Gennady Makanin Explained

Gennady Makanin
Birth Name:Gennady Semenovich Makanin
Birth Date:19 May 1938
Death Date:2017
Nationality:Russian
Fields:mathematics
Workplaces:Steklov Institute of Mathematics
Alma Mater:Moscow State University
Known For:Makanin's algorithm (1977)[1] [2]
Makanin-Razborov algorithm[3] [4]
Makanin-Razborov diagrams[5]

Gennady (or Gennadii or Gennadiy) Semenovich Makanin (1938–2017) was a Russian mathematician, awarded the 2010 I. M. Vinogradov Prize for a series of papers on the problem of algorithmically recognizing the solvability of arbitrary equations in free groups and semigroups.

Education and career

At Moscow State University he received his undergraduate degree and in 1967 his Russian Candidate of Sciences degree (PhD). His dissertation К проблеме тождества в конечно-определённых группах и полугруппах (On the identity problem in finitely-presented groups and semigroups) was supervised by Andrey Markov Jr. and Sergei Adian.[6]

Makanin spent his career (since 1966) working at the Steklov Institute of Mathematics (since 2013 as a freelance employee). From the Steklov Institute of Mathematics he received in 1977 his Russian Doctor of Sciences degree (similar to habilitation) with dissertation Проблема разрешимости уравнений в свободной полугруппе (The problem of solvability of equations in a free semigroup). On the basis of his 1977 dissertation he was an invited speaker at the 1978 International Congress of Mathematicians in Helsinki.[7] [8]

He gained international recognition for his research on combinatorial group theory and algorithmic problems in the theory of semigroups. Zlil Sela, Eliyahu Rips, and others have made important applications of Makanin-Razborov diagrams to geometric group theory.

In 1982 Makanin published a complete solution (an algorithm with proof of validity) to the problem of recognizing the solvability of equations in a free group. An English translation was published in 1983.[9] In 1984 (followed by English translation in 1985) he published a proof, using techniques similar to those in his 1982 paper, of the decidability, for any free group, of two different formal theories generated by that free group.[10]

Remarks on Makanin's research

Martin Davis and Julia Robinson worked unsuccessfully on the problem which was eventually solved in 1977 by Makanin:

Yuri Matiyasevich published a generalization of what he called the "celebrated theorem of G. S. Makanin about decidability of word equations".[11]

Selected publications

References

  1. Diekert, Volker. Makanin's algorithm for solving word equations with regular constraints. 1998. elib, University of Stuttgart. 10.18419/opus-2419.
  2. Book: Gutiérrez. Claudio. Solving equations in strings: On Makanin's algorithm. In: Lucchesi C.L., Moura A.V. (eds.) LATIN '98: Theoretical Informatics (3rd Latin American Symposium — Campinals, Brazil, April 20–24, 1998 Proceedings). Lecture Notes in Computer Science, vol. 1380. Springer. Berlin; Heidelberg. 1998. 1380. 358–373. 0302-9743. 10.1007/BFb0054336. 978-3-540-64275-6.
  3. G. S. MakaninEquations in a free group. (Russian), Izvestia Akademii Nauk SSSR, Seriya Matematischeskaya, vol. 46 (1982), no. 6, pp. 1199 - 1273
  4. A. A. Razborov.Systems of equations in a free group. (in Russian) Izvestia Akademii Nauk SSSR, Seriya Matematischeskaya, vol. 48 (1984), no. 4, pp. 779 - 832.
  5. Sela, Z.. Word equations I: Pairs and their Makanin-Razborov diagrams. 2016. math.GR. 1607.05431.
  6. A translation of G. S. Makanin's 1966 Ph.D. thesis "On the Identity Problem for Finitely Presented Groups and Semigroups. Nyberg-Brodda, Carl-Fredrik. 2021. math.GR. 2102.00745.
  7. Web site: Makanin, G. S.. ICM Plenary and Invited Speakers.
  8. Book: Eight Lectures Delivered at the International Congress of Mathematicians in Helsinki, 1978. American Mathematical Society Translations: Series 2. 1981. 117. American Mathematical Society. 10.1090/trans2/117. 9780821830697. 665105. (print); (online)
  9. Makanin, Gennady S.. Equations in a free group. Mathematics of the USSR-Izvestiya. 21. 3. 1983. 483–. 10.1070/IM1983v021n03ABEH001803. 1983IzMat..21..483M.
  10. Makanin, Gennadiı Semyonovich. Decidability of the universal and positive theories of a free group. Mathematics of the USSR-Izvestiya. 25. 1. 1985. 75. 10.1070/IM1985v025n01ABEH001269. 1985IzMat..25...75M.
  11. Book: Matiyasevich Y.. 1997. Some decision problems for traces. Adian S.. Nerode A.. Logical Foundations of Computer Science. LFCS 1997. Lecture Notes in Computer Science, vol. 1234. 248–257. Springer. Berlin; Heidelberg. 10.1007/3-540-63045-7_25.

External links

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