I. Ivanov-Pogodaev

• PhD
• Senior Researcher at Moscow Institute of Physics and Technology

This project explores the possibility of transforming a square into a circle using parallel shifts, focusing on various geometric and combinatorial methods. The study begins with a series of puzzles and questions (Part A) addressing the feasibility of reassembling different shapes into circles and other figures, under specific cutting rules. It pro...

In the famous works “Tilings, substitution systems and dynamical systems generated by them” by S. Mozes and “Matching rules and substitution tilings” by Ch. Goodman-Strauss, important properties of tiling on the plane connecting the language of substitution systems and local rules are investigated. In particular, it is proved that with almost any s...

The paper is devoted to the study of combinatorial determinacy properties of a family of substitution complexes consisting of quadrilaterals glued side-to-side with each other. These properties are useful in constructing algebraic structures with a finite number of defining relations. In particular, this method was used when constructing a finitely...

The paper discusses various methods for constructing finitely presented algebraic objects. In the first part, the main object of consideration is the automata approach, when a letter in an algebraic structure is considered as a finite automata. This approach is quite effective, in particular, it is used to construct examples of finitely presented s...

A. Введение Теория определимости-это часть математической логики, занимающаяся проблемами определения понятий, в частности одного понятия через другие. Понятие определения относится к числу столь же базовых понятий математической логики и теоретической информатики, математики и существования человека, как и понятия: доказательства, вычисления, слож...

The talk is devoted to discrete Didona problem

Данная работа посвящена обхождению лабиринта коллективом конечных автоматов. Эта часть теории автоматов породила довольно широкий спектр различных задач [1], [2], в том числе связанных с задачами теории сложности вычислений и теорией вероятностей. Оказывается, что рассмотрение сложных алгебраических объектов, таких как бернсайдовы группы, может быт...

The present paper is devoted to traversing a maze by a collective of automata. This part of automata theory gave rise to a fairly wide range of diverse problems ([1], [2]), including those related to problems of the theory of computational complexity and probability theory. It turns out that the consideration of complicated algebraic objects, such...

This is the second paper 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. In the first part we constructed the sequence of complexes with some set of properties. Namely, all these complexes are uniform elliptic: any two points $A$ and...

Проблемы Бернсайдовского типа относятся к числу центральных в современной алгебре. Фундаментальный вклад П.С. Новикова и С.И. Адяна в данную проблематику, революционизировал комбинаторную алгебру. В настоящее время все примеры бесконечных периодических групп являются бесконечно определенными. Известна проблема построения конечно представленной беск...

It is shown that the Cayley graph of infinite periodic groups cannot be circumvented by any system of finite automata. If there is a non-periodic element such a detour exists. At the same time, a system of one Robot and three stones is enough if aperiodic element exist.

An infinite finitely presented nilsemigroup with identity x 9 = 0 is constructed. This construction answers the question of L.N. Shevrin and M.V. Sapir. The proof is based on the construction of a sequence of geometric complexes, each obtained by gluing several simple 4-cycles (squares). These complexes have certain geometric and combinatorial prop...

This paper is devoted to construction of finitely presented infinite nil semigroup with identity x9=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 cal...

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. This is the short version of the paper presenting the main ideas of the construction. The work was carried out with the help of the Russian Science Foundation Grant N 17-11...

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 ca...

https://www.turgor.ru/lktg/2018/ 30-я Летняя конференция Международного математического Турнира городов 30th Summer conference of the International mathematical Tournament of towns Даховская, Республика Адыгея, Россия / Dakhovskaya, Adygey, Russia / 03.08.2018-11.08.2018

This work provides an example of a finitely presented semigroup with zero containing an infinite ideal of the form I = L, where L is a generator of , such that every word in generators representing an element of I is square free (i.e. any word of the type LXYYZ, for non-empty Y, equals zero in ).

1. Our main result is the proof of Kontsevich Conjecture on the existence of a natural isomorphism between the automorphism group of the Weyl algebra (the algebra of polynomial differential operators) and the group of polynomial symplectomorphisms of the affine space of corresponding dimension. The proof by A.Ya. Belov and A.M. Elishev and Jie Tai...

This work presents a sample construction of two algebras both with the ideal of relations defined by a finite Gröbner basis. For the first algebra the question whether a given element is nilpotent is algorithmically unsolvable, for the second one the question whether a given element is a zero divisor is algorithmically unsolvable. This gives a nega...

This work presents an example of a finitely presented semigroup $S$ containing an infinite nonnilpotent nil ideal $LS$, whose elements do not have a square (i.e. any word of the type $LXYYZ$ equals zero.)

This work presents a sample construction of an algebra with the ideal of relations defined by a finite Gr\"obner basis for which the question whether a given element is nilpotent is algorithmically unsolvable. This gives a negative answer to a question raised by Latyshev.

Доклад посвящен решению проблемы Шеврина-Сапира о существовании бесконечной конечно определенной ниль-полугуппы. При этом удается добиться выполнимости тождества x9=0x9=0. По всей видимости, данная техника может оказаться полезной также для аналогичных проблем в теории колец и групп (с неограниченной экспонентой). Данная техника восходит к знаменит...

The talk is devoted to the construction of finitely presented infinite nil-semigroups. In order to do it, we use some geometric methods for the construction.

, eds. Zlil Sela (The Hebrew University) Michah Sageev (Technion), HUJI, Jerusalaem,

pp. 30; 149

This project on the , Problem 2. Aperiodic tilings was proposed by T.Fernique, I.Ivanov-Pogodaev, A.Belov, I.Mitrofanov. My oun motivation was based on the relations between tillings and finite presented nill-semigroup construction

Pavages : systèmes dynamiques, combinatoire, théorie des nombres, décidabilité, géométrie discrète, géométrie non-commutative

Version vith nice pictures

Одиннадцатая летняя конференция Международного математического Турнира Городов, 1999, Конференция проходила с 1-го по 8-е августа 1999 года в детском оздоровительном комплексе “РУСИЧИ” МВД РФ под г. Малоярославец Калужской обл

We present structures comprised of identical convex polyhedra which are interlocked geometrically. These sets cannot be disassembled by removing individual polyhedra by translations and/or rotations. The shapes that permit interlocking arrangements include all five platonic solids. Criteria for interlocking based on transformations of the cross-sec...

This work presents a sample construction of an algebra with the ideal of relations defined by a finite Gröbner basis for which the question whether this element is a zero divisor is algorithmically unsolvable. This gives the negative answer to a question raised by V. N. Latyshev.

Общая постановка. n жадных (завистливых) разбойников делят добычу. Мы считаем, что каждое подмножество сокровищ каждый разбойник оценивает по своему разумению в долларах. Оценка всегда неотрицательна, и если часть сокровищ разбита на две непересекающиеся части A=A1 U A2, A1 З A2=Ж, то оценка части A равна сумме оценок частей A1 и A2. Добыча считает...