Anatolii Moiseevich Vershik (on his 80th birthday)

On 28 December 2013 the remarkable mathematician Anatolii Moiseevich Vershik, head of a laboratory in the St. Petersburg Division of the Steklov Mathematical Institute of the Russian Academy of Sciences and professor at St. Petersburg State University, observed his 80th birthday. The stages of Vershik’s research, teaching, and public activities were described in a paper dedicated to him on his 60th birthday,1 including his work in measure theory on topological vector spaces, ergodic theory, operator algebras, representation theory, linear programming, non-holonomic dynamics, probability theory, and combinatorics. In this tribute we focus on his work in the last two decades. Representation theory has remained one of the main directions of his research. A long series of his joint papers with M. I. Graev [203], [204], [210], [229], [230], [239], [249], [252], [257] continues the line of research initiated by his well-known paper [26] on unitary representations of the measurable current groups SL(2, R) . New in these papers is a study of the realization of representations by means of the so-called infinite-dimensional Lebesgue measure L. This measure L is connected with Poisson–Dirichlet distributions and is a remarkable example of a σ-finite measure with an infinite-dimensional symmetry group. The separate papers [178], [222], [227] are devoted to the measure L and related topics. The paper [146] explains in detail the ergodic method [30] applied to the problem of describing unitarily invariant measures on the space of infinite Hermitian matrices. As a by-product, it gives a new proof of Schoenberg’s classical theorem on the classification of translation-invariant totally positive kernels on the line.

[1]  A. Vershik.  Two Ways to Define Compatible Metrics on the Simplex of Measures , 2014 .

[2]  Гуров Михаил Николаевич.  О преобразовании Фурье одной осциллирующей функции , 2014 .

[3]  А. В. Галажинский.  Геометрия экстремальных черных дыр вблизи горизонта и интегрируемые системы , 2014 .

[4]  A. Vershik,et al.  Virtual continuity of measurable functions of several variables and embedding theorems , 2013 .

[5]  A. Vershik.  Long History of the Monge-Kantorovich Transportation Problem , 2013 .

[6]  On classification of measurable functions of several variables , 2013, 1301.5724.

[7]  A. Vershik.  Between "very large" and "infinite" : The asymptotic representation theory , 2013 .

[8]  SMOOTH AND NON-SMOOTH AF -ALGEBRAS AND PROBLEM ON INVARIANT MEASURES , 2013 .

[9]  A. Vershik,et al.  Finite traces and representations of the group of infinite matrices over a finite field , 2012, 1209.4945.

[10]  A. Vershik,et al.  Infinite-Dimensional Schur–Weyl Duality and the Coxeter–Laplace Operator , 2012, 1209.4800.

[11]  Leonid Vital'evich Kantorovich (on the 100th anniversary of his birth) , 2012 .

[12]  M. Graev,et al.  The Poisson model of the Fock space and representations of current groups , 2012 .

[13]  Geometry and dynamics of admissible metrics in measure spaces , 2012 .

[14]  A.Vershik,et al.  Geometry and Dynamics of Admissible Metrics in Measure Spaces , 2012, 1205.1174.

[15]  A. Vershik,et al.  Special representations of nilpotent Lie groups and the associated Poisson representations of current groups , 2012, 1204.0477.

[16]  A. Vershik,et al.  Stable states and representations of the infinite symmetric group , 2012, 1204.0457.

[17]  A. Vershik,et al.  Special representations of the groups U(∞, 1) and O(∞, 1) and the associated representations of the current groups U(∞, 1)X and O(∞, 1)X in quasi-poisson spaces , 2012, Functional Analysis and Its Applications.

[18]  A. Vershik,et al.  C*-Algebras Associated with Endomorphisms and Polymorphisms of Compact Abelian Groups , 2012, 1202.5960.

[19]  Фарит Сахапович Замалиев.  Численные эксперименты в исследованиях пространственной работы сталежелезобетонных перекрытий , 2012 .

[20]  A. Vershik.  The pascal automorphism has a continuous spectrum , 2011 .

[21]  A. Vershik,et al.  Totally nonfree actions and infinite symmetric group , 2011, 1109.3413.

[22]  A. Vershik,et al.  Description of the characters and factor representations of the infinite symmetric inverse semigroup , 2011, 1102.4425.

[23]  Anatoly M. Vershik,et al.  Nonfree actions of countable groups and their characters , 2010 .

[24]  Anatoly M. Vershik,et al.  Numerical experiments in problems of asymptotic representation theory , 2010 .

[25]  A. Vershik,et al.  Rational approximation of maximal commutative subgroups of $${GL(n,\mathbb{R})}$$ , 2010 .

[26]  A. Vershik.  Orbit theory, locally finite permutations and morse arithmetic , 2010, 1004.2623.

