Material to Scientific Biography

The following material is presented:

1. 1993 Steele Prizes, Career Award, Notices of the American Mathematical Society, 40, 975 - 977 ,

2. E. B. Dynkin, by Mark I. Freidlin, The Dynkin Festschrift, Markov Processes and their Applications , Mark I. Freidlin, Editor; Progr. Probab., 34; Birkhäuser, Boston 1994, ix-xvi ,

3. Evgenii Borisovich Dynkin (on his seventieth birthday), by R.L. Dobrushin, A.L.Onishchik, V.A.Uspenskii, N.D.Vvedenskaya, Russian Mathematical Surveys, 49:4 (1994), 183-191; English Translation of Uspekhi Mat. Nauk 49:4 (1994), 183-188

4. E.B. Dynkin's Seminar on Lie Groups and Lie Algebras, by S.G. Gindikin, In: Lie Groups and Lie Algebras: E. B. Dynkin Seminar, S. G. Gindikin and E. B. Vinberg, Editors, American Mathematical Society Translations, Series 2, Volume 169, American Mathematical Society, 1995, Providence R. I., ix-xi


See, in addition:

Audio and video recordings of Dynkin, stored in the Dynkin Collection of Mathematics Interviews

On the Work of E.B. Dynkin in the Theory of Lie Groups, by F.I.Karpelevich, A.L.Onishchik, and E.B. Vinberg in: Lie Groups and Lie Algebras: E. B. Dynkin Seminar, S. G. Gindikin and E. B. Vinberg, Editors, American Mathematical Society Translations, Series 2, Volume 169, American Mathematical Society, 1995, Providence R. I. [Reprinted in Selected Papers of E. B. Dynkin with commentary, A. A. Yushkevich, G. M. Seitz, A. L. Onishchik, Editors; American Math. Society/International Press (2000)]

On the 70th birthday of E. B. Dynkin, by N. V. Krylov, S. E. Kuznetsov, A. V. Skorokhod, A. A. Yushkevich, Teor. Veroyatnost. i Primen., 39 (1994), 796-798
{English transl. Theory Probab. Appl., 39 (1994), 654-656}

Comments on Dynkin's work by G. M. Seitz, B. Kostant, D. Vogan, A.L. Onishchik, A. Gabrielov, K. Gottfried, Y. Ne'eman, A. A. Yushkevich, I. V. Evstigneev, S. E. Kuznetsov, P.-A. Meyer, M. F. Olshanetsky and J.-F. Le Gall in the volume  Selected Papers of E. B. Dynkin with commentary,
A. A. Yushkevich, G. M. Seitz, A. L. Onishchik, Editors; American Math. Society/International Press (2000)


1. 1993 Steele Prizes, Career Award

Citation

Eugene B. Dynkin has made major contributions to the theory of Lie algebras and to probability theory. Dynkin's most famous contribution to the theory of Lie algebras was his use of the "Coxeter-Dynkin diagrams" to describe and classify the Cartan matrices of semisimple Lie algebras. This work was done while Dynkin was still a student at Moscow University.

Even though Dynkin has directed many Ph.D. theses in Lie group/algebra theory, most of his career has been devoted to probability theory. Dynkin has laid much of the foundations of the general theory of Markov processes as we know it today. His books ( Foundations of the Theory of Markov Processes, Moscow 1959, translation published by Pergamon 1960; and Markov Processes, vols. I and II, Moscow, 1963, translation published by Springer-Verlag, 1963) have had a tremendous influence. He formulated and proved the strong Markov property (in 1956, together with his student Yushkevich; this was done almost at the same time and independently of Hunt's introduction of the strong Markov property). Dynkin proved the measurability of certain hitting times (again almost at the same time as Hunt and independently of him). He developed the semigroup theory of Markov processes and characterized Markov processes by the generator of their semigroup. He also showed the usefulness of what is now known as "Dynkin's formula". This formula, which expresses expectations of functionals of the Markov process as an integral involving its generator, has become a standard and indispensable tool which is still used all the time. Dynkin further studied such topics as excessive functions, Martin boundary, additive functionals, entrance and exit laws, random time change, control theory, and mathematical economics.

Around 1980 Dynkin interpreted and vastly generalized an identity which had first come up in the context of quantum field theory. In his hands it became a remarkable relation between occupation times of a Markov process and a related Gaussian random field. This identity has led to many deep studies, by Dynkin himself as well as a host of others, of the properties of local times of Markov processes as well as to the detailed study of multiple points or self-intersections of Brownian motion.

In the last few years Dynkin has obtained exciting results in the theory of "superprocesses". This is a class of measure-valued Markov processes, which in many cases can be constructed as a suitable scaled limit of branching processes. These processes can be used to give probabilistic solutions to certain nonlinear PDE's in a way which is analogous to the classical solution of the Dirichlet problem by means of Brownian motion. Dynkin has used this to relate analytic properties of solutions of such PDE's (e.g., removability of singularities) to probabilistic properties of superprocesses.

Even though Dynkin has dealt with quite concrete probability problems, one of his strengths is his ability to build general theories and an apparatus to answer broad questions (e.g., characterize a certain natural class of additive functionals of a Markov process, or find all superprocesses with a branching property).

The list of Dynkin's Ph.D. students in Moscow is a "Who's Who" in Russian probability theory. In Moscow he has been extremely active in a special high school for gifted students in mathematics. He has also had many Ph.D. students since he immigrated to the U.S. He has been invited several times to speak at the International Congress of Mathematicians, and he is a member of the National Academy of Sciences.

This career award is in recognition of Dynkin's foundational contributions to two areas of mathematics over a long period and his production of outstanding research students in both countries to whose mathematical life he contributed so richly.


Biographical Note

Dynkin was born in Leningrad on May 11, 1924. After receiving his Ph.D. from Moscow University in 1948 he ultimately became professor at the same institution and held that position until 1968. He was a senior research fellow at the Academy of Sciences, Moscow (1968-1976). Since 1977 he has been a professor of mathematics at Cornell University.

