Russian Mathematical Surveys最新文献

{"title":"On the resolution of singularities of one-dimensional foliations on three-manifolds","authors":"J. Rebelo, H. Reis","doi":"10.1070/RM9993","DOIUrl":"https://doi.org/10.1070/RM9993","url":null,"abstract":"This paper is devoted to the resolution of singularities of holomorphic vector fields and one-dimensional holomorphic foliations in dimension three, and it has two main objectives. First, within the general framework of one-dimensional foliations, we build upon and essentially complete the work of Cano, Roche, and Spivakovsky (2014). As a consequence, we obtain a general resolution theorem comparable to the resolution theorem of McQuillan–Panazzolo (2013) but proved by means of rather different methods. The other objective of this paper is to consider a special class of singularities of foliations containing, in particular, all the singularities of complete holomorphic vector fields on complex manifolds of dimension three. We then prove that a much sharper resolution theorem holds for this class of holomorphic foliations. This second result was the initial motivation for this paper. It relies on combining earlier resolution theorems for (general) foliations with some classical material on asymptotic expansions for solutions of differential equations. Bibliography: 34 titles.","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59005926","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":"Chaos and integrability in -geometry","authors":"A. Bolsinov, A. Veselov, Y. Ye","doi":"10.1070/RM10008","DOIUrl":"https://doi.org/10.1070/RM10008","url":null,"abstract":"We review the integrability of the geodesic flow on a threefold admitting one of the three group geometries in Thurston’s sense. We focus on the case. The main examples are the quotients , where is a cofinite Fuchsian group. We show that the corresponding phase space contains two open regions with integrable and chaotic behaviour, with zero and positive topological entropy, respectively. As a concrete example we consider the case of the modular threefold with the modular group . In this case is known to be homeomorphic to the complement of a trefoil knot in a 3-sphere. Ghys proved the remarkable fact that the lift of a periodic geodesic on the modular surface to produces the same isotopy class of knots as that which appears in the chaotic version of the celebrated Lorenz system and was studied in detail by Birman and Williams. We show that these knots are replaced by trefoil knot cables in the integrable limit of the geodesic system on . Bibliography: 60 titles.","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59011883","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":"Local groups in Delone sets: a conjecture and results","authors":"N. Dolbilin, M. Shtogrin","doi":"10.1070/RM10037","DOIUrl":"https://doi.org/10.1070/RM10037","url":null,"abstract":"In the framework of a new approach to the concept of local symmetry in arbitrary Delone sets we obtain new results for such sets without any restrictions. These results have important consequences for lattices and regular systems. A conjecture about the crystal kernel is stated, which generalises significantly the classical theorem on the non-existence of a five-fold symmetry in three-dimensional lattices. The following theorems related to the foundations of geometric crystallography are proved.","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59012763","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":"On compatible diagonal metrics","authors":"Alexander Mikhailovich Gagonov, Олег Иванович Мохов","doi":"10.1070/RM10031","DOIUrl":"https://doi.org/10.1070/RM10031","url":null,"abstract":"In this note the well-known important problem of a complete description of compatible diagonal metrics is solved. In 2000 (see [1] and [2]) Mokhov obtained a complete explicit description of pairs of compatible metrics for which all eigenvalues are distinct. In the case of distinct eigenvalues such a pair of metrics can be simultaneously diagonalized and, as shown in [1] and [2], it is compatible if and only if the Nijenhuis tensor of the affinor associated with this pair of metrics vanishes. This made it possible in [1] and [2] to describe all such compatible metrics explicitly. The general case of pairs of compatible diagonal metrics that have coincident eigenvalues has remained unexplored despite its importance for applications. This case is completely investigated in this work. The general case of describing all pairs of compatible metrics remains an open problem to date. Compatible metrics play an important role in the theory of integrable systems, the Hamiltonian and bi-Hamiltonian theory of systems of hydrodynamic type, integrable hierarchies, the theory of Frobenius manifolds and their generalizations, the theory of multidimensional Poisson brackets, differential geometry and mathematical physics (see [3]–[12] and the review paper [13]). The general notion of compatible metrics was introduced by Mokhov in [1] and [2], and was motivated by the study of the compatibility conditions for local and non-local Poisson structures of hydrodynamic type, the theory of which was developed by Dubrovin and Novikov ([14], local theory) and Mokhov and Ferapontov ([15] and [16], non-local theory) for the purposes of the theory of systems of hydrodynamic type. Recall that a pair of