scholarly journals Poisson algebras via model theory and differential-algebraic geometry

2017 ◽  
Vol 19 (7) ◽  
pp. 2019-2049 ◽  
Author(s):  
Jason Bell ◽  
Stéphane Launois ◽  
Omar León Sánchez ◽  
Rahim Moosa
Keyword(s):  
Algebraic Geometry ◽  
Model Theory ◽  
Poisson Algebras

scholarly journals Zariski Geometries and Quantum Mechanics

2021 ◽  
Author(s):  
Milan Zanussi
Keyword(s):  
Quantum Mechanics ◽  
Algebraic Geometry ◽  
Model Theory ◽  
Topological Structure ◽  
Classification Theorem ◽  
Zariski Topology ◽  
Specific Class ◽  
Mathematical Structures ◽  
Topological Structures ◽  
Structure Setting

Model theory is the study of mathematical structures in terms of the logical relationships they define between their constituent objects. The logical relationships defined by these structures can be used to define topologies on the underlying sets. These topological structures will serve as a generalization of the notion of the Zariski topology from classical algebraic geometry. We will adapt properties and theorems from classical algebraic geometry to our topological structure setting. We will isolate a specific class of structures, called Zariski geometries, and demonstrate the main classification theorem of such structures. We will construct some Zariski structures where the classification fails by adding some noncommuting structure to a classical one. Finally we survey an application of these nonclassical Zariski structures to computation of formulas in quantum mechanics using a method of structural approximation developed by Boris Zilber.

Non-Archimedean Tame Topology and Stably Dominated Types (AM-192)

2017 ◽  
Author(s):  
Ehud Hrushovski ◽  
François Loeser
Keyword(s):  
Algebraic Geometry ◽  
Model Theory ◽  
Stability Theory ◽  
Analytic Geometry ◽  
Previous Knowledge ◽  
Real Numbers ◽  
Algebraic Dynamics ◽  
Theoretic Foundations ◽  
Rational Polytope ◽  
Deformation Retraction

Over the field of real numbers, analytic geometry has long been in deep interaction with algebraic geometry, bringing the latter subject many of its topological insights. In recent decades, model theory has joined this work through the theory of o-minimality, providing finiteness and uniformity statements and new structural tools. For non-archimedean fields, such as the p-adics, the Berkovich analytification provides a connected topology with many thoroughgoing analogies to the real topology on the set of complex points, and it has become an important tool in algebraic dynamics and many other areas of geometry. This book lays down model-theoretic foundations for non-archimedean geometry. The methods combine o-minimality and stability theory. Definable types play a central role, serving first to define the notion of a point and then properties such as definable compactness. Beyond the foundations, the main theorem constructs a deformation retraction from the full non-archimedean space of an algebraic variety to a rational polytope. This generalizes previous results of V. Berkovich, who used resolution of singularities methods. No previous knowledge of non-archimedean geometry is assumed and model-theoretic prerequisites are reviewed in the first sections.

scholarly journals Algebraic geometry over groups III: Elements of model theory

2005 ◽  
Vol 288 (1) ◽  
pp. 78-98 ◽  
Author(s):  
Alexei Kvaschuk ◽  
Alexei Myasnikov ◽  
Vladimir Remeslennikov
Keyword(s):  
Algebraic Geometry ◽  
Model Theory

scholarly journals Notes on Quasiminimality and Excellence

2004 ◽  
Vol 10 (3) ◽  
pp. 334-366 ◽  
Author(s):  
John T. Baldwin
Keyword(s):  
Algebraic Geometry ◽  
Model Theory ◽  
Special Kind ◽  
Order Logic ◽  
First Order Logic ◽  
Structure Theory ◽  
Mathematical Structures ◽  
First Order ◽  
Elementary Class ◽  
Abstract Elementary Class

AbstractThis paper ties together much of the model theory of the last 50 years. Shelah's attempts to generalize the Morley theorem beyond first order logic led to the notion of excellence, which is a key to the structure theory of uncountable models. The notion of Abstract Elementary Class arose naturally in attempting to prove the categoricity theorem for Lω1,ω(Q). More recently, Zilber has attempted to identify canonical mathematical structures as those whose theory (in an appropriate logic) is categorical in all powers. Zilber's trichotomy conjecture for first order categorical structures was refuted by Hrushovski, by the introducion of a special kind of Abstract Elementary Class. Zilber uses a powerful and essentailly infinitary variant on these techniques to investigate complex exponentiation. This not only demonstrates the relevance of Shelah's model theoretic investigations to mainstream mathematics but produces new results and conjectures in algebraic geometry.

Model Theory and Algebraic Geometry

2003 ◽  
pp. 291-311
Author(s):  
Annalisa Marcja ◽  
Carlo Toffalori
Keyword(s):  
Algebraic Geometry ◽  
Model Theory

scholarly journals ALGEBRAIC LOGIC AND LOGICALLY-GEOMETRIC TYPES IN VARIETIES OF ALGEBRAS

2012 ◽  
Vol 12 (02) ◽  
pp. 1250146 ◽  
Author(s):  
BORIS PLOTKIN ◽  
ELENA ALADOVA ◽  
EUGENE PLOTKIN
Keyword(s):  
Algebraic Geometry ◽  
Model Theory ◽  
Affine Space ◽  
Algebraic Logic ◽  
Geometric Properties ◽  
First Order ◽  
Galois Correspondence ◽  
Universal Algebraic Geometry

The main objective of this paper is to show that the notion of type which was developed within the frames of logic and model theory has deep ties with geometric properties of algebras. These ties go back and forth from universal algebraic geometry to the model theory through the machinery of algebraic logic. We show that types appear naturally as logical kernels in the Galois correspondence between filters in the Halmos algebra of first order formulas with equalities and elementary sets in the corresponding affine space.

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