He received the Prize of the Moscow Mathematical Society (1951) and also served on the council and as vice-president of that organization. He is a member of the National Academy of Sciences; the American Academy of Arts and Sciences; the Institute of Mathematical Statistics; and the Bernoulli Society for Mathematical Statistics and Probability. His research interests include Lie theory and probability theory, stochastic processes, optimal control, probabilistic models of economic growth and equilibrium, and sufficient statistics.

Response

I feel deeply honored to have been awarded a Steele Prize by the American Mathematical Society.

I came to the United States from the Soviet Union in 1977. Since then I have taught at Cornell University, a great center in probability theory. I found here kind and friendly colleagues; gorgeous scenery of forests, lakes, and waterfalls; and a few bright graduate students with whom I have started a seminar of the Moscow type. The most exciting was a new feeling of freedom and independence of big and little bosses - something which I never enjoyed in my previous life.

I had a great advantage to spend my formative years in an extremely stimulating atmosphere of the Moscow mathematical school; to be a pupil of such great mathematicians as A. N. Kolmogorov, I. M. Gelfand, I. G. Petrovskii; and later to enjoy interaction with brilliant young mathematicians who started their research at my seminars, among them are (in chronological order) R. L. Dobrushin, F. I. Karpelevich, A. V. Skorokhod, F. A. Berezin, A. A. Yushkevich, I. V. Girsanov, Ya. G. Sinai, A. L. Onishchik, R. Z. Khasminskii, V. A. Volkonskii, M. I. Freidlin, M. G. Shur, A. A. Kirillov, A. D. Wentzell, E. B. Vinberg, V. N. Tutubalin, N. V. Krylov, S. A. Molchanov, M. B. Malyutov, G. A. Margulis, S. E. Kuznetsov, I. V. Evstigneev, M. I. Taksar. On the other hand, the life was hard under the oppression of a totalitarian regime. I was eleven when my family was exiled from Leningrad to Kasakhstan and I was thirteen when my father, one of millions of Stalin's victims, disappeared in the Gulag. It was almost a miracle that I was admitted (at the age of sixteen) to Moscow University. Every step in my professional career was difficult because the fate of my father, in combination with my Jewish origin, made me permanently undesirable for the party authorities at the university. Only special efforts by A. N. Kolmogorov, who put, more than once, his influence at stake, made it possible for me to progress through the graduate school to a teaching position at Moscow University.

My poor vision made me unfit for the draft, so during the war I continued my undergraduate study. The motherly attitude of S. A. Yanovskaya (a professor at Moscow University) softened the hardship of the most difficult years and allowed me to concentrate on mathematics. (Her support and encouragement are gratefully remembered by a number of other mathematicians and logicians.) I worked at Gelfand's seminar on Lie groups and at Kolmogorov's seminar on Markov chains. Both were important for my development as a research mathematician.

Gelfand requested that I review the H. Weyl - Van der Waerden papers on semisimple Lie groups. I found them very difficult to read, and I tried to find my own ways. It came to my mind that there is a natural way to select a set of generators for a semisimple Lie algebra by using simple roots (i.e., roots which cannot be represented as a sum of two positive roots). Since the angle between any two simple roots can be equal only to pi/2, 2pi/3, 3pi/4, 5pi/6, a system of simple roots can be represented by a simple diagram. An article was submitted to Matematicheskii Sbornik in October 1944. Only a few years later, when recent literature from the West reached Moscow, I discovered that similar diagrams have been used by Coxeter for describing crystallographic groups.

Lie algebras remained my main field for about ten years. I used simple roots and the corresponding diagrams to investigate automorphisms and semisimple subalgebras of Lie algebras. After coming to the West I learned that these results have been used by a number of physicists to deal with elementary particles. (I was flattered when Yuval Ne'eman told me that his work on this subject was based on my dissertation, which he had read in one of the London libraries.)

A few times I was lucky to find a new approach which simplified an important theory. One of them is related to the celebrated Campbell - Hausdorff theorem claiming that the formal series log(exey) can be expressed in terms of commutators. In 1947 I found a simple explicit expression: it is sufficient to replace all multiplications by commutators and then to divide each monomial by its degree. My debut in probability theory was made about the same time as my debut in algebra. In 1945 I solved a problem on Markov chains posed by Kolmogorov. In 1948 I became an assistant professor at Kolmogorov's Probability Chair, and I continued to work on probability and statistics parallel to algebra (the results on exponential families and sufficient statistics are probably the best known). Beginning in the middle of the 1950s, I switched almost completely to work on stochastic calculus, especially on Markov processes. Some detail about this part of my research can be found in the citation.

Contacts with my friends and colleagues in the Soviet Union were severed when I moved to the United States. Now, after the end of the Cold War, they flourish again. We started a joint project with S. E. Kuznetsov and A. V. Skorokhod on the structure of branching measure - valued processes. During my recent visits to Moscow we had two very emotional reunions with a large group of scientist who were my pupils in the 1960s in a special Moscow high school for mathematically gifted students.

Reprinted from Notices of the American Mathematical Society, 40, 975 - 977; by permission of the American Mathematical Society. Copyright 1994 by the American Mathematical Society.

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2. Eugene B. Dynkin

[Numbers in brackets [\_] refer to the List of Dynkin's Research Papers ]

Eugene B. Dynkin was born in Leningrad on May 11, 1924. Russia had only recently emerged from the revolution and a devastating civil war. But the bloody repressions of the 1930s were already looming. Dynkin was eleven when his family was banished to a remote part of Kasakhstan. Two years later, his father disappeared into the Gulag - one of the innocent millions to fall victim to Stalinist terror. They have been oficially denounced as "people's enemies" and the life of their children was rather bleak. Eugene at 10 with his father are pictured below in Leningrad, 1934.

