``Pure’’ mathematics.

1. Theory of Polynomial Identities. Spechts problem: proof of local representability of relatively free algebras and finite basis property of systems of identities with a finite set of generators in associative algebras over arbitrary Noetherian commutative rings. Resolution of Malcevs problems of 1968. In the case of the infinite number of generators a counterexample to Spechts problem -- an infinitely based system of identities over an arbitrary field of positive characteristic -- was constructed. Proof of decidability of the Malcev-Tarski-von Neumann (posed in 1967) for identities in associative algebra. Proof of rationality of Hilbert series of relatively free algebras (a problem of Procesi, 1970). Construction of an example of a representable algebra that has transcendent Hilbert series.

Prospects. Proof of local finite basis property and local representability of Lie algebras that admit Capelli identities, proof of local representability in the case of rings asymptotically close to associative, in which the radical is separated away from the semisimple part, in characteristic zero. Work on the problem of Latyshev regarding the finiteness of the basis of obstructions for $T$-ideals. The construction of counterexamples to Spechts problem has led to understanding the trace of the skew-symmetric form and to the concept of the "minus sign" in characteristic two (and over rings) and accordingly has opened a way to the development of supertheory.

In light of the rationality of Hilbert series for relatively free algebras and examples of representable algebras with transcendent series, the inquiry into linear bases of algebras is of especial interest. A monomial algebra is representable if and only if the set of its non-zero words is a set of subwords of a finite collection of series of elements $x_1^{k_1}\cdots x_s^{k_s}$; the set of vectors $k_1,\dots,k_s$ obeys a system of exponentially diophantine inequalities of the form $\sum_I P_I(k_1,\dots,k_s) \lambda_{i_1}^{k_1}\cdots\lambda_{i_s}^{k_s} \ne 0$. Therefore, over a field of characteristic zero the isomorphism decidability problem for monomial subalgebras of the matrix algebra with polynomial entries is negative; on the other hand, due to joint work with student A.A. Chilikov , which describes the set of solutions of a system of exponentially diophantine equations with bases of exponents lying in a field of positive characteristic , this problem turns out to be decidable. The investigation of bases of representable algebras is also of considerable interest. The author has shown that the Gelfand-Kirillov dimension of these algebras is integer and coincides with their essential height . It seems also worthwhile to obtain polynomial estimates in Shirshovs height theorem (subexponential estimates have been obtained jointly with M.I. Kharitonov, ).

2. Geometric ring theory. Together with E. Rips, A.Atkarskaya, E.Plotkin an example of a some version of small cancellation theory for rings was constructed, papers are in process ; also an example of an infinitely dimensional skew field, any two whose entries form a system of its generators (Tarski monster) has been given, which yields an example of an infinite critical ring. At the same time it has been shown that any infinite critical ring is a skew field.

Prospects: development of a ring analogue of the theory of hyperbolic groups. This area is inhabited by various rather beautiful problems.

Thanks to the resolution by Rips and Juhasz of the problem of construction of an Engel group of index $n$ which is not locally soluble, a theory of groups of non-positive curvature with flats -- i.e. commuting fragments -- has emerged. That theory allows to tackle the ring case, as the flats exhibit additive-like behavior.

3. Affine algebraic geometry. Jointly with M.L. Kontsevich it has been demonstrated that the Jacobian conjecture implies the Dixmier conjecture on Weyl algebra endomorphisms. Together with student A.M. Elishev and with Jie-Tai Yu it has been shown that all the authomorphisms of Ind-scheme automorphisms are inner (a conjecture of Plotkin).

Together with Jie-Tai Yu it has been shown that wild automorphisms of $K[x,y,z]$ cannot be lifted to $z$-automorphisms of the corresponding free associative algebra, and that $z$-automorphisms of $K<x,y,z>$ are stably tame.

Further research includes theory of $IND$-schemes, Kontsevichs isomorphism conjecture, and the lifting problem of wild automorphisms.

Of significance is inquiry into matrix-valued polynomials. Due to progress made in the Kaplansky-Lvov problem (with L. Rowen and S. Malev ) it is possible to describe the range of multilinear polynomial maps in the general case (modulo Zariski closure). The investigation of the case when the size of the matrices involved is much greater than the degree of the polynomial may aid the proof of the freeness theorem, and also construct an example of algebraically closed skew field in arbitrary characteristic (a generalization of results due to Makar-Limanov).

