Tribute to Anatoly Moiseevich Vershik

The Russian mathematician Anatoly Moiseevich Vershik passed away on February 14, 2024. Anatoly Vershik was Head of the Laboratory of Representation Theory and Dynamical Systems at the St. Petersburg Department of the Steklov Institute of Mathematics and Chair of Mathematical Analysis at the Department of Mathematics and Mechanics at St. Petersburg State University. From 1998 to 2008 he was President of the St. Petersburg Mathematical Society. Anatoly Vershik made significant contributions to various areas of mathematics, including representation theory, dynamical systems, ergodic theory, asymptotic combinatorics, operator algebras, and measure theory. His students include Alexander Barvinok, Anna Erschler and Sergey Fomin. Anatoly Vershik was a regular visitor at IHES in the 1990s, with his last visit to the Institute in 2012. IHES mourns the loss of a truly remarkable mathematician and respected leader in the Russian mathematical community. Misha Gromov, Professor Emeritus at IHES, wrote a few words to pay him tribute.

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“Anatoly Vershik died on February 14, 2024. Throughout his life, he consistently generated new mathematical ideas and generously shared them with his friends and students. Over time, several of these ideas—some developed by Vershik himself, and others by his students and colleagues—have evolved into established mathematical theories.

(I recall that—around 1960 or 1961—I overheard Vladimir Abramovich Rokhlin expressing great expectations regarding the possible outcomes of Vershik’s original point of view on mathematics.)

The last time I spoke to Vershik was over the phone on December 28, 2023. During our conversation, we discussed his new preprint on the classification of measurable functions in k variables, which he had sent me a couple of weeks earlier. In it, he elaborated on his old idea of using ergodic theory in this context.

Interestingly enough, one of the first problems he shared with me (in 1964 or 1965) was the description of the combinatorial structure of convex hulls of orbits of products of permutation groups $\prod_{n_1} \times \dots \times \prod_{n_k}$, which act on the tensor products $\mathbb{R}_{n_1} \hspace{-.25em} \otimes \dots \otimes \mathbb{R}_{n_k}$.

Around the same time, I also learned about Vershik’s (isoperimetric) Følner function. The introduction of this function was motivated by Følner’s criterion for amenability. The study of Vershik’s Følner function has since become an active branch of research in infinite group theory.

Over the years, I have often spoken with Vershik, and what I remember most are his elegant ideas and probabilistic descriptions of the universal Urysohn spaces, as well as the classification of metric measure spaces. Vershik was interested in everything in life; he welcomed truth and courageously fought against wrongs. At 90 years of age, he died too young.

His friend, Misha Gromov.”

Bibliography

A.I. Barvinok and A.M. Vershik. Convex hulls of orbits of representations of finite groups and combinatorial optimization. Functional Analysis Its Applications 22, 224–225 (1988)

T. Ceccherini-Silberstein and M. D’Adderio. Topics in groups and geometry. Growth, amenability, and random walks. Springer (2021)

A. Erschler. Poisson-Furstenberg boundaries, large-scale geometry and growth of groups. Proceedings of the International Congress of Mathematicians Hyderabad, India (2010)

M. Gromov. Entropy and Isoperimetry for Linear and non-Linear Group Actions. Groups, Geometry, and Dynamics 2, no. 4, 499-593 (2008)

A.M. Vershik. Countable groups that are close to finite ones, Appendix in F.P. Greenleaf “Invariant Means on Topological Groups and their Applications”, Moscow, Mir, 1973 (in Russian), a revised English translation “Amenability and approximation of infinite groups” was published in Selecta Mathematica Sovietica 2, no. 4, 311–330 (1982)

A.M. Vershik. Classification of Measurable Functions of Several Variables and Invariantly Distributed Random Matrices. Functional Analysis and Its Applications 36, 93–105 (2002)

A.M. Vershik. Random metric spaces and universality. Russian Mathematical Surveys 59 259 (2004)

A.M. Vershik. Classification of measurable functions of several variables and matrix distributions. arXiv:2311.01281 (2023)

A.M. Vershik, G.A. Veprev and P.B. Zatitskii. Dynamics of metrics in measure spaces and scaling entropy. arXiv:2311.14550 (2023)

T. Zheng. Asymptotic behaviors of random walks on countable groups. Proceedings of the International Congress of Mathematicians (2022)