We review the main results obtained in the Yaroslavl branch of Martin Greendlinger’s algebraic school from the middle 1970s up to the present.

Outlines of the history of researches on the Combinatorial Group Theory in the Ivanovo State University and an overview of the results obtained from 60-s of the last century up to the present. The results that are presented concern mainly to the study of property of residual finiteness of groups and of its various generalizations as applied to free constructions of groups and to the one-relator groups.

The paper is devoted to subexponential estimations in Shirshov’s Height theorem. A word W is n-divisible, if it can be represented in the following form: W = W0W1 · · · Wn such that W1 ≺ W2 ≺ · · · ≺ Wn. If an affine algebra A satisfies polynomial identity of degree n then A is spanned by non n-divisible words of generators a1 ≺ · · · ≺ al . A. I. Shirshov proved that the set of non n-divisible words over alphabet of cardinality l has bounded height h over the set Y consisting of all the words of degree 6 n−1. We show, that h < Φ(n, l), where Φ(n, l) = 296l · n 12 log3 n+36 log3 log3 n+91 . Let l, n и d > n be positive integers. Then all the words over alphabet of cardinality l which length is greater than Ψ(n, d, l) are either n-divisible or contain d-th power of subword, where Ψ(n, d, l) = 227l(nd) 3 log3 (nd)+9 log3 log3 (nd)+36 . In 1993 E. I. Zelmanov asked the following question in Dniester Notebook: “Suppose that F2,m is a 2-generated associative ring with the identity x m = 0. Is it true, that the nilpotency degree of F2,m has exponential growth?” We give the definitive answer to E. I. Zelmanov by this result. We show that the nilpotency degree of l-generated associative algebra with the identity x d = 0 is smaller than Ψ(d, d, l). This imply subexponential estimations on the nilpotency index of nil-algebras of an arbitrary characteristics. Original Shirshov’s estimation was just recursive, in 1982 double exponent was obtained, an exponential estimation was obtained in 1992.

Our proof uses Latyshev idea of Dilworth theorem application. We think that Shirshov’s height theorem is deeply connected to problems of modern combinatorics. In particular this theorem is related to the Ramsey theory. We obtain lower and upper estimates of the number of periods of length 2, 3,(n−1) in some non n-divisible word. These estimates are differ only by a constant 

By M. P. Mineev’s method of basic and auxiliary systems, we prove a theorem on a number of solutions of some inhomogeneous Diophantine equation with variables from a lacunar sequence of natural numbers. Using Bernoulli numbers, we obtain a polynomial-type expression for the number of basic systems. The estimates for the number of auxiliary systems are also given.

We present an universality theorem for the periodic zeta-function which is defined by a Dirichlet series with periodic coefficients satisfying a certain dependence condition. This simplifies the problem and allows to elucidate the universality of the periodic zeta-function.

In this article the authors set two main objectives: first, to describe the main periods of life of Konstantin A. Rybnikov, Professor of Moscow State University named after M. V. Lomonosov and, second, to give a brief analysis of Konstantin A. Rybnikov’s scientific and pedagogical activity, which had considerable influence on the development of scientific schools in the area of history and methodology of mathematics. K. A. Rybnikov has also bring the significant contribution into combinatorial analysis area, where he was one of the first domestic experts and organizers of international scientific seminars. 23 students schooled by Konstantin A. Rybnikov have defended their dissertations, and two of them subsequently have got a Doctor of Physic and Mathematical Sciences degree. Professor K. A. Rybnikov’s archive search and analysis of mathematical manuscripts of Karl Marx also deserve to be highlighted in the article. These researches have aroused great interest in the wider circles of mathematicians, historians and philosophers. Professor K. A. Rybnikov’s biographical information in the article is accompanied by extracts from his personal memoirs of Professor K. A. Rybnikov.

The author shares his quite subjective reminiscences about some professors who worked at Mathematical Faculty of L.N.Tolstoy Tula State Pedagogical Institution during the second half of 1960s. The main part of the article consists of recollections and reflections about formation of the algebraic school of Professor Martin Greendlinger in Tula and about the role of city algebraic seminar created by him.