[27]  Fedor Petrov,et al.  Uncountable graphs and invariant measures on the set of universal countable graphs , 2008, Random Struct. Algorithms.

[28]  P. Moree,et al.  Dynamical numbers : interplay between dynamical systems and number theory : a special program, May 1-July 31, 2009 : international conference, July 20-24, 2009, Max Planck Institute for Mathematics, Bonn, Germany , 2010 .

[29]  Анатолий Моисеевич Вершик,et al.  Описание Характеров И Фактор-представлений Бесконечной Симметрической Инверсной Полугруппы * , 2010 .

[30]  A. Vershik.  Dynamics of metrics in measure spaces and their asymptotic invariants , 2009 .

[31]  A.Vershik.  Dynamics of metrics in measure spaces and their asymptotic invariants , 2009, 0912.2123.

[32]  A. Vershik,et al.  Rational approximation of the maximal commutative subgroups of GL(n,R) , 2009 .

[33]  M. Graev,et al.  Integral models of representations of the current groups of simple Lie groups , 2009 .

[34]  I. Gelfand.  Israel Moiseevich Gelfand , Part , 2009 .

[35]  Boundaries of braid groups and the Markov-Ivanovsky normal form , 2008 .

[36]  B. Solomyak,et al.  The adic realization of the Morse transformation and the extension of its action to the solenoid , 2008, 0811.3324.

[37]  M. Graev,et al.  Integral models of unitary representations of current groups with values in semidirect products , 2008, 0809.1387.

[38]  A.Vershik,et al.  The Word and Geodesic Problems in Free Solvable Groups , 2008, 0807.1032.

[39]  A. Vershik,et al.  Invariant Measures for the Continual Cartan Subgroup , 2009 .

[40]  The behavior of the Laplace transform of the invariant measure on the hypersphere of high dimension , 2008 .

[41]  Anatoly M. Vershik,et al.  Globalization of the partial isometries of metric spaces and local approximation of the group of isometries of the Urysohn space , 2008, 0803.4300.

[42]  K. Schmidt,et al.  Algebraic polymorphisms , 2007, Ergodic Theory and Dynamical Systems.

[43]  A. Vershik,et al.  Integral models of representations of current groups , 2008 .

[44]  A New Approach to the Representation Theory of the Symmetric Groups, IV. Z2-graded groups and algebras; projective representations of the group Sn , 2008 .

[45]  A. Vershik.  L.V.Kantorovich and Linear Programming , 2007, 0707.0491.

[46]  A.Vershik,et al.  Four Drafts of The Representation Theory of the Group of Infinite Matrices over Finite Fields , 2007, 0705.3605.

[47]  A. Vershik.  Does there exist a lebesgue measure in the infinite-dimensional space? , 2007, math-ph/0703033.

[48]  The local stationary presentation of the alternating groups and the normal form , 2007 .

[49]  A. Vershik.  On F. A. Berezin and his work on representations of current groups , 2007 .

[50]  A. Vershik,et al.  Compactness of the congruence group of measurable functions in several variables , 2007 .

[51]  A. Vershik.  Krein duality, positive 2-algebras, and dilation of comultiplications , 2007 .

[52]  Espaces métriques linéairement rigides , 2007 .

[53]  A. Vershik,et al.  Induced representations of the infinite symmetric group and their spectral theory , 2007 .

[54]  G. Felder.  The work of Andrei Okounkov , 2006, math/0609847.

[55]  Анатолий Моисеевич Вершик,et al.  ИНДУЦИРОВАННЫЕ ПРЕДСТАВЛЕНИЯ БЕСКОНЕЧНОЙ СИММЕТРИЧЕСКОЙ ГРУППЫ И ИХ СПЕКТРАЛЬНАЯ ТЕОРИЯ , 2007 .

[56]  A. Vershik,et al.  Linearly rigid metric spaces and the embedding problem , 2006, math/0611049.

[57]  M. Graev,et al.  Structure of the complementary series and special representations of the groups O(n,1) and U(n,1) , 2006 .

[58]  On the Fourier transform on the infinite symmetric group , 2006 .

[59]  N. V. Tsilevich,et al.  On different models of representations of the infinite symmetric group , 2006, Adv. Appl. Math..

[60]  A. Vershik.  A new approach to the representation theory of the symmetric groups, III: Induced representations and the Frobenius--Young correspondence , 2006 .

[61]  A. Vershik,et al.  Traces on infinite-dimensional brauer algebras , 2006 .

[62]  A. Vershik,et al.  Twisted Burnside theorem for type II_1 groups: an example , 2006, math/0606161.

[63]  N. Tsilevich,et al.  Induced Representations of Infinite Symmetric Group , 2006, math/0605166.

[64]  A. Vershik,et al.  Fluctuations of the Maximal Particle Energy of the Quantum Ideal Gas and Random Partitions , 2006 .