Applied Mathematics.

1. Mechanics and combinatorial geometry. Invention of self-interlocking structures and development of the corresponding theory , together with collaborators. A fault in a small grain cannot develop, its growth stops at the boundary. At the same time there exist configurations of convex bodies (in particular, regular polyhedra) which support and lock each other. This observation may allow one to create composite materials resistant to high pressure.

It is already in use in military industry , in particular in development of bullet-proof vests. Further research of self-interlocking structures and their applications to engineering is planned.

2. Methods of construction of finitely presented objects in general algebra. Together with student I.A. Ivanov-Pogodaev the solution to the Shevrin-Sapir problem has been devised -- an example of an infinite finitely presented nil semigroup with identity $x^9=0$ has been constructed.

The problem was initially posed in the renowned Sverdlovsk notebook. At present there are few effective methods of constructing finitely presented objects. For our solution of the nil semigroup problem we have developed a new method for such constructions, which utilizes aperiodic tilings. In particular, words are viewed as paths in a geometric uniformly elliptic space which has aperiodic nature and a number of specific properties. The Shevrin-Sapir problem requires methods related to geometry of aperiodic tilings on uniformly elliptic spaces.

This problem is of significance in computer science: a letter in an alphabet corresponds to a finite automaton, while a word corresponds to a chain of locally interacting automata. The problem is in coordinating the behavior of a system of finite automata under invertible transforms starting from arbitrary initial conditions.

Prospects. We plan on achieving the identity $x^2=0$, as well as working on the unbounded exponent. We plan to adapt the method of construction of finitely presented objects in rings and in groups (for example, obtain meaningful progress in Latyshevs problem on finitely presented non-nilpotent nil ring).

For a number of problems, such as for the construction of finitely presented semigroup with recursive GK dimension, it is sufficient to restrict oneself to purely finite automata-related methods, i.e. to working on chains of locally interacting automata (cf. Ivanov-Pogodaevs thesis , which precedes the resolution of the Shevrin-Sapir problem).

This situation bears resemblance to ideas of Gacs, Kurdyumov and Levin on the behavior of systems of finite automata on a line.

It is therefore of significant interest to apply the geometric methods which allow transition from the one-dimensional case to the multi-dimensional one in order to clarify the renowned theorem of Gacs - an example of two randomly evolving stationary systems of cellular automata on a line with distinct statistics ( positive rates conjecture). The beginning of the joint effort with Ivanov-Pogodaev was connected with the problem of A. Toom (related to image processing) on the planar evolution of patterns , which had emerged thanks to the Positive Rates Conjecture.

It may be considered worthwhile to obtain an example of a finitely presented semigroup with GK dimension equal to $2.5$.

3. Combinatorics of words and Rauzy schemes, theorems of Vershik-Lifshitz type. Together with student A.L. Chernyatiev a criterion for a symbolic dynamical system to be generated by an interval exchange transform has been obtained, which has thus given an answer to a question asked by Rauzy in 1979. (Later for almost periodic words satisfying the i.d.o.c.-condition S. Ferenci and independently L. Zamboni have obtained an analogous result). Together with student I.V. Mitrofanov a theorem of Vershik-Lifshitz type, which gives a criterion for almost periodic superword to be defined by HDOLL-system, has been proved (for the first time progress in this area was made in 1986). This has allowed Mitrofanov to resolve several well-known problems posed by A.A. Muchnik, Yu. L. Pritykin and A.L. Semenov - namely to establish the decidability of the check for periodicity, as well as almost-periodicity, of an HDOLL-system. An analogous result has been independently obtained by F. Durand. It has also been shown that if the leading coefficient in the polynomial $P(x)$ is irrational, and $W$ is a word composed of the first numbers in the binary decomposition of the fractional part of $\{P(n)\}$, then the number of subwords of length $n$ of the word $W$ is a polynomial of $n$, for $n$ sufficiently large .

