The Zamolodchikov tetrahedron equation inherits almost all the richness of structures and topics in which the Yang–Baxter equation is involved. At the same time, this transition symbolizes the growth of the order of the problem, the step from the Yang–Baxter equation to the local Yang–Baxter equation, from the Lie algebra to the 2-Lie algebra, from ordinary knots in to 2-knots in . These transitions are followed in several examples, and there are also discussions of the manifestation of the tetrahedron equation in the long-standing question of integrability of the three-dimensional Ising model and a related model of neural network theory: the Hopfield model on a two-dimensional lattice. Bibliography: 82 titles.
{"title":"Tetrahedron equation: algebra, topology, and integrability","authors":"D. Talalaev","doi":"10.1070/RM10009","DOIUrl":"https://doi.org/10.1070/RM10009","url":null,"abstract":"The Zamolodchikov tetrahedron equation inherits almost all the richness of structures and topics in which the Yang–Baxter equation is involved. At the same time, this transition symbolizes the growth of the order of the problem, the step from the Yang–Baxter equation to the local Yang–Baxter equation, from the Lie algebra to the 2-Lie algebra, from ordinary knots in to 2-knots in . These transitions are followed in several examples, and there are also discussions of the manifestation of the tetrahedron equation in the long-standing question of integrability of the three-dimensional Ising model and a related model of neural network theory: the Hopfield model on a two-dimensional lattice. Bibliography: 82 titles.","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":"3 1","pages":"685 - 721"},"PeriodicalIF":0.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59011948","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Petr Anatol'evich Borodin, Il'dar Abdullovich Ibragimov, B. Kashin, Valery Vasil'evich Kozlov, Aleksandr Viktorovich Kolesnikov, S. V. Konyagin, E. D. Kosov, O. Smolyanov, N. A. Tolmachev, D. Treschev, Alexander Shaposhnikov, Stanislav Valer'evich Shaposhnikov, A. Shiryaev, A. Shkalikov
The prominent mathematician Vladimir Igorevich Bogachev, Professor at the Department of the Theory of Functions and Functional Analysis of the Faculty of Mechanics and Mathematics at Lomonosov Moscow State University, Professor at the Faculty of Mathematics of the HSE University, and Professor at the Department of Mathematics of the Faculty of Informatics and Applied Mathematics at St Tikhon’s Orthodox University, celebrated his sixtieth birthday on 14 February 2021. He was born in Moscow. His parents worked for defence industry and were involved directly in launching Earth satellites and ballistic missiles. After graduating from Moscow secondary school no. 19 with a gold medal, where B. L. Geidman was his mathematics teacher, Bogachev enrolled at the Faculty of Mechanics and Mathematics at Moscow State University, and later started postgraduate studies there with O. G. Smolyanov as his scientific advisor. He completed his postgraduate studies ahead of time, and in 1986, after defending his PhD thesis, begun to work at the same Faculty. Bogachev is a major expert in measure theory, the theory of probability, infinitedimensional analysis, and partial differential equations. He has solved a number of difficult problems stated by well-known mathematicians, and has obtained fundamental results in the theory of Gaussian distributions, investigated the differentiability properties of measures, and developed a new line of research in the theory of Fokker–Planck–Kolmogorov equations. His first papers, published in the early 1980s, concerned measure theory in infinite-dimensional spaces and the theory of differentiable measures, where he continued the research of his advisor Smolyanov. Bogachev gained recognition by successfully solving three problems posed by Aronszajn in the theory of infinite-dimensional probability distributions. Aronszajn proposed the following definition as an infinite-dimensional analogue of a set with Lebesgue measure zero.