[65]  P. J. Cameron,et al.  Some isometry groups of the Urysohn space , 2004, Ann. Pure Appl. Log..

[66]  A. Vershik,et al.  ${{C}^{*}}$ -Algebras of Irreversible Dynamical Systems , 2002, Canadian Journal of Mathematics.

[67]  Anatoly M. Vershik.  Universality and Randomness for the Graphs and Metric Spaces , 2006 .

[68]  Анатолий Моисеевич Вершик,et al.  Структура дополнительных серий и особых представлений групп $O(n,1)$ и $U(n,1)$@@@Structure of the complementary series and special representations of the groups $O(n,1)$ and $U(n,1)$ , 2006 .

[69]  I. Singer,et al.  The unity of mathematics : in honor of the ninetieth birthday of I.M. Gelfand , 2006 .

[70]  A. Vershik.  Gelfand-Tsetlin Algebras, Expectations, Inverse Limits, Fourier Analysis , 2006 .

[71]  A. Vershik,et al.  Markov Measures on Young Tableaux and Induced Representations of the Infinite Symmetric Group , 2005 .

[72]  A. Vershik.  Polymorphisms, Markov processes, quasi-similarity , 2005 .

[73]  A. Vershik.  Towards the definition of metric hyperbolicity , 2005 .

[74]  A Commutative Model of a Representation of the Group O(n, 1)X and a Generalized Lebesgue Measure in the Space of Distributions , 2005 .

[75]  A.M.Vershik,et al.  The Basic Representation of the Current Group O(n,1)^X in the L^2 space over the generalized Lebesgue Measure , 2005, math/0503404.

[76]  A. Vershik,et al.  Institute for Mathematical Physics the Basic Representation of the Current Group O(n,1) Space over the Generalized Lebesgue Measure , 2022 .

[77]  A. Vershik,et al.  A New Approach to the Representation Theory of the Symmetric Groups. II , 2005 .

[78]  Anatoly M. Vershik,et al.  Kantorovich Metric: Initial History and Little-Known Applications , 2005 .

[79]  Anatoly M. Vershik,et al.  Geometrical Approach to the Free Solvable Groups , 2005, Int. J. Algebra Comput..

[80]  A. Vershik.  Science and Totalitarianism , 2005 .

[81]  A. Vershik.  Polymorphisms, Markov Processes, and Quasi-similarity , 2005 .

[82]  P.J.Cameron,et al.  Some isometry groups of Urysohn space , 2004 .

[83]  M. Yor,et al.  On the Markov–Krein Identity and Quasi-Invariance of the Gamma Process , 2004 .

[84]  A. Vershik,et al.  Random Metric Spaces and Universality , 2004 .

[85]  A. Vershik.  Random and Universal Metric Spaces , 2004 .

[86]  A. Vershik.  Kolmogorov's Example (A Survey of Actions of Infinite-Dimensional Groups with an Invariant Probability Measure) , 2004 .

[87]  A. Vershik,et al.  Harmonic analysis on the infinite symmetric group , 2003, math/0312270.

[88]  A. Vershik,et al.  Asymptotics of the Uniform Measures on Simplices and Random Compositions and Partitions , 2003 .

[89]  A. Vershik,et al.  Canonical Semigroups of States and Cocycles for the Group of Automorphisms of a Homogeneous Tree , 2003 .

[90]  A. Vershik,et al.  Fock factorizations, and decompositions of the L2 spaces over general Lévy processes , 2003, math/0304282.

[91]  A. Vershik.  Numerical Invariants of Exponential Groups and Locally Free Approximation of the Braid Group , 2003 .

[92]  Strange factor representations of type II_1 and pairs of dual dynamical systems , 2003 .

[93]  A. Vershik,et al.  Frobenius-Schur Functions , 2001, math/0110077.

[94]  A M Vershik,et al.  Fock factorizations, and decompositions of the L2 spaces over general Lévy processes , 2003 .

[95]  A. Vershik.  Two lectures on the asymptotic representation theory and statistics of Young diagrams , 2003 .

[96]  A. Vershik,et al.  Classification of Measurable Functions of Several Variables and Invariantly Distributed Random Matrices , 2002 .

[97]  A. Vershik.  Graded Lie Algebras and Dynamical Systems , 2002 .

[98]  A. Vershik.  Numerical Characteristics of Groups and Corresponding Relations , 2001 .

[99]  M. Yor,et al.  An Infinite-Dimensional Analogue of the Lebesgue Measure and Distinguished Properties of the Gamma Process , 2001 .

[100]  A. Vershik,et al.  Topology, Ergodic Theory, Real Algebraic Geometry: Rokhlin’s Memorial , 2001 .