It is planned to apply the technique involving Rauzy schemes and theorems of Vershik-Lifshitz type to the Pizot conjecture and to the study of points in Rauzy fractals, as well as to further inquiry into HDOLL-systems, in particular to the problem of coincidence of superwords and coincidence of languages. R. Schwartz gave a plenary talk at the International Congress of Mathematicians 2022. "The cases n = 8;10;12 also have a self-similar structure. Without having a reference, I have the sense that the case n = 7 is somewhat understood in the sense that there are some regions of renormalization. I think that the cases n = 9; 11 are not understood at all. G. Hughes has made beautiful and detailed pictures of outer billiards on regular polygons. These pictures (and earlier ones) suggest

Conjecture 6.5. Outer billiards on the regular n-gon have an aperiodic orbit if n , 3;4;6. I think that this is not known aside from n = 5;8;10;12, and perhaps n = 7.

Theorem (Timorin, Belov, Beletsky Rukhovich). For any outer billiard, the circumference of a regular n-gon for n\ne 3,4,6 there is an aperiodic point.

4. Statistical geometry. Several applications of statistical geometry to geology are known. Faults can be modeled by random planes. A method for finding explicit formulae for probabilistic distributions in planar problems of statistical geometry, in particular the distribution of area and perimeter with planar distribution by a Poisson field of lines, has been developed.

5. Mathematical education as a tool for inquiry into perception and cognition, artificial intelligence.

Educational practises, in particular creation of problems for mathematical olympiads, is important first and foremost as a classification tool for ideas and stereotypes in problem solving and for cognition research. Existing olympiad-related publications , as well as classification parameters represent steps in this direction. This research seems useful in connection with the work on the AI. As was pointed out by a well-known linguist A.V. Gladkiy , a kind of "literary" investigation of mathematical texts bears resemblance to NLP, as well as known works of V. Propp on historical roots of fairy tales.

From this point of view there is interest in combinatorics, in particular in combinatorial geometry. The results obtained in the pedagogical area (cf. on olympiad problem-solving) are significant to verification of idea and stereotype extraction.

Further work in this direction is planned.

Born June 9, 1963. Birthplace: Moscow, USSR. Parents: Dr. Belova Maya Mikhailovna and Dr. Kanel Yakov Isaakovich, both mathematicians. Graduated from Lyceum No. 2 ("Lycei Vtoraya Shkola" - one of Moscows strongest mathematics-oriented public schools); in 1985 graduated with academic distinction from the Moscow State Pedagogical Institute. Thesis titled "On Shirhovs basis of relatively free algebra of complexity nn", project was conducted under supervision of S.V. Ptchelintsev. Also attended the Peoples University and studied under D.B. Fuchs. In 1985 enrolled as a postgraduate student in Moscow Mining Institute, graduated in 1988. In 1992 defended PhD (candidate of sciences) thesis at the Otto Schmidt Institute for Earth Physics, thesis titled " Statistical geometry and equilibrium of block massifs", doctoral advisor R.L. Salganik, official opponents V.B. Kolmanovskii and E.I. Dinaburg, reviewers N.N. Anoschenko, V.A. Bukrinskii, I.P. Dobrovolskii, and M.Ya. Kelbert. Thesis main result consisted of the proof of the area and perimetra distribution law for plane partitions by random sets of straight lines as well as for other planar problems. Investigation of block massif geometry, as introduced in the thesis, have paved the way to the invention of self-interlocking structures.

In 1988-1990 held a position at the Moscow Mining Institute, also in 1990-2001 worked as an instructor at the Moscow City Hall of Pioneers and Schoolchildren. Actively (from 1986 till 2002) participated in the Moscow State University seminar in ring theory (presided by V.N. Latyshev, A.V. Mikhalev and E.S. Golod) and, before Latyshevs relocation to Ulyanovsk, in Latyshevs weekly Friday seminar. In 2002 A. K.-B. defended his D.Sc. (Doctor of Physical and Mathematical Sciences -- Russias highest habilitation degree) thesis, speciality 01.01.06 "Mathematical logic, algebra and number theory"; title of the thesis: "Algebras with polynomial identities: representations and combinatorial methods" (advisors: V.N. Latyshev, A.V. Mikhalev; official opponents: A.R. Kemer, S.V. Ptchelintsev, A.V. Yakovlev). Main results of the thesis: proof of local representability for varieties of associative algebras over commutative Noetherian rings, and construction of varieties of associative algebras with infinite bases. A. K.-B. has successfully supervised eight Ph.D. (I.A. Ivanov-Pogodaev, A.L. Chernyatiev, M.I. Kharitonov, and I.V. Mitrofanov -- jointly with A.V. Mikhalev) (S. Malev, jointly with L. Rowen), P.D.Rukhovich (jointly with A.L.Semenov), A.M.Elishev, Zhang Wenchao.

