Related papers: Model Theory, Arithmetic & Algebraic Geometry
Mostly aimed at an audience with backgrounds in geometry and homological algebra, these notes offer an introduction to derived geometry based on a lecture course given by the second author. The focus is on derived algebraic geometry, mainly…
This report on the topics in the title was written for a lecture series at the Southwestern Center for Arithmetic Algebraic Geometry at the University of Arizona.It may serve as an introduction to certain conjectural relations between…
In algebraic geometry there is the notion of a height pairing of algebraic cycles, which lies at the confluence of arithmetic, Hodge theory and topology. After explaining a motivating example situation, we introduce new directions in this…
In the paper the main attention is paid to conditions on algebras from a given variety which provide coincidence of their algebraic geometries. The main part here play the notions mentioned in the title of the paper.
The aim of this project is to attach a geometric structure to the ring of integers. It is generally assumed that the spectrum $\mathrm{Spec}(\mathbb{Z})$ defined by Grothendieck serves this purpose. However, it is still not clear what…
In this paper, for a given finitely generated algebra (an algebraic structure with arbitrary operations and no predicates) A we study finitely generated limit algebras of A, approaching them via model theory and algebraic geometry. Along…
We give an introduction to the study of algebraic hypersurfaces, focusing on the problem of when two hypersurfaces are isomorphic or close to being isomorphic. Working with hypersurfaces and emphasizing examples makes it possible to discuss…
Many statistical models are algebraic in that they are defined by polynomial constraints or by parameterizations that are polynomial or rational maps. This opens the door for tools from computational algebraic geometry. These tools can be…
In this article, the notion of a mathematical model in science is attempted to be enlightened from several points of view. In particular, it is shown that mathematical models are introduced differently and used differently in different…
We will present several examples in which ideas from ergodic theory can be useful to study some problems in arithmetic and algebraic geometry.
Many statistical models are algebraic in that they are defined in terms of polynomial constraints, or in terms of polynomial or rational parametrizations. The parameter spaces of such models are typically semi-algebraic subsets of the…
In this position paper, we promote the study of function spaces parameterized by machine learning models through the lens of algebraic geometry. To this end, we focus on algebraic models, such as neural networks with polynomial activations,…
The topics of Convexity and Concavity and Envelopes are central in Complex Analysis and extensively investigated. The aim of this paper is to find a possible counterpart in Algebraic Geometry. The article presents preliminary results on…
This thesis studies arithmetic of linear algebraic groups. It involves studying the properties of linear algebraic groups defined over global fields, local fields and finite fields, or more generally the study of the linear algebraic groups…
In these notes we aim at bringing together design theory and projective geometry over a ring. Both disciplines are well established, but the results on the interaction between them seem to be rare and scattered over the literature. Thus our…
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…
Here we briefly describe some topics along the lines of projective spaces and related geometric constructions connected to linear algebra, which provide fundamental examples in classical geometry and analysis.
We define and investigate a geometric object, called an associative geometry, corresponding to an associative algebra (and, more generally, to an associative pair). Associative geometries combine aspects of Lie groups and of generalized…
We define and investigate a geometric object, called an associative geometry, corresponding to an associative algebra (and, more generally, to an associative pair). Associative geometries combine aspects of Lie groups and of generalized…
This is a survey on appearances of reflection groups, real and complex, in algebraic geometry. We also include a brief introduction into the theory of reflection groups.