[101]  A. Vershik.  Randomization of Algebra and Algebraization of Probability , 2001 .

[102]  Anatoly M. Vershik,et al.  The Limit Shape and Fluctuations of Random Partitions of Naturals with Fixed Number of Summands , 2001 .

[103]  Anatoly M. Vershik,et al.  Dynamic theory of growth in groups: Entropy, boundaries, examples , 2000 .

[104]  Anatoly M. Vershik,et al.  Statistical Properties of Locally Free Groups¶with Applications to Braid Groups¶and Growth of Random Heaps , 2000 .

[105]  A. Vershik,et al.  Graded Lie algebras whose Cartan subalgebra is the algebra of polynomials in one variable , 2000 .

[106]  A. Vershik,et al.  A Local Limit Theorem for Random Strict Partitions , 2000 .

[107]  Ofer Zeitouni,et al.  Large deviations in the geometry of convex lattice polygons , 1999 .

[108]  A. Vershik,et al.  C-algebras and algebras in plancherel duality , 1999 .

[109]  N. V. Tsilevich,et al.  Quasi-invariance of the gamma process and multiplicative properties of the Poisson-Dirichlet measures , 1999 .

[110]  Юлия Вацлавовна Нагребецкая.  О границе разрешимости бесконечной симметрической группы , 1999 .

[111]  Anatolii Moiseevich Vershik,et al.  The universal Urysohn space, Gromov metric triples and random metrics on the natural numbers , 1998 .

[112]  N. Sidorov,et al.  Bijective coding of automorphisms of the torus and binary quadratic forms , 1998 .

[113]  Nikita Sidorov,et al.  Ergodic properties of the Erdös measure, the entropy of the goldenshift, and related problems , 1998 .

[114]  A. Vershik,et al.  Bijective Arithmetic Codings of Hyperbolic Automorphisms of the 2-Torus, and Binary Quadratic Forms , 1998, math/9810085.

[115]  Richard Kenyon,et al.  Arithmetic construction of sofic partitions of hyperbolic toral automorphisms , 1998, Ergodic Theory and Dynamical Systems.

[116]  Boris Tsirelson,et al.  Examples of Nonlinear Continuous Tensor Products of Measure Spaces and Non-Fock Factorizations , 1998 .

[117]  Славский Виктор Владимирович,et al.  Оценки кривизн левоинвариантных римановых метрик трёхмерных компактных простых групп Ли , 1998 .

[118]  Yu. A. Davydov,et al.  Réarrangements convexes des marches aléatoires , 1998 .

[119]  Amir Dembo,et al.  Large deviations for integer partitions , 1998 .

[120]  Sergei Evdokimov,et al.  Algebras in plancherel duality and algebraic combinatorics , 1997 .

[121]  Amitai Regev,et al.  Asymptotics of Young Diagrams and Hook Numbers , 1997, Electron. J. Comb..

[122]  A. Vershik.  Limit distribution of the energy of a quantum ideal gas from the viewpoint of the theory of partitions of natural numbers , 1997 .

[123]  A. Vershik,et al.  A new approach to representation theory of symmetric groups , 1996 .

[124]  V. Buchstaber,et al.  Combinatorial algebras and multivalued involutive groups , 1996 .

[125]  A. Vershik,et al.  Random walks on braid groups: Brownian bridges, complexity and statistics , 1996 .

[126]  COMMUNICATIONS OF THE MOSCOW MATHEMATICAL SOCIETY: An inductive method of expounding the representation theory of symmetric groups , 1996 .

[127]  A. Vershik,et al.  Statistical mechanics of combinatorial partitions, and their limit shapes , 1996 .

[128]  G. Olshanski,et al.  ERGODIC UNITARILY INVARIANT MEASURES ON THE SPACE OF INFINITE HERMITIAN MATRICES , 1996, math/9601215.

[129]  A M Vershik,et al.  An inductive method of expounding the representation theory of symmetric groups , 1996 .

[130]  Сергей Алексеевич Евдокимов,et al.  Комбинаторные алгебры и многозначные инволютивные группы , 1996 .

[131]  Canonical States on the Group of Automorphisms of a Homogeneous Tree , 1996 .

[132]  A. Vershik.  Asymptotic Combinatorics and Algebraic Analysis , 1995 .

[133]  A. Vershik,et al.  Random walks on multiconnected manifolds and conformal field theory , 1994 .

[134]  A. Vershik,et al.  The limit shape of convex lattice polygons and related topics , 1994 .

[135]  L. Bokut,et al.  Opinion: To guard the future of soviet mathematics , 1992 .

[136]  Г. Джеймс.  Теория представлений симметрических групп , 1982 .

[137]  Sergeĭ Mikhaĭlovich Nikolʹskiĭ.  Приближение функций многих переменных и теоремы вложения , 1969 .