO POWERS IN ALGEBRAICALLY CLOSED FIELDS

We study structures on the fields of characteristic zero obtained by introducing (multi-valued) operations of raising to power. Using Hrushovski–Fraisse construction we single out among the structures exponentially-algebraically closed once and prove, under certain Diophantine conjecture, that the first order theory of such structures is model complete and every its completion is superstable.

Exponentially Closed Fields and the Conjecture on Intersections with Tori

Abstract 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.

The quantum harmonic oscillator as a Zariski geometry

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.

On Counting Generalized Colorings

It is well known that the number of proper k-colorings of a graph is a polynomial in k. We investigate in this talk under what conditions a numeric graph invariant which is parametrized with parameters k 1 , ..., k m is a polynomial in these parameters. We give a sufficient conditions for this to happen which is general enough to encompass all the graph polynomials which are definable in Second Order Logic. This not only covers the various generalizations of the Tutte polynomials, Interlace polynomials, Matching polynomials, but allows us to identify new graph polynomials related to combinatorial problems discussed in the literature.

The geometric semantics of algebraic quantum mechanics

In this paper, we will present an ongoing project that aims to use model theory as a suitable mathematical setting for studying the formalism of quantum mechanics. We argue that this approach provides a geometric semantics for such a formalism by means of establishing a (non-commutative) duality between certain algebraic and geometric objects.