{"title":"Index of minimal surfaces in the 3-sphere","authors":"Egor Aleksandrovich Morozov, Alexei Viktorovich Penskoi","doi":"10.4213/rm10094e","DOIUrl":"https://doi.org/10.4213/rm10094e","url":null,"abstract":"","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136368166","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}
Diophantine exponents are some of the simplest quantitative characteristics responsible for the approximation properties of linear subspaces of a Euclidean space. This survey is aimed at describing the current state of the area of Diophantine approximation that studies Diophantine exponents and relations they satisfy. We discuss classical Diophantine exponents arising in the problem of approximating zero with the set of the values of several linear forms at integer points, their analogues in Diophantine approximation with weights, multiplicative Diophantine exponents, and Diophantine exponents of lattices. We pay special attention to the transference principle. Bibliography: 99 titles.
{"title":"Geometry of Diophantine exponents","authors":"Oleg Nikolaevich German","doi":"10.4213/rm10089e","DOIUrl":"https://doi.org/10.4213/rm10089e","url":null,"abstract":"Diophantine exponents are some of the simplest quantitative characteristics responsible for the approximation properties of linear subspaces of a Euclidean space. This survey is aimed at describing the current state of the area of Diophantine approximation that studies Diophantine exponents and relations they satisfy. We discuss classical Diophantine exponents arising in the problem of approximating zero with the set of the values of several linear forms at integer points, their analogues in Diophantine approximation with weights, multiplicative Diophantine exponents, and Diophantine exponents of lattices. We pay special attention to the transference principle. Bibliography: 99 titles.","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136368159","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}
Left-invariant optimal control problems on Lie groups are an important class of problems with a large symmetry group. They are theoretically interesting because they can often be investigated in full and general laws can be studied by using these model problems. In particular, problems on nilpotent Lie groups provide a fundamental nilpotent approximation to general problems. Also, left-invariant problems often arise in applications such as classical and quantum mechanics, geometry, robotics, visual perception models, and image processing. The aim of this paper is to present a survey of the main concepts, methods, and results pertaining to left-invariant optimal control problems on Lie groups that can be integrated by elliptic functions. The focus is on describing extremal trajectories and their optimality, the cut time and cut locus, and optimal synthesis. Bibliography: 162 titles.
{"title":"Left-invariant optimal control problems on Lie groups that are integrable by elliptic functions","authors":"Y. Sachkov","doi":"10.4213/rm10063e","DOIUrl":"https://doi.org/10.4213/rm10063e","url":null,"abstract":"Left-invariant optimal control problems on Lie groups are an important class of problems with a large symmetry group. They are theoretically interesting because they can often be investigated in full and general laws can be studied by using these model problems. In particular, problems on nilpotent Lie groups provide a fundamental nilpotent approximation to general problems. Also, left-invariant problems often arise in applications such as classical and quantum mechanics, geometry, robotics, visual perception models, and image processing. The aim of this paper is to present a survey of the main concepts, methods, and results pertaining to left-invariant optimal control problems on Lie groups that can be integrated by elliptic functions. The focus is on describing extremal trajectories and their optimality, the cut time and cut locus, and optimal synthesis. Bibliography: 162 titles.","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70328887","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}
This survey is devoted to a number of achievements in the theory of extremal problems in geometric function theory. The approaches to the solution of problems under consideration and the methods used are based on conformal isomorphisms and on the theory of univalent functions developed since the beginning of the 20th century. Results on integral means of conformal mappings of a disc are presented and, in particular, Dolzenko's inequality for rational functions is extended to arbitrary domains with rectifiable boundaries. Investigations in the field of Bohr-type inequalities are described. An emphasis is made on integral inequalities of Hardy and Rellich type, in which the analytic properties of inequalities are intertwined with geometric characteristics of the boundaries of domains. Results related to the solution of the Vuorinen problem on the behaviour of conformal moduli under unlimited dilations of the plane are presented. Formulae for the variation of Robin capacity are obtained. One-parameter families of rational and elliptic functions whose critical values vary in accordance with a prescribed law are characterized. The last results on Smale's conjecture and Smale's dual conjecture are described. Bibliography: 149 titles.