By Dynkin's own accounting, it was "almost a miracle" that he was admitted to Moscow State University to study mathematics in 1940. His poor eyesight exempted him from the draft, and Dynkin continued his undergraduate work during the war. The motherly attentions of Professor S. A. Yanovskaya helped ease the hardships of that time, allowing him to concentrate on school. The great mathematicians A. N. Kolmogorov and I. M. Gelfand were his teachers. He also worked in a seminar of I. G. Petrovsky. Dynkin thrived despite adversity, obtaining his first original results in mathematics while still an undergraduate. In 1948 he received his Ph.D. degree and became an Assistant Professor of Kolmogorov's Probability Chair.

It would be difficult to overstate the stimulating atmosphere of the Mathematics Department of Moscow University during these years. Since the 1930s, it was one of the great world centers in mathematics. At the end of World War II (because of the role of nuclear weapons) physics and mathematics have gained a special status in the USSR. The pressure of the totalitarian ideology in these fields was somewhat weaker than in other sciences. The professions of mathematician and physicist became highly regarded, and many promising young people applied for admission to the mathematics departments. At Moscow University, the competition was especially high (and not always fair). The leading mathematicians, such as Kolmogorov and Petrovskii (who became the rector of Moscow University in 1951) have done their best to help the brightest young people.

In spite of his outstanding achievements in research and his clear ability as a teacher, every step in Dynkin's career was difficult: the enemy status of his late father, coupled with his Jewish origin, made him "permanently undesirable" to the Communist Party operatives at the university. Only Kolmogorov's influence and reputation made possible for Dynkin to get a teaching position at Moscow University. However, all Kolmogorov's efforts to promote him to Professorship failed until the political atmosphere has improved after Stalin's death in 1953. (Dynkin became Professor in 1954.)

From the very beginning, Dynkin started to work at two seminars, Gelfand's on the Lie groups and Kolmogorov's on Markov chains. He discovered new results in both directions, however, most of his research in 1944-1954 was concentrated on Lie groups.

Dynkin's most famous contribution to the theory of Lie groups is, of course, the introduction and use of the notions of simple roots and the corresponding diagrams. He invented these notions when preparing, at Gelfand's seminar, an exposition of the H. Weyl - Van der Waerden papers on semi-simple Lie groups. The corresponding article [2] was submitted to Matematicheskii Sbornik in October 1944. Only a few years later, when the war was over and recent literature from the West reached Moscow, did Dynkin learn that similar diagrams had been used by Coxeter for describing crystallographic groups.

Dynkin's approach simplifies enormously the classification of semi-simple groups. It permits the description of the automorphisms and semi-simple subalgebras of Lie algebras [5,13,14,16,19-21], ideas that allow Dynkin to give a complete solution to the classical problem of geometry: classification of the primitive homogeneous spaces with semi-simple groups of motions. These publications became a handbook for a generation of mathematicians and physicists. Dynkin's diagrams arise also in representation theory and are useful in a number of physical problems concerning elementary particles.

Another of Dynkin's discoveries in this area was the formula for log(exey) for non-commuting x and y in terms of commutators [6],[9]. The existence of such an expression was known, but in 1947 Dynkin found a very simple and explicit form: It is sufficient to replace all multiplications in the formal expansion by commutators and then to divide each monomial by its degree. This result allowed the simplification of the construction of a Lie group from a Lie algebra and the generalization of the construction to arbitrary normed fields [4,7,10].

Dynkin also obtained a number of deep results in the topological theory of Lie groups [24,25,27-30,32 ].

Due to Dynkin, a whole school in Lie groups arose in Moscow in the 1950s. It is enough to mention F. I. Karpelevich, F. A. Berezin, A. L. Onishchik, A. A. Kirillov, and E. B. Vinberg, who obtained their first results on Lie groups in Dynkin's seminar. At a later stage, the work of the seminar has been greatly enriched by the active participation of I. I. Pyatetskii-Shapiro.

As already noted, Dynkin started to work in probability as far back as his undergraduate studies. In fact, his first published paper deals with a problem arising in Markov chains theory. The most significant among his earliest probabilistic results concern sufficient statistics. In [15] and [17], Dynkin described all families of one-dimensional probability distributions admitting non-trivial sufficient statistics. These papers have considerably influenced the subsequent research in this field. But Dynkin's most famous results in probability concern the theory of Markov processes.

Following Kolmogorov, Feller, Doob, and Ito, Dynkin opened a new chapter in the theory of Markov processes. He created the fundamental concept of a Markov process as a family of measures corresponding to various initial times and states and he defined time homogeneous processes in terms of the shift operators thetat. In a joint paper with his student A. Yushkevich [40], a general definition of the strong Markov property was introduced and it was proved that all right continuous Feller processes have this property. (About the same time, the strong Markov property of the Brownian motion was proved by G. A. Hunt. For Markov processes in a countable space, it has been obtained earlier by J. L. Doob.) Dynkin established tests for continuity and for the existence of right and left limits of paths in terms of the transition function [23]. [One year later similar tests have been found independently by J. R. Kenney.] He introduced an important class of standard processes (wider than the class of Hunt processes). Dynkin's characteristic operator [36] describes a Markov process by its local behaviour in space rather than in time. The relation between the characteristic and the infinitesimal operators was deduced from an identity which is commonly called the "Dynkin formula" in the literature. At the same time, Dynkin started to investigate additive functionals of Markov processes and related transformations of processes [51-53], [56], [58]. His own results and the results obtained by his school in the late 50s and the early 60s have been summarized in two monographs, Foundations of the Theory of Markov Processes (Moscow, 1959), and Markov Processes (Moscow, 1963). The books were translated into many languages and had a tremendous influence. Here Dynkin laid much of the foundations of the general theory of Markov processes as we know it today. Markov theory became not only a consumer of analytic methods but a powerful tool in classical analysis, especially in such areas as potential theory and the theory of partial differential equations. Dynkin was one of the first who recognized the great potential of Ito's differential equations as a tool in theory of partial differential equations. Participants of his seminar started to work in this direction as early as 1959.

In 1959 Dynkin married and Irene G. Dynkin became his devoted and wise friend for life.