{"title":"Vladimir Igorevich Bogachev","authors":"Petr Anatol'evich Borodin, Il'dar Abdullovich Ibragimov, B. Kashin, Valery Vasil'evich Kozlov, Aleksandr Viktorovich Kolesnikov, S. V. Konyagin, E. D. Kosov, O. Smolyanov, N. A. Tolmachev, D. Treschev, Alexander Shaposhnikov, Stanislav Valer'evich Shaposhnikov, A. Shiryaev, A. Shkalikov","doi":"10.1070/RM9997","DOIUrl":"https://doi.org/10.1070/RM9997","url":null,"abstract":"The prominent mathematician Vladimir Igorevich Bogachev, Professor at the Department of the Theory of Functions and Functional Analysis of the Faculty of Mechanics and Mathematics at Lomonosov Moscow State University, Professor at the Faculty of Mathematics of the HSE University, and Professor at the Department of Mathematics of the Faculty of Informatics and Applied Mathematics at St Tikhon’s Orthodox University, celebrated his sixtieth birthday on 14 February 2021. He was born in Moscow. His parents worked for defence industry and were involved directly in launching Earth satellites and ballistic missiles. After graduating from Moscow secondary school no. 19 with a gold medal, where B. L. Geidman was his mathematics teacher, Bogachev enrolled at the Faculty of Mechanics and Mathematics at Moscow State University, and later started postgraduate studies there with O. G. Smolyanov as his scientific advisor. He completed his postgraduate studies ahead of time, and in 1986, after defending his PhD thesis, begun to work at the same Faculty. Bogachev is a major expert in measure theory, the theory of probability, infinitedimensional analysis, and partial differential equations. He has solved a number of difficult problems stated by well-known mathematicians, and has obtained fundamental results in the theory of Gaussian distributions, investigated the differentiability properties of measures, and developed a new line of research in the theory of Fokker–Planck–Kolmogorov equations. His first papers, published in the early 1980s, concerned measure theory in infinite-dimensional spaces and the theory of differentiable measures, where he continued the research of his advisor Smolyanov. Bogachev gained recognition by successfully solving three problems posed by Aronszajn in the theory of infinite-dimensional probability distributions. Aronszajn proposed the following definition as an infinite-dimensional analogue of a set with Lebesgue measure zero.","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":"76 1","pages":"1149 - 1157"},"PeriodicalIF":0.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59006655","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
V. Alexeev, C. Birkar, F. Bogomolov, Y. Zarhin, V. Nikulin, Dmitri Orlov, A. N. P. Y. G. Prokhorov, M. Reid, A. Tikhomirov, I. Cheltsov
On 18 May 2020, Vyacheslav Shokurov, a great scientist, leading researcher at the Steklov Mathematical Institute of the Russian Academy of Sciences, and Professor of Mathematics at the Johns Hopkins University in Baltimore, turned 70 years old. Vyacheslav Shokurov is a world-leading expert in birational algebraic geometry, who has completely reshaped this area of modern mathematics. His novel research, which often used amazing approaches, underlies many current trends in this area. The impact he has had on higher-dimensional birational geometry with his deep insight, new methods, and prophetic conjectures (many of them still open) cannot be overestimated. Vyacheslav Shokurov was born in Moscow. He was educated in High School no. 2, one of the best mathematical schools in Moscow at the time. Many former students of this school became famous scientists in their later lives. Among his mathematics teachers in the school were several faculty members of Moscow State University, and some students from this university were assisting. One of these
{"title":"Vyacheslav Vladimirovich Shokurov","authors":"V. Alexeev, C. Birkar, F. Bogomolov, Y. Zarhin, V. Nikulin, Dmitri Orlov, A. N. P. Y. G. Prokhorov, M. Reid, A. Tikhomirov, I. Cheltsov","doi":"10.1070/RM10002","DOIUrl":"https://doi.org/10.1070/RM10002","url":null,"abstract":"On 18 May 2020, Vyacheslav Shokurov, a great scientist, leading researcher at the Steklov Mathematical Institute of the Russian Academy of Sciences, and Professor of Mathematics at the Johns Hopkins University in Baltimore, turned 70 years old. Vyacheslav Shokurov is a world-leading expert in birational algebraic geometry, who has completely reshaped this area of modern mathematics. His novel research, which often used amazing approaches, underlies many current trends in this area. The impact he has had on higher-dimensional birational geometry with his deep insight, new methods, and prophetic conjectures (many of them still open) cannot be overestimated. Vyacheslav Shokurov was born in Moscow. He was educated in High School no. 2, one of the best mathematical schools in Moscow at the time. Many former students of this school became famous scientists in their later lives. Among his mathematics teachers in the school were several faculty members of Moscow State University, and some students from this university were assisting. One of these","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":"76 1","pages":"553 - 556"},"PeriodicalIF":0.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59011800","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Results due to Keldysh on the convergence of Bieberbach polynomials and the density of polynomials in spaces of analytic functions are considered. Their further development and relevance in the contemporary context of constructive complex analysis are discussed. Particular focus is placed on Mergelyan’s conjecture on the rate of convergence in a domain with smooth boundary, which is still open. Bibliography: 20 titles.
