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| PEOPLE |
| Belov Aleksei Yakovlevich |
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| Professor |
| Doctor of physico-mathematical sciences (2002) |
``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. F. Rukhovich has developed a computer program that proves the existence of Rauzy induction, self-similarity and aperiodic points; he also has proved the existence of aperiodic points in an outer billiard with regular dodecagon.
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.
ISTINA