Dynkin has made fundamental contributions to the boundary theory of Markov processes - a beautiful branch of mathematics lying on the border between probability theory, analysis and geometry. The foundations were laid by Martin, Doob and Hunt. Dynkin started from concrete problems: the evaluation of the Martin boundary for symmetric spaces of negative curvature [57], [59], [67], for random walks on non-Abelian groups [55] and for the Brownian motion subject to the boundary condition with oblique derivative [62], [63]. In connection with the first problem, he conjectured that the Martin boundary of any n-dimensional complete Riemannian manifold of negative curvature has dimension n-1. [In 1983, Anderson and Schoen proved that this is true if the curvature is bounded from above and from below by two strictly negative constants.] All possible boundary conditions for several types of Markov processes have been investigated in [66],[68],[70],[71],[72]. In [79],[80],[81],[82], [83],[84], [86],[90], Martin theory was greatly simplified and extended to the most general Markov processes. The central role is played by a remarkable result on convex cones K in the space of probability measures. Namely, it was proved that, if K admits a sufficient statistic of a certain type, then every element of K can be decomposed, in a unique way, into extremal elements. This can be considered as a measure-theoretical version of the celebrated Choquet't theorem which is stated in topological terms. It can be applied not only to Martin theory, but also to ergodic theory (the representation of invariant measures through ergodic measures), to exchangeable random variables, to Gibbs systems and others. [The original idea has been stated in [86]; the most complete exposition is presented in [86].]

During the 1960s and earlier 1970s, Dynkin worked also in the following directions:

(1) stochastic parallel transport [73] (together with the pioneering Ito's work of 1962, this paper has influenced all subsequent work on stochastic calculus in manifolds);

(2) theory of duality based on construction of Markov processes with random birth and death times [90],[93],[95],[104],[107] [one of farther developments were "Kuznetsov's measures" widely used in the modern literature];

(3) the last exit decompostion [75],[76],[85] [investigated later also by Getoor and Sharpe, Maisonneuve and others];

(4) description of additive functionals by their characteristic (spectral) measures [96], [101], [105];

(5) optimal stopping of a Markov process based on a beautiful application of excessive functions [60], a game version of optimal stopping problem [78];

(6) optimal control under incomplete data [65]; stochastic concave dynamic programming [91] [the results have been covered in the monograph of E. B. Dynkin and A. A. Yushkevich, Controlled Markov Processes, Moscow 1975].

(7) economic growth and economic equilibrium under uncertainty [88], [94],[99],[100], [103] [the work was done when Dynkin worked with a group of young collaborators (I. V. Evstigneev, S. E. Kuznetsov, S. M. Natanzon, A. I. Ovseevich and M. I. Taksar) at the Central Institute for Economics and Mathematics, Academy of Sciences of the USSR].

At the end of 1976, Dynkin left the USSR. The decision to leave was very hard: pupils, friends, and youth were left behind. To apply for emigration was a great risk, especially for an outstanding scientist: many such applicants have been denied the exit visas, they have losed their jobs and lived for years as outcasts of the Soviet society. Dynkin took the risk because the life in the USSR became more and more unbearable, and the Dynkins' only daughter had already left for Israel.

In 1977, Dynkin became a Professor at Cornell University in Ithaca, New York, where Dynkin said he "found kind and friendly colleagues, gorgeous scenery of forests, lakes, and waterfalls .... The most exciting was a new feeling of freedom and independence of big and little bosses, something which I never enjoyed in my previous life."

These changes led to a new blossoming of Dynkin's activity in mathematics. Around 1980, Dynkin started a series of papers on random fields. Motivated by the work of Nelson on Euclidean quantum field theory, Dynkin associated, in [109], a Gaussian field phi with an arbitrary symmetric Markov process X, he expressed the correlation function of phi in terms of additive functionals of X and he used this expression to prove that phi has a Markov property on a pair of sets A, B if and only if X can not pass from A to B without crossing the intersection of A and B. In [116],[119] he established an "isomorpism theorem" - a remarkable identity which relates the field phi and the occupations times for X. [A very special case of this identity had first occurred in the context of quantum field theory.] "Dynkin's isomorphism theorem" has led to many deep studies of the properties of local times by Dynkin himself as well as by a host of others. In [120], [123],[126],[130] and [131] Dynkin introduced and investigated random fields associated with multiple points of the Brownian motion. His starting points were polynomials of the occupation field which correspond to Wick's powers of the Gaussian field phi . The "self-intersection gauges" introduced in [130] and [131] have been used by Le Gall to describe an asymptotic expansion of the Lebesgue measure of a thin Brownian sausage.

Dynkin continued his work on additive functionals [111],[124], on excessive functions and measures [108] and he has developed a probabilistic theory of Dirichlet spaces [113],[114].

In the last few years Dynkin has obtained exciting results on branching measure-valued processes. In [137],[139],[144] he constructed a wide class of such processes by a passage to the limit from systems of small particles which move independently according to the law of a Markov process; and produce, at their death time, a random offspring. Dynkin suggested to call them superprocesses. In [138],[141] and [144] he established connections between superdiffusions (i.e., superprocesses corresponding to diffusions and a certain class of non-linear partial differential equations similar to the well-known connections between diffusions and linear PDE's. These connections have been used by him to investigate the path properties of superdiffusions. On the other hand, they lead to a better understanding of such phenomena as blowing up of solutions of PDE at the boundary of a domain, character of isolated singularities etc. In a recent joint paper [145] a result on the structure of general branching measure-valued processes was obtained. Namely, it was proved that, under usual regularuty assumptions, every critical branching measure-valued process with finite second moments is a superprocess.

E. B. Dynkin's merits as a teacher should especially be mentioned. An excellent lecturer, his talks are always well organized and a bit theatrical. But the most interesting thing was his seminars. As a rule, he started a seminar for freshman undergraduate students and brought them up to the graduate school level and further. In the seminars for beginners, many topics were considered, from differential geometry, topology, and Lie groups to PDEs, stochastic processes, and statistics. Dynkin carefully chose the material to avoid nonessential difficulties. The atmosphere of the seminar was informal and highly competitive. Dynkin frequently stopped the speaker (often he himself was the speaker) and asked a participant to continue or to solve a more or less simple problem connected with the talk. Many unsolved problems were suggested, and soon first new results were obtained by the participants. One had to be young, persistent and, maybe, a bit ambitious to survive such an atmosphere. Perhaps the specific atmosphere of the Mathematics Department of Moscow University in the 1950s and 1960s contributed to the success of those seminars.