{"title":"Convergence of Bieberbach polynomials: Keldysh’s theorems and Mergelyan’s conjecture","authors":"A. Aptekarev","doi":"10.1070/RM9991","DOIUrl":"https://doi.org/10.1070/RM9991","url":null,"abstract":"Results due to Keldysh on the convergence of Bieberbach polynomials and the density of polynomials in spaces of analytic functions are considered. Their further development and relevance in the contemporary context of constructive complex analysis are discussed. Particular focus is placed on Mergelyan’s conjecture on the rate of convergence in a domain with smooth boundary, which is still open. Bibliography: 20 titles.","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":"76 1","pages":"379 - 387"},"PeriodicalIF":0.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59005731","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
V. S. Atabekyan, L. Beklemishev, V. Guba, I. Lysenok, A. Razborov, A. L. Semenov
This is a survey of results on the Burnside problem and properties of Burnside groups, the finite basis problem for group identities, periodic products of groups and Malcev’s problem, construction of groups with special properties (Tarski monsters), constructive bounds in the Burnside- Magnus problem, and algorithmic problems: the problem of recognition of group properties, the word problem for semigroups with one relation, and semi-Thue systems. The focus is on the most important results obtained in papers of Adian and his students. Bibliography: 81 titles.
{"title":"Questions in algebra and mathematical logic. Scientific heritage of S. I. Adian","authors":"V. S. Atabekyan, L. Beklemishev, V. Guba, I. Lysenok, A. Razborov, A. L. Semenov","doi":"10.1070/RM9980","DOIUrl":"https://doi.org/10.1070/RM9980","url":null,"abstract":"This is a survey of results on the Burnside problem and properties of Burnside groups, the finite basis problem for group identities, periodic products of groups and Malcev’s problem, construction of groups with special properties (Tarski monsters), constructive bounds in the Burnside- Magnus problem, and algorithmic problems: the problem of recognition of group properties, the word problem for semigroups with one relation, and semi-Thue systems. The focus is on the most important results obtained in papers of Adian and his students. Bibliography: 81 titles.","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":"38 1","pages":"1 - 27"},"PeriodicalIF":0.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59005444","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
V. Atabekyan, L. Beklemishev, V. Buchstaber, S. Goncharov, V. Guba, Y. Ershov, V. Kozlov, I. Lysenok, S. Novikov, Y. Osipov, M. Pentus, V. Podolskii, A. Razborov, V. Sadovnichii, A. L. Semenov, A. Talambutsa, D. Treschev, L. N. Shevrin
Academician Sergei Ivanovich Adian (1 January 1931 —5 May 2020), one of the most prominent Russian mathematicians, was born in the mountain village of Kushchi, in the Dashkasan district of the Azerbaijan Soviet Socialist Republic, which lies 40 kilometers away from the town of Ganja (which was soon renamed Kirovabad, but now is Ganja again). His father Ivan Arakelovich Adian was a carpenter, a son of a herdsman, and his mother Lusik was a daughter of Konstantin Truzyan, a peasant. Two years later Sergei Adian’s parents moved to Kirovabad. By the beginning of World War II they had four children. In 1941, during the first days of the war the father was conscripted and was soon killed when his unit was surrounded. Sergei, like his parents, did not speak Russian, but in 1938 they sent him to the Russian secondary school no. 11 in Kirovabad, where his mathematical abilities became obvious quite early. When he graduated, the public education department of Kirovabad applied to have him included in the Azerbaijan quota of graduates sent to study at Moscow State University. The application was declined (it was mainly ethnic Azerbaijanis that were accepted), and as a result he enrolled in
{"title":"Sergei Ivanovich Adian","authors":"V. Atabekyan, L. Beklemishev, V. Buchstaber, S. Goncharov, V. Guba, Y. Ershov, V. Kozlov, I. Lysenok, S. Novikov, Y. Osipov, M. Pentus, V. Podolskii, A. Razborov, V. Sadovnichii, A. L. Semenov, A. Talambutsa, D. Treschev, L. N. Shevrin","doi":"10.1070/RM9989","DOIUrl":"https://doi.org/10.1070/RM9989","url":null,"abstract":"Academician Sergei Ivanovich Adian (1 January 1931 —5 May 2020), one of the most prominent Russian mathematicians, was born in the mountain village of Kushchi, in the Dashkasan district of the Azerbaijan Soviet Socialist Republic, which lies 40 kilometers away from the town of Ganja (which was soon renamed Kirovabad, but now is Ganja again). His father Ivan