The complexes with property of uniform ellipticity

This paper is devoted to construction of finitely presented infinite nil semigroup with identity $x^9=0$. This construction answers to the problem of Lev Shevrin and Mark Sapir. The paper is quite long so the proof is separated into geometric, combinatorial and finalization parts. In the first part we construct uniformly elliptic space. Space is called {\it uniformly elliptic} if any two points $A$ and $B$ at the distance of $D$ can be connected by the system of geodesics which form a disc with width $\lambda\cdot D$ for some global constant $\lambda>0$. In the second part we study combinatorial properties of the constructed complex. Vertices and edges of this complex coded by finite number of letters so we can consider semigroup of paths. Defining relations correspond to pairs of equivalent short paths on the complex. Shortest path in sense of natural metric correspond nonzero words in the semigroup. Words which are not presented as paths on complex and words correspond to non shortest paths can be reduced to zero. In the third part we make a finalization. In particular, we show that word containing ninth degree word can be reduced to zero by defining relations. The present paper contains first part of the proof. This work was carried out with the help of the Russian Science Foundation Grant N 17-11-01377. The first author is the winner of the contest Young Mathematics of Russia .Comment: 32 pages, 12 figures, in Russia

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

This paper surveys results related to well-known works of B. Plotkin and V. Remeslennikov on the edge of algebra, logic and geometry. We start from a brief review of the paper and motivations. The 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 Gr\"obner 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.Comment: In this review we partially used results of arXiv:1512.06533, arXiv:math/0512273, arXiv:1812.01883 and arXiv:1606.01566 and put them in a new contex