Among the participants of Dynkin's seminars in probability in Moscow who obtained their first results in these seminars were (in chronological order): R. L. Dobrushin, A. V. Skorohod, A. A. Yushkevich, R. Z. Khasminskii, V. A. Volkonskii, I. V. Girsanov, Ya. G. Sinai, A. D. Wentzell, M. I. Freidlin, M. G. Shur , V. N. Tutubalin, K. Dambis, N. V. Krylov, M. B. Malyutov, S. A. Molchanov, G. A. Margulis, A. L. Rozental, S. E. Kuznetsov, I. V. Evstigneev, and M. I. Taksar. Most became Dynkin's graduate students. As it was written in the citation to E. B. Dynkin's Steele Prize Career Award from the American Mathematical Society, "The list of Dynkin's Ph.D. students in Moscow is a Who's Who in Russian probability theory." At Cornell Dynkin also started a seminar of the "Moscow type."

Teaching regular seminars and organizing special competitions - Mathematical Olympiads - for high school students was a well respected tradition at Moscow University. The selection process was also quite effective at identifying promising students; most of the winners of those olympiads became good mathematicians, and many of them became first class mathematicians. While still a graduate student, Dynkin started such a seminar for high school students. N. N. Chentsov, F. I. Karpelevich, V. A. Uspenskii, F.A. Berezin, R. A. Minlos, and A. A. Yushkevich were among participants of this seminar. Later Dynkin worked mostly with university students, but in the early 1960s he became involved in the work of a special school for mathematically gifted children, where he created an exciting atmosphere. Dozens of Dynkin's students from that school became professional mathematicians. He is also the author of several books for high school students, written with great skill and mastery.

Dynkin's contacts with pupils and colleagues in the Soviet Union were severed when he moved to the United States. But through all those years, Dynkin kept a deep concern in developments in the USSR and, especially, in the fate of Soviet mathematicians. Since the end of the Cold War, he has visited Russia several times and started a joint project with his former students and colleagues. Many of the best of Dynkin's pupils from Moscow are now university professors in the USA.

The mathematical community of the United States and of the world greatly appreciates Dynkin's contributions. He is a member of the National Academy of Sciences of the U.S.A. and a Fellow of the American Academy of Arts and Sciences. He has given many invited talks at the International Congresses of Mathematicians. Recently, he received the prestigious Steele Prize Career Award. As is written in the citation, "This career award is in recognition of Dynkin's foundational contributions to two areas of mathematics over a long period and his production of outstanding research students in both countries to whose mathematical life he contributed so richly."

E. B. Dynkin continues to produce first-class new results and outstanding research students. He takes an active part in mathematical life all over the world.

We wish Eugene B. Dynkin many healthy active years and new achievements in mathematics.

By Mark I. Freidlin

Reprinted from "The Dynkin Festschrift: Markov Processes and their Applications", pages ix-xvi; by permission of Birkhäuser Boston. Copyright 1994 by Birkhäuser Boston.

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3. Evgenii Borisovich Dynkin (on his seventieth birthday)

May 1994 saw the birthday of the eminent mathematician Evgenii Borisovich Dynkin, a scholar whom algebraists recognize as a famous algebraist, and specialists in probability theory as a famous probabilist.

Dynkin was born in Leningrad on 11 May 1924. As an 11 -year old boy, he and his entire family were banished to Kazakhstan and, when he was 13, his father disappeared in a Gulag. Nevertheless in 1940 Dynkin succeeded in entering the Mechanics and Mathematics Faculty of Moscow University [MGU] where he became a student of A. N. Kolmogorov. In the acceptance speech he delivered when receiving the Steele prize (awarded by the American Mathematical Society) in 1993, Dynkin appraises his admission to university as "`almost a miracle".

Professor S. A. Yanovskaya of MGU noticed the gifted boy, saw that he was in need of help, and gave him mathematical encouragement (even to the extent of taking him and his mother into her own home). This allowed Dynkin to overcome the difficulties he had had in his life and, starting with these student years, to concentrate on mathematics. More than once, Kolmogorov had to use his authority to tip the balance of the scales in order to ensure Dynkin's advancement from student to professor.

While a student, Dynkin began to study in Gel'fand's seminar on Lie groups and in Kolmogorov's on Markov chains. Participation in these seminars also shaped the range of Dynkin's scientific interests: algebra, analysis, the theory of probability. It is significant that this very first student paper (written jointly with N. A. Dmitriev and published in 1945) On the characteristic roots of stochastic matrices lies at the junction of the three disciplines mentioned: it deals with the problem of the location of the named roots in the complex plane. Dynkin himself always emphasized (and taught his students) that one of the greatest values in mathematics is its unity as revealed in the links between its various fields.

On graduating from the Mechanics and Mathematics Faculty in 1945, Dynkin became Kolmogorov's research student in the same Faculty. He earned his Ph.D. in 1948 and began working there in the department of probability theory led (until 1966) by Kolmogorov. He stayed there for 20 years (1948-1968), being a professor from 1954.

Even in his student years one of Evgenii Borisovich's most remarkable features emerged - his outstanding teaching ability, his capacity for gathering pupils and students around him and launching them into creative work.

Dynkin's activities as a teacher began with school mathematics circles. Stalin's totalitarianism left little scope for free intellectual work and even less for unfettered public activities. Yet on the few islands of freedom, the cornfields yielded a rich harvest. In the intellectual sphere, mathematics and chess were such islands. The school mathematics circles at MGU can serve as one example from the public domain. They grew up in the prewar years and were interrupted by the war. Their heyday came at the end of the war and during the immediate postwar period.