Arakelovich Adian was a carpenter, a son of a herdsman, and his mother Lusik was a daughter of Konstantin Truzyan, a peasant. Two years later Sergei Adian’s parents moved to Kirovabad. By the beginning of World War II they had four children. In 1941, during the first days of the war the father was conscripted and was soon killed when his unit was surrounded. Sergei, like his parents, did not speak Russian, but in 1938 they sent him to the Russian secondary school no. 11 in Kirovabad, where his mathematical abilities became obvious quite early. When he graduated, the public education department of Kirovabad applied to have him included in the Azerbaijan quota of graduates sent to study at Moscow State University. The application was declined (it was mainly ethnic Azerbaijanis that were accepted), and as a result he enrolled in","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":"76 1","pages":"177 - 181"},"PeriodicalIF":0.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59005554","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Multipoint formulae for inverse scattering at high energies","authors":"R. Novikov","doi":"10.1070/RM9994","DOIUrl":"https://doi.org/10.1070/RM9994","url":null,"abstract":"","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":"76 1","pages":"723 - 725"},"PeriodicalIF":0.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59006044","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Separation of variables for type [IMG align=ABSMIDDLE alt=$ D_n$]tex_rm_5265_img1[/IMG] Hitchin systems on hyperelliptic curves","authors":"P. I. Borisova","doi":"10.1070/RM9935","DOIUrl":"https://doi.org/10.1070/RM9935","url":null,"abstract":"","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":"76 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59004916","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
According to Dirac, changes in the equations of motion related to additional external forces performing no work can be described in terms of deformations of the Poisson bracket. It is natural to ask whether or not Dirac’s ideas are valid in non-holonomic mechanics. We discuss this question here by taking the Chaplygin ball as an example. We consider the linear Lie–Poisson bracket on the Lie algebra e∗(3):
{"title":"Chaplygin ball in a solenoidal field","authors":"A. Borisov, A. Tsiganov","doi":"10.1070/RM9930","DOIUrl":"https://doi.org/10.1070/RM9930","url":null,"abstract":"According to Dirac, changes in the equations of motion related to additional external forces performing no work can be described in terms of deformations of the Poisson bracket. It is natural to ask whether or not Dirac’s ideas are valid in non-holonomic mechanics. We discuss this question here by taking the Chaplygin ball as an example. We consider the linear Lie–Poisson bracket on the Lie algebra e∗(3):","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":"76 1","pages":"546 - 548"},"PeriodicalIF":0.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59004771","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let LG(n) be the Lagrangian Grassmannian parameterizing the Lagrangian linear subspaces of the 2n-dimensional complex symplectic vector space. It has a Plücker embedding to a projective space P, so that for H = OP(1) we have Pic(LG(n)) = ZH. Let X ⊂ LG(n) be a smooth Fano complete intersection of degrees d1, . . . , dk. We have ∑k i=1 di < n + 1, and dk+1 = n + 1 − ∑k i=1 di is the Fano index of X. Let pi, i = 1, . . . , n, be formal variables. Consider the series
设LG(n)为参数化2n维复辛向量空间的拉格朗日线性子空间的拉格朗日格拉斯曼函数。它有一个plencker嵌入到射影空间P中，因此对于H = OP(1)我们有Pic(LG(n)) = ZH。设X∧LG(n)是一个光滑的Fano完全交(d1，…)dk。我们有∑k1 = 1di < n +1, dk+1 = n +1−∑k1 = 1di是x的Fano指数，设pi, i=1，…， n是形式变量。考虑这个系列
{"title":"Landau–Ginzburg models of complete intersections in Lagrangian Grassmannians","authors":"V. Przyjalkowski, K. Rietsch","doi":"10.1070/RM9984","DOIUrl":"https://doi.org/10.1070/RM9984","url":null,"abstract":"Let LG(n) be the Lagrangian Grassmannian parameterizing the Lagrangian linear subspaces of the 2n-dimensional complex symplectic vector space. It has a Plücker embedding to a projective space P, so that for H = OP(1) we have Pic(LG(n)) = ZH. Let X ⊂ LG(n) be a smooth Fano complete intersection of degrees d1, . . . , dk. We have ∑k i=1 di < n + 1, and dk+1 = n + 1 − ∑k i=1 di is the Fano index of X. Let pi, i = 1, . . . , n, be formal variables. Consider the series","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":"76 1","pages":"549 - 551"},"PeriodicalIF":0.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59005263","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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