{"title":"Extremal problems in geometric function theory","authors":"Farit Gabidinovich Avkhadiev, Ilgiz Rifatovich Kayumov, Semen Rafailovich Nasyrov","doi":"10.4213/rm10076e","DOIUrl":"https://doi.org/10.4213/rm10076e","url":null,"abstract":"This survey is devoted to a number of achievements in the theory of extremal problems in geometric function theory. The approaches to the solution of problems under consideration and the methods used are based on conformal isomorphisms and on the theory of univalent functions developed since the beginning of the 20th century. Results on integral means of conformal mappings of a disc are presented and, in particular, Dolzenko's inequality for rational functions is extended to arbitrary domains with rectifiable boundaries. Investigations in the field of Bohr-type inequalities are described. An emphasis is made on integral inequalities of Hardy and Rellich type, in which the analytic properties of inequalities are intertwined with geometric characteristics of the boundaries of domains. Results related to the solution of the Vuorinen problem on the behaviour of conformal moduli under unlimited dilations of the plane are presented. Formulae for the variation of Robin capacity are obtained. One-parameter families of rational and elliptic functions whose critical values vary in accordance with a prescribed law are characterized. The last results on Smale's conjecture and Smale's dual conjecture are described. Bibliography: 149 titles.","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136368164","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}
In this survey we describe the history and current state of one of the key areas in the qualitative theory of elliptic partial differential equations related to the strong maximum principle and the boundary point principle (normal derivative lemma). Bibliography: 234 titles.
{"title":"The normal derivative lemma and surrounding issues","authors":"D. Apushkinskaya, A. Nazarov","doi":"10.1070/RM10049","DOIUrl":"https://doi.org/10.1070/RM10049","url":null,"abstract":"In this survey we describe the history and current state of one of the key areas in the qualitative theory of elliptic partial differential equations related to the strong maximum principle and the boundary point principle (normal derivative lemma). Bibliography: 234 titles.","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2022-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47121450","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}
This paper focuses on Richard Thompson’s group , which was discovered in the 1960s. Many papers have been devoted to this group. We are interested primarily in the famous problem of amenability of this group, which was posed by Geoghegan in 1979. Numerous attempts have been made to solve this problem in one way or the other, but it remains open. In this survey we describe the most important known properties of this group related to the word problem and representations of elements of the group by piecewise linear functions as well as by diagrams and other geometric objects. We describe the classical results of Brin and Squier concerning free subgroups and laws. We include a description of more modern important results relating to the properties of the Cayley graphs (the Belk–Brown construction) as well as Bartholdi’s theorem about the properties of equations in group rings. We consider separately the criteria for (non-)amenability of groups that are useful in the work on the main problem. At the end we describe a number of our own results about the structure of the Cayley graphs and a new algorithm for solving the word problem. Bibliography: 69 titles.
{"title":"R. Thompson’s group and the amenability problem","authors":"V. Guba","doi":"10.1070/RM10040","DOIUrl":"https://doi.org/10.1070/RM10040","url":null,"abstract":"This paper focuses on Richard Thompson’s group , which was discovered in the 1960s. Many papers have been devoted to this group. We are interested primarily in the famous problem of amenability of this group, which was posed by Geoghegan in 1979. Numerous attempts have been made to solve this problem in one way or the other, but it remains open. In this survey we describe the most important known properties of this group related to the word problem and representations of elements of the group by piecewise linear functions as well as by diagrams and other geometric objects. We describe the classical results of Brin and Squier concerning free subgroups and laws. We include a description of more modern important results relating to the properties of the Cayley graphs (the Belk–Brown construction) as well as Bartholdi’s theorem about the properties of equations in group rings. We consider separately the criteria for (non-)amenability of groups that are useful in the work on the main problem. At the end we describe a number of our own results about the structure of the Cayley graphs and a new algorithm for solving the word problem. Bibliography: 69 titles.","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2022-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43005488","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":"Elements of hyperbolic theory on an infinite-dimensional torus","authors":"S. Glyzin, A. Kolesov","doi":"10.1070/rm10058","DOIUrl":"https://doi.org/10.1070/rm10058","url":null,"abstract":"","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59014106","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}