At that time these circles were a unique phenomenon. For those who led them (the undergraduates and research students of the faculty), they provided the opportunity to do something that was reasonable and yet was without bureaucratic overtones. For the participants (the schoolchildren) they provided an atmosphere of creative democratic collaboration. Democratic traditions were being carefully preserved. Thus the leaders of the circle could be addressed by their first names (rather than by first name plus patronymic, as at school). Such details made a strong impression on the schoolchildren. Of course, the personality of the leader played a decisive role. And one of those vivid personalities was Evgenii Borisovich (then familiarly called Zhenya, in keeping with the tradition just mentioned). In the spring of 1945, one of the signatories of this article (Uspenskii) in fact met him when he was a fifth year student and leader of one of the circles (more precisely, "General Section of the School Mathematics Circle at MGU"). In the 1945/46 and 1946/47 school years, Zhenya continued to run the circle when he had already become a research student.

In 1947 several active participants in the circle became undergraduates in the Faculty of Mechanics and Mathematics at MGU and the circle itself was smoothly transformed into the seminar "Selected problems of contemporary mathematics" for the first year students. This was how Dynkin's celebrated seminar for students in the faculty came into existence: some time later it was given the new title "Selected problems of algebra and analysis" and in 1955 was divided into two daughter seminars - for algebra and for probability theory.

In its first years the atmosphere in Dynkin's seminar inherited that of the school circle. The activities were carried on emotionally. The dialogue of the participants with one another and with the instructor did not finish at the end of the session but continued as a rule out on the street. It became a tradition to accompany one's teacher - in a sort of animated saunter along Gor'kii Street - for at least a part of his way home. (At that time Dynkin lived at the beginning of Leningrad Prospect while the faculty was in one of the old university buildings on Mokhovaya Street, opposite the Manege.) All this reminds one somewhat - if one can judge from their reminiscences - of the behaviour of the famous Luzitanians of the 1920s (and it is appropriate to recall that Dynkin, through Kolmogorov, is a scientific grandson of N. N. Luzin). The attitude of the students towards Dynkin was one of gratitude, respect, at times even fervour; just as two of us (Vvedenskaya and Uspenksii) were happy when in the summer of 1949 we unexpectedly met our teacher on the beach at Riga! In his turn, Dynkin's attitude towards his students was extremely solicitous - including both the selection of the topic for the first publication and a selfless editing of immature versions of its text. He could, however, also be quite strict.

Here is a list (possibly incomplete) of mathematicians who began their academic work in Dynkin's seminars and under his supervision and can therefore be counted as his pupils: R. L. Dobrushin, F. I. Karpelevich, E. E. Balash, I. Z. Rozenknop, V. A. Uspenskii, N. N. Chentsov, F. A. Berezin, N. D. Vvedenskaya, A. A. Yushkevich, A. V. Skorokhod, V. M. Zolotarev, I. V. Girsanov, Ya G. Sinai, A. L. Onishchik, L. V. Seregin, V. A. Volkonskii, R. Z. Khas'minskii, A. D. Venttsel'(= Wentzell), E. B. Vinberg, K. Dambis, A. A. Kirillov, V. N. Tutubalin, M. I. Freidlin, M. G. Shur, N. V. Krylov, S. A. Molchanov, M. B. Malyutov, G. A. Margulis, E. L. Nol'de, A. L. Rozental', S. E. Kuznetsov, I. V. Evstigneev, M. I. Taksar, Yu. I. Kifer, S. M. Natanzon, S. A. Pirogov.

In 1951 Dynkin was awarded his Doctorate. On that occasion the students presented him with a construction made from clay and wire, a space graph representing a diagram of the roots of a Lie algebra. Nowadays the term Dynkin graph or Dynkin diagram (see below) has become firmly established in science: it occurs for example as the title of one of the sections in N. Bourbaki's Elements of Mathematics (the book Lie groups and algebras, Chapter VI.4.2) and also as the title of the recently published Volume 1548 of the series Lecture Notes in Mathematics.

Dynkin had already begun to take an interest in Lie groups and algebras back in his student years. Work in this field of mathematics occupied a comparatively short period of his life (1944-1955) but it was an extremely intensive and productive phase. This is apparent in the list of his fundamental achievements from this time: the discovery of simple roots and the Dynkin diagram, the finding of an explicit form for the coefficients in a Campbell-Hausdorff series, a description of maximal subalgebras of simple complex Lie algebras, the classification of semisimple subalgebras of exceptional complex Lie algebras, a description of primitive homology and cohomology classes of the classical compact Lie groups.

The role of the simple roots of a semisimple complex Lie algebra (that is, the positive roots not representable as a sum of two positive roots) was discovered by Dynkin in 1944 while studying papers by H. Weyl and B. L. Van der Waerden, when at the suggestion of I. M. Gel'fand he was preparing a review on the structure and classification of semisimple Lie algebras. It turned out that, starting from the system Pi of simple roots of a semisimple Lie algebra G , it is possible to construct a system of generators of this algebra, and the relations between these generators are completely determined by the angles between the simple roots and the ratios of their lengths. In this way, the system Pi, considered as a system of vectors in Euclidean space, determines the Lie algebra G uniquely to within isomorphism. The system Pi may be depicted by a graph whose vertices correspond to the simple roots and are coloured in two colours depending on their lengths, while the edges are assigned multiplicities of 0, 1, 2 or 3 according to the angle between the corresponding roots, which can be equal to either pi/2, 2pi/3, 3pi/4 or 5pi/6. This graph is now called the Dynkin diagram of the Lie algebra G . The classification of simple complex Lie algebras thus leads to a combinatorial problem whose solution is easily obtained using simple facts of Euclidean geometry. Nowadays the method of simple roots lies at the heart of the structural theory of Lie algebras, algebraic groups, Kac-Moody algebras, and other algebraic entities close to these.