Alexei Kanel-Belov studies various problems related to the finite basis property of systems of identities, automorphisms of algebraic varieties, combinatorics of words and symbolic dynamical systems, particularly IET. Together with I.A. Ivanov- Pogodaev he has solved the Shevrin -- Sapir problem on the existence of finitely presented infinite nil semigroup. A. K.-B. is involved in a joint research endeavor on ring theory with E. Rips, A.Atkarskaya, E.Plotkin -- in particular they managed to construct a skew field which is finitely generated as a ring, as well as an analog of the Tarski monster. The construction utilizes a variation of the Novikov--Adian method and the theory of the canonical form, as recently developed by Rips. Further research in this direction is expected to be focused on the creation of a ring version of small cancellation theory. Also A.K.-B. has been for a while interested in self-interlocking structures and their applications to creation of novel hybrid materials, particularly applications to superhigh pressure systems. (cf. megagrant http://hybrid-nano-lab.misis.ru/index.php.)

Another area of A. Kanel-Belovs lasting activity is training and engagement with high-school and university students with high aptitude for mathematics. A. K.-B. had been an active participant in the organisation of the Moscow Mathematical Olympiad, was the chairman of the International Internet Olympiad jury committee. During his time, A.K.-B. was appointed coach of various student mathematical olympiad teams throughout the world: Jacobs University Bremen, Israel National Team, South University of Science and Technology (China) among others.

A.K.-B. is currently an editor of the following academic journals: "Fundamentalnaya i prikladnaya matematika", "Chebyshevskii sbornik", "Kvant", "Matematicheskoye prosvescheniye" (editor of the problems section), "Matematicheskoye obrazovaniye".

1. Ilya Ivanov-Pogodayev, Alexey Kanel-Belov, Construction of infinite finitely presented nillsemigroup, 2014 , 160 pp., 131 figures, in Russian, arXiv: 1412.5221          
2. A. Ya. Belov, “The local finite basis property and local representability of varieties of associative rings”, Izv. Math., 74:1 (2010), 1–126                      
3. A. J. Kanel-Belov, A. V. Dyskin, Y. Estrin, E. Pasternak, I. A. Ivanov-Pogodaev, “Interlocking of convex polyhedra: towards a geometric theory of fragmented solids”, Mosc. Math. J., 10:2, http://www.ams.org/distribution/mmj/vol10-2-2010/kanel-belov-etal.pdf (2010), 337–342 http://olympiads.mccme.ru/mmo/2000/mmo2000.htm, arXiv: 0812.5089                
4. A. Ya. Kanel-Belov, M. L. Kontsevich, “The Jacobian conjecture is stably equivalent to the Dixmier conjecture”, Mosc. Math. J., 7:2 (2007), 209–218 , arXiv: math/0512171            
5. A. Ya. Belov, A. L. Chernyatev, Opisanie mnozhestva slov, porozhdaemykh perekladyvaniem otrezkov, Dep. v VINITI, No 1048-B2007 ot 09.11.07, VINITI, Moskva, 2007 , 18 pp., prinyato v pechat 09.10.07, ref. 23 items, Describing the set of words generated by interval exchange transformations https://www.lirmm.fr/arith/wiki/MathInfo2010/...  
6. Kanel–Belov, Alexei; Rowen, Louis Halle, Combinatorial aspects in polynomial identities, Research Notes in Mathematics, 9, A K Peters, Ltd., Wellesley, MA, 2005 , 378 pp.    
7. A. Ya. Belov, “On non-Spechtian varieties”, Fundam. Prikl. Mat., 5:1 (1999), 47–66      
8. A. Ya. Belov, “On the rationality of Hilbert series of relatively free algebras”, Russian Math. Surveys, 52:2 (1997), 394–395