Nonstandard analysis, deformation quantization and some logical aspects of (non)commutative algebraic geometry

Abstract

The connections between algebraic geometry and mathematical logic are extremely important. This short survey is related to ideas that contained in the well-known works of B. Plotkin and V. Remeslennikov and their followers. We assume that lots of questions still require further illumination. One of the goals of this paper is to narrow the gap and to draw attention to this topic.We start from a brief review of the paper and motivations. First sections deal with model theory. In Section 2.1 we describe the geometric equivalence, the elementary equivalence, and the isotypicity of algebras. We look at these notions from the positions of universal algebraic geometry and make emphasis on the cases of the first order rigidity. In this setting Plotkin's problem on the structure of automorphisms of (auto)endomorphisms of free objects, and auto-equivalence of categories is pretty natural and important. Section 2.2 is dedicated to particular cases of Plotkin's problem. Section 2.3 is devoted to Plotkin's problem for automorphisms of the group of polynomial symplectomorphisms. This setting has applications to mathematical physics through the use of model theory ( non-standard analysis) in the studying of homomorphisms between groups of symplectomorphisms and automorphisms of the Weyl algebra.The last two sections deal with algorithmic problems for noncommutative and commutative algebraic geometry. Section 3.1 is devoted to the Gröbner basis in non-commutative situation. Despite the existence of an algorithm for checking equalities, the zero divisors and nilpotency problems are algorithmically unsolvable. Section 3.2 is connected with the problem of embedding of algebraic varieties; a sketch of the proof of its algorithmic undecidability over a field of characteristic zero is given. Dedicated to the 70-th anniversary of A.L. Semenov and to the 95-th anniversary of B.I. Plotkin. Contents KANEL-BELOV, CHILIKOV, IVANOV-POGODAEV, MALEV, PLOTKIN, YU, AND ZHANG 3.1. Finite Gröbner basis algebras with unsolvable nilpotency problem and zero divisors problem 22 3.2. On the Algorithmic Undecidability of the Embeddability Problem for Algebraic Varieties over a Field of Characteristic Zero 25 References 27