While a research student at MGU, Dynkin took up another classical problem in the theory of Lie groups. It concerns the formal power series log(exey) of two non-commutating variables x, y. As Campbell and Hausdorff had shown at the beginning of this century, it can be represented as a series consisting of "Lie monomials" in x, y, that is, expressed in terms of x, y with the aid of the commutator operation [u,v] = uv-vu. Written in this way, the Campbell-Hausdorff series gives an expression (in canonical coordinates) for the multiplication operation in an arbitrary Lie group in terms of the commutator operation in the associated Lie algebra. An explicit form of this expression remained unknown until 1947, when Dynkin found the unexpectedly simple solution of the problem. It turned out that the expresison being sought will be obtained if in every term of the power series one replaces the operation of multiplication by a commutator and divides each term by its degree. In particular, this theorem made it possible to develop the theory of Banach Lie groups over an arbitrary complete normed field of characteristic 0. The results obtained constituted Dynkin's Ph.D. thesis, submitted in 1948.

The method of simple roots, mentioned above, made possible considerable simplification and improvement in the enunciation and proof of a range of classical theorems concerning the structure of semisimple complex Lie algebras (the classification of automorphisms, the theory of finite-dimensional linear representations). In 1950-51, using this method, Dynkin obtained fundamental results on the classification of the subalgebras of simple complex Lie algebras.

More than a century earlier Sophus Lie had raised the problem of the classification of primitive local groups of transformations, that is, transitive local Lie groups of transformations that do not leave any non-trivial fibration invariant. In the language of Lie algebras, this is equivalent to a classification of the maximal subalgebras of finite-dimensional Lie algebras G. V. V. Morozov had reduced the problem to the case when G is simple. In 1943 he had explicitly listed all the nonsemisimple maximal subalgebras of simple complex Lie algebras. (In 1951 this classification was simplified by Dynkin's student F. I. Karpelevich, who showed that any nonsemisimple maximal subalgebra of a semisimple complex Lie algebra belongs to the class of subalgebras now called parabolic and he described this class in terms of simple roots.) In 1950 Dynkin obtained a simple proof of A. I. Mal'tsev's results on the classification of semisimple subalgebras of orthogonal and symplectic Lie algebras and then gave a description of all maximal subalgebras of the classical complex Lie algebras. The most difficult was the case in which the subalgebra is simple and irreducible (that is, does not leave any proper subspace invariant); in that case the result obtained can be expressed in the following way: almost any irreducible simple complex subalgebra of the full linear Lie algebra GLn(C) is maximal in one of the classical Lie algebras SLn(C), SOn(C) or (for even n) SPn(C). The exceptions to this rule (comprising 4 series and 14 isolated subalgebras) are explicitly listed. This work formed the substance of the 1951 doctoral dissertation. In that same year Dynkin succeeded in solving the very difficult problem of classifying semisimple subalgebras of exceptional complex Lie algebras whose structure at that time had been very little studied. In particular, all the semisimple maximal subalgebras were described and this together with the results mentioned above gave a complete solution to Sophus Lie's problem in the case of the field of complex numbers. For this series of papers on the classification of subalgebras of semisimple Lie algebras Dynkin received the prize of the Moscow Mathematical Society in 1951.

Dynkin's first papers on the topology of compact Lie groups appeared in 1952. As Hopf and Samelson had shown, the cohomology algebra and homology algebra (or Pontryagin algebra) of a compact Lie group G over the field of rational numbers, which are dual to one another, are Grassmann algebras and it is possible to choose as free generators of these the primitive elements, that is, those orthogonal to the decomposable elements of the dual algebra. In the work mentioned (a detailed account of which came out in 1953) an explicit form of the primitive integral generators was described in the case when G is one of the classical groups. In particular, the primitivity of the generators of the Pontryagin algebra, constructed by L. S. Ponotryagin in 1939, was proved. For the description of the primitive cohomology classes use was made of the relation between the symmetric and skew-symmetry invariants of the associated group, which coincides, as was subsequently elucidated, with the transgression in the principal fibration of the group G. In papers of 1954 and 1955, explicit formulae were found for the coefficients that describe the natural mapping of primitive homology classes of a subgroup into primitive homology classes of the group, and these have fundamental importance in the study of the topology of the corresponding homogeneous space.

This phase of Dynkin's creative work came at a difficult period in the life of our country when national science was forcibly isolated from that of the world. Personal contacts with mathematicians from abroad were practically impossible and even if foreign journals appeared in libraries they did so with a considerable delay. Only several years later did Dynkin find out that graphs similar to his diagrams were independently devised by H. S. M. Coxeter for describing groups generated by reflections. Despite the loud pseudopatriotic propaganda, Dynkin never lost faith in the unity of world mathematics and educated his students in the same spirit. When the political climate changed in 1953 he quickly set about establishing mathematical contacts with foreign mathematicians. Thanks to his efforts, papers by A. Borel, Leray and Koszul on the topology of fibre spaces and homogeneous spaces of Lie groups appeared in Moscow. An important role in the development of national topology was played by the seminar he organized at MGU, in the leadership of which P. S. Aleksandrov and I. M. Gel'fand also took part.

Dynkin's last publication with an algebraic theme was dated 1959. Since that time the dominant focus of his creative work has been the theory of probability. We have already noted his student paper in 1945: from 1949 onwards papers in this field have appeared annually.

It may be assumed that Dynkin's definitive scientific choice in favour of probability theory was due not only to his participation in Kolmogorov's seminar on Markov chains but also to the fact that Kolmogorov himself had managed to do what hardly anyone else would have succeeded in doing at that time: to retain in his department (that of probability theory) a talented student with an " improper " anketa"" (biographical particulars in the Soviet jargon). As a result of this conjunction of circumstances, probability theory gained Dynkin - and without doubt it was lucky to do so.