contravariant (Riemannian or pseudo-Riemannian) metrics g 1 (u) and g ij 2 (u) is called almost compatible [1], [2] if for any linear combination g λ1,λ2(u) = λ1g ij 1 (u) + λ2g ij 2 (u) of these metrics, where λ1 and λ2 are arbitrary constants, the same linear relation holds for the Christoffel symbols corresponding to these metrics (the compatibility condition for the Levi-Civita connections of these metrics): Γ λ1,λ2;k(u) = λ1Γ ij 1;k(u) + λ2Γ ij 2;k(u), where Γ λ1,λ2;k(u) = g is λ1,λ2 (u)Γjλ1,λ2;sk(u), Γ ij 1;k(u) = g is 1 (u)Γ j 1;sk(u), and Γ ij 2;k(u) = g 2 (u)Γ j 2;sk(u). A pair of almost compatible metrics g ij 1 (u) and g ij 2 (u) is called compatible [1], [2] if for any linear combination g λ1,λ2(u) = λ1g ij 1 (u) + λ2g ij 2 (u) of these metrics, where λ1 and λ2 are arbitrary constants, the same linear relation holds for the Riemann curvature tensors corresponding to these metrics (the compatibility condition for the curvatures of these metrics):","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59012914","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":"Theory of homotopes with applications to mutually unbiased bases, harmonic analysis on graphs, and perverse sheaves","authors":"A. Bondal, I. Zhdanovskiy","doi":"10.1070/RM9983","DOIUrl":"https://doi.org/10.1070/RM9983","url":null,"abstract":"This paper is a survey of contemporary results and applications of the theory of homotopes. The notion of a well-tempered element of an associative algebra is introduced, and it is proved that the category of representations of the homotope constructed by a well-tempered element is the heart of a suitably glued -structure. The Hochschild and global dimensions of homotopes are calculated in the case of well-tempered elements. The homotopes constructed from generalized Laplace operators in Poincaré groupoids of graphs are studied. It is shown that they are quotients of Temperley–Lieb algebras of general graphs. The perverse sheaves on a punctured disc and on a 2-dimensional sphere with a double point are identified with representations of suitable homotopes. Relations of the theory to orthogonal decompositions of the Lie algebras into a sum of Cartan subalgebras, to classifications of configurations of lines, to mutually unbiased bases, to quantum protocols, and to generalized Hadamard matrices are discussed. Bibliography: 56 titles.","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45792580","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":"The Dickman–Goncharov distribution","authors":"S. Molchanov, V. Panov","doi":"10.1070/RM9976","DOIUrl":"https://doi.org/10.1070/RM9976","url":null,"abstract":"In the 1930s and 40s, one and the same delay differential equation appeared in papers by two mathematicians, Karl Dickman and Vasily Leonidovich Goncharov, who dealt with completely different problems. Dickman investigated the limit value of the number of natural numbers free of large prime factors, while Goncharov examined the asymptotics of the maximum cycle length in decompositions of random permutations. The equation obtained in these papers defines, under a certain initial condition, the density of a probability distribution now called the Dickman–Goncharov distribution (this term was first proposed by Vershik in 1986). Recently, a number of completely new applications of the Dickman–Goncharov distribution have appeared in mathematics (random walks on solvable groups, random graph theory, and so on) and also in biology (models of growth and evolution of unicellular populations), finance (theory of extreme phenomena in finance and insurance), physics (the model of random energy levels), and other fields. Despite the extensive scope of applications of this distribution and of more general but related models, all the mathematical aspects of this topic (for example, infinite divisibility and absolute continuity) are little known even to specialists in limit theorems. The present survey is intended to fill this gap. Both known and new results are given. Bibliography: 62 titles.","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41688728","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":"Spinning tops and magnetic orbits","authors":"S. Novikov","doi":"10.1070/RM9977","DOIUrl":"https://doi.org/10.1070/RM9977","url":null,"abstract":"A number of directions were initiated by the author and his students in their papers of 1981–1982. However, one of them, concerning the properties of closed orbits on the sphere and in the groups and , has not been sufficiently developed. This paper revives the discussion of these questions, states unsolved problems, and explains what was regarded as fallacies in old papers. In general, magnetic orbits have been poorly discussed in the literature on dynamical systems and theoretical mechanics, but Grinevich has pointed out that in theoretical physics one encounters similar situations in the theory related to particle accelerators such as proton cyclotrons. It is interesting to look at Chap. III of Landau and Lifshitz’s Theoretical physics, vol. 2, Field theory (Translated into English as The classical theory of fields [12]. where mathematical relatives of our situations occur, but the physics is completely different and there are actual strong magnetic fields. Bibliography: 12 titles.","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44194390","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":"Bifurcations