Every reductive complex Lie algebra g is equipped with the canonical Poisson structure {φ,ψ}(x) = (x, [dxφ, dxψ]), where φ and ψ are smooth functions on g, while dxφ and dxψ are treated as elements of g, which is identified with g∗ using the invariant inner product. Moreover, for every a ∈ g, a Poisson structure ‘with frozen argument’ is defined: {φ,ψ}a(x) = (a, [dxφ, dxψ]). An approach described in [1] makes it possible to work with Poisson structures using the language of linear algebra. The Poisson brackets { · , · }a and { · , · } are regarded as skew-symmetric bilinear forms fa and fx over the field K = C(g) on the space g⊗K of rational vector fields on g, where the element a is fixed and x is a generic element. Namely, if φ and ψ are polynomials, then dφ and dψ can be regarded as elements of g⊗K, and then {φ,ψ}(x) = fx(dφ, dψ) and {φ,ψ}a(x) = fa(dφ, dψ). The above approach can be used to solve an important problem in Hamiltonian mechanics, namely, the search for complete families of functions in bi-involution, that is, maximal families of functions commuting with respect to both Poisson brackets. The polynomials φ1, . . . , φs define a complete family of functions in bi-involution with respect to { · , · }a and { · , · } if and only if their differentials dφ1, . . . , dφs form a basis of a bi-Lagrangian subspace (that is, a maximal subspace which is isotropic with respect to both bilinear forms). Thus, to obtain a complete family of functions in bi-involution, it suffices to find a basis for a bi-Lagrangian subspace and ‘integrate with respect to x’. If a basis is found (we call it canonical) in which the matrices of both forms fa and fx are reduced simultaneously to the canonical Jordan–Kronecker form (with blocks of two types, Jordan and Kronecker, see [1]), then a basis of the bi-Lagrangian subspace is formed by the second halves of the bases in each block. The second (‘larger’) halves of the bases in Kronecker blocks span a subspace L, which is the intersection of all bi-Lagrangian subspaces for the forms fa and fx. In [4] a basis of the subspace L and the corresponding functions in bi-involution were constructed for the Lie algebras gln and sp2n. In [3] the Kronecker part of a canonical basis and the corresponding part of the complete system of functions in bi-involution were constructed for the Lie algebra gln. Now we introduce our notation and formulate the result. Let {λ1, . . . , λs} be distinct eigenvalues of a matrix A ∈ gln. Assume that Jordan cells of order nk,1 ⩾ · · · ⩾ nk,ik correspond to an eigenvalue λk. We write
{"title":"On a canonical basis of a pair of compatible Poisson brackets on a symplectic Lie algebra","authors":"A. A. Garazha","doi":"10.1070/RM10035","DOIUrl":"https://doi.org/10.1070/RM10035","url":null,"abstract":"Every reductive complex Lie algebra g is equipped with the canonical Poisson structure {φ,ψ}(x) = (x, [dxφ, dxψ]), where φ and ψ are smooth functions on g, while dxφ and dxψ are treated as elements of g, which is identified with g∗ using the invariant inner product. Moreover, for every a ∈ g, a Poisson structure ‘with frozen argument’ is defined: {φ,ψ}a(x) = (a, [dxφ, dxψ]). An approach described in [1] makes it possible to work with Poisson structures using the language of linear algebra. The Poisson brackets { · , · }a and { · , · } are regarded as skew-symmetric bilinear forms fa and fx over the field K = C(g) on the space g⊗K of rational vector fields on g, where the element a is fixed and x is a generic element. Namely, if φ and ψ are polynomials, then dφ and dψ can be regarded as elements of g⊗K, and then {φ,ψ}(x) = fx(dφ, dψ) and {φ,ψ}a(x) = fa(dφ, dψ). The above approach can be used to solve an important problem in Hamiltonian mechanics, namely, the search for complete families of functions in bi-involution, that is, maximal families of functions commuting with respect to both Poisson brackets. The polynomials φ1, . . . , φs define a complete family of functions in bi-involution with respect to { · , · }a and { · , · } if and only if their differentials dφ1, . . . , dφs form a basis of a bi-Lagrangian subspace (that is, a maximal subspace which is isotropic with respect to both bilinear forms). Thus, to obtain a complete family of functions in bi-involution, it suffices to find a basis for a bi-Lagrangian subspace and ‘integrate with respect to x’. If a basis is found (we call it canonical) in which the matrices of both forms fa and fx are reduced simultaneously to the canonical Jordan–Kronecker form (with blocks of two types, Jordan and Kronecker, see [1]), then a basis of the bi-Lagrangian subspace is formed by the second halves of the bases in each block. The second (‘larger’) halves of the bases in Kronecker blocks span a subspace L, which is the intersection of all bi-Lagrangian subspaces for the forms fa and fx. In [4] a basis of the subspace L and the corresponding functions in bi-involution were constructed for the Lie algebras gln and sp2n. In [3] the Kronecker part of a canonical basis and the corresponding part of the complete system of functions in bi-involution were constructed for the Lie algebra gln. Now we introduce our notation and formulate the result. Let {λ1, . . . , λs} be distinct eigenvalues of a matrix A ∈ gln. Assume that Jordan cells of order nk,1 ⩾ · · · ⩾ nk,ik correspond to an eigenvalue λk. We write","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59012690","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":"Spectrum of a convolution operator with potential","authors":"D. Borisov, E. Zhizhina, Andrey L. Piatnitski","doi":"10.1070/rm10038","DOIUrl":"https://doi.org/10.1070/rm10038","url":null,"abstract":"","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59012783","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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