At first a search went on for a suitable sphere of application for Dynkin's outstanding academic potential. But even his first papers in probability theory were an important contribution to the science. Already then, in the material on the theory of systems of equivalent random variables (that is, systems of random variables whose joint distributions are invariant with respect to all permutations of these variables), he had introduced a fruitful idea, which was developed in his subsequent papers and in those of several other authors. In various problems in probability theory, a convex set of probability measures on function spaces arises naturally. The problem is to of describe the extreme points of this set and to represent any point of the set as an integral mean value of its extreme points. The traditional approach for functional analysis (the key word - Choquet's theorems) requires the introduction of a topology and yet there is no unique natural topology on the space of measures. Dynkin discovered the possibility of a purely measure-theoretic approach to this problem, under which the extremal measures are interpreted as measures with trivial behaviour of the trajectories at infinity. This technique works well in many problems including describing Markov processes with given transition probabilities (studied by Dynkin) and also the theory of Gibbs random fields (developed later).

Another example is the coefficient of ergodicity of a stochastic matrix. In the general mathematical language, this is the norm of the corresponding operator restricted to the space of differences of the probability distributions. It is convenient for describing the ergodic behaviour of Markov chains. Sometimes its introduction is attributed to one of the authors of this article (Dobrushin) but in fact this ideas was suggested by Dynkin in one of his early papers on probability theory. During this period Dynkins also obtained interesting results on the conditions for continuity and for the absence of discontinuities of the second kind in the trajectories of Markov processes, and on limit theorems for sums of independent variables with infinite mathematical expectation, but he soon (in about 1955) found his culminating theme - the general theory of Markov processes, to which he has devoted almost all his subsequent academic life.

The flesh of mathematics consists of theorems and definitions. To argue which is the more important is as senseless as to solve the classical question of which came first, the chicken or the egg. Definitions are empty without theorems based on them; theorems cannot be formulated without definitions. Many important theorems are due to Dynkin, but what he was an especially outstanding master of was the creation of correct systems of definitions (that is, ones that subsequently became standard in mathematical literature) and, what should not be underestimated, adequate systems of notations. Dynkin has found the theory of Markov process during a period when its reconstruction on the basis of contemporary mathematical ideas became imminent. The ideas of the Great Mathematical Revolution at the beginning of our century penetrated very slowly into probability theory which continued for a long time living its own separate sectarian life. Traces of this can be seen even now in its archaic terminology: a measurable set in probability language is an event, a measurable function is a random variable, its integral is mathematical expectation. The potential for the fusion of probability theory with contemporary mathematics arose after Kolmogorov's discovery of the possiblity of expresisng probability theory in the language of measure theory, and yet right up to the time when Dynkin started work on the theory of Markov processes this theory had been restricted to an examination of unrelated, even if important, particular classes of processes, and its mathematical machinery to classical analysis and differential equations.

Any important change in science is, of course, the result of the combined work of the international community of scholars, but there is no doubt that Dynkin's contribution to an understanding of the theory of Markov processes was decisive. Many concepts that have already become generally accepted and elementary first appeared in his papers: the treatment of a Markov process as a consistent family of measures depending on the initial state and the initial moment of time, the strong Markov property, Dynkin's formula that connects two approaches to the description of the infinitesimal behavior of the process: local in time and local in space - and much else. A general theory of Markov processes was created based on the apparatus of functional analysis (measure theory, the theory of semigroups of operators and so on). Two brilliant books by Dynkin ( Foundations of the theory of Markov processes, 1959, and Markov processes, 1963) summed up this heroic period of developing that theory and became classics of probability theory. The thoroughness of the exposition, characteristic of Dynkin, is striking. For example, the first book contains the concept of Dynkin's classes of sets, maybe the only recent innovation in the presentation of the foundations of measure theory. This concept is now established as standard. It makes much easier a verification of the properties that an arbitrary function measurable with respect to some sigma-algebra of sets must have.

The new point of view on the theory of Markov processes revealed a mass of interesting mathematical problems; the possibility arose of a fruitful interaction between the theory of Markov processes on the one hand and the theory of partial differential equations and potential theory on the other. A boom in investigations in this newly discovered domain arose, but even against its background Dynkin's subsequent contributions to the theory of Markov processes proved to be very important. His later results on the theory of boundary-value problems for Markov processes (Martin boundaries), on the problem of optimal stopping, on the classification of additive functionals of Markov processes, and so on, have a permanent place in probability theory.

Dynkin's important results on the theory of random fields should be mentioned: here he revealed unexpected and deep relations between Gaussian random fields and symmetric Markov processes.

In recent years, Dynkin has turned his attention to an investigation of a fresh and important new class of random processes. This are the so-called superprocesses, whose states are measures. Their construction combines the ideology of diffusion Markov processes and branching Markov processes in which, however, the branching proceeds by infinitely small steps. The significance of this class of random processes is evident even from the fact that, while the diffusion processes give a stochastic representation for the solutions of linear parabolic equations, the superprocesses allow one to obtain such a representation also for certain non-linear equations of parabolic type. The work Dynkin put into this subject and the results may be compared with his classical contribution to the foundations of the theory of Markov processes. A forthcoming book by Dynkin contains a systematic exposition of the theory of superprocesses.

In 1968 Dynkin's work at Moscow University was compulsorily interrupted and from 1968 to 1976 he was a senior scientific worker at the Central Economics and Mathematics Institute at the USSR Academy of Sciences. During his short spell of work there he organized a group of young workers together with whom he obtained important results in the theory of economic growth and economic equilibrium that culuminated in the first Soviet report on this topic at the International Mathematics Congress in Vancouver (to which, incidentally, in the usual way, he was not allowed to go).

In 1976 Dynkin emigrated from USSR and in 1977 became a professor at Cornell University (USA). Since 1978 he has been a member of the American Academy of Arts and Sciences and since 1985 a member of the National Academy of Sciences of the USA.

Dynkin has more than once returned to Russia - most recently as an invited speaker at the International Congress "Kolmogorov and Modern Mathematics" (St. Petersburg, May-June 1993). We hope that he will come back many more times.

At the age of 70, Dynkin is young in his enthusiasm for mathematics and we anticipate many more unexpected ideas and results from him.

By R. L. Dobrushin, A. L. Onishchik, V. A. Uspenskii, N. D. Vvedenskaya. Translated by