in spatially distributed chains of two-dimensional systems of equations","authors":"S. Kaschenko","doi":"10.1070/RM9986","DOIUrl":"https://doi.org/10.1070/RM9986","url":null,"abstract":"u̇j = Auj + F (uj) + D[uj+1− 2uj + uj−1], j = 1, . . . , N ; u0 ≡ uN , uN+1 ≡ u1. (2) We associate the element uj(t) with the value of a function u(t, xj) of two variables, where xj = 2πjN−1 is the angular coordinate. The main assumption is that the number N of elements in (2) is sufficiently large, so that the parameter ε = 2πN−1 is sufficiently small: 0 < ε ≪ 1. This gives reason to switch from the discrete system (2) to the following system, which is continuous with respect to the spatial variable x:","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42177421","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":"Mikhail Aleksandrovich Shubin","authors":"M. Braverman, V. Buchstaber, M. Gromov, V. Ivrii, Y. Kordyukov, P. Kuchment, V. Maz'ya, S. Novikov, T. Sunada, L. Friedlander, A. Khovanskii","doi":"10.1070/RM9968","DOIUrl":"https://doi.org/10.1070/RM9968","url":null,"abstract":"The prominent mathematician Mikhail Aleksandrovich Shubin passed away after a long illness on 13 May 2020. He was born on 19 December 1944 in Kuibyshev (now Samara) and raised by his mother and grandmother. His mother, Maria Arkadievna, was an engineer at the State Bearing Factory, where she was hired in 1941 after graduating from the Faculty of Mechanics and Mathematics at the Moscow State University. At that time the factory was evacuated from Moscow to Kuibyshev. She worked at the factory for many years as the Head of the Physics of Metals Laboratory. Later, she defended her Ph.D. thesis and moved to Kuibyshev Polytechnical Institute, where she worked as an associate professor. In his school years Shubin was mainly interested in music. He had absolute pitch. After finishing music school, he seriously considered entering a conservatory. However, in high school he developed an interest to mathematics, was successful in olympiads, and eventually decided to apply to the Faculty of Mechanics and Mathematics at the Moscow State University. He was admitted there in 1961. When the time came to choose an adviser, he became a student of M. I. Vishik. After graduating, he began postgraduate work there, and in 1969 defended his Ph.D. thesis. In the thesis he derived formulae for the index of matrix-valued Wiener–Hopf operators. In particular, for the study of families of such operators, he had to generalize a theorem of Birkhoff stating that a continuous matrix-valued function M(z) defined on the unit circle |z| = 1 can be factored as M(z) = A+(z)D(z)A−(z), where A+(z) and A−(z) are continuous and have analytic continuations to the interior of the unit circle and its exterior (infinity included), respectively, and D(z) is a diagonal matrix with entries zj on the diagonal, with integer nj . Shubin considered the problem of what happens when the matrix M depends continuously on an additional parameter t. The Birkhoff factorization cannot be made continuous","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46819886","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":"Valerii Vasil’evich Kozlov","authors":"S. Bolotin, A. V. Borisov, A. Karapetyan, B. Kashin, E. I. Kugushev, Anatolii Iserovich Neishtadt, Dmitri Orlov, D. Treschev","doi":"10.1070/RM9949","DOIUrl":"https://doi.org/10.1070/RM9949","url":null,"abstract":"On 1 January 2020 the prominent researcher and academician of the Russian Academy of Sciences Valerii Vasil’evich Kozlov observed his 70th birthday. Kozlov has made fundamental contributions to diverse areas of mathematics and mechanics: the theory of Hamiltonian systems, stability theory, the mechanics of non-holonomic systems, statistical mechanics. He has published about 300 papers on mathematics and mechanics and 8 monographs which are now classical. In this one article it is impossible to give even a brief account of all the directions of his research. Kozlov was born on 1 January 1950 in the village of Kostyli, in the Mikhailovskoe District of the Ryazan Oblast. His mother Ol’ga Arkhipovna was a teacher of mathematics, and his father Vasilii Nestorovich was a train-driver, and a veteran of World War II, from the first days when the Soviet Union was attacked until Victory Day. Valerii started his early school education in his native small village (where nobody lives now). There was only a primary school there, with one female teacher, who gave simultaneous lessons to grades I and III in the morning and to grades II and IV in the afternoon. As an 8-year boy, Kozlov moved with his parents to Lyublino-Dachnaya, close to Moscow. When the Moscow Ring Road was built (in 1961) this settlement, like many others, found itself inside the expanding Moscow. In this way Kozlov became a Moscow resident. During his last two years in secondary school he became deeply interested in mathematics and physics. Three times a week he travelled to lessons at a volunteer physics-mathematics evening school under the auspices of the Bauman Moscow State Technical School (now Technical University). This proved to be a remarkable school! (It was founded in 1962 and still exists.) Most teachers were students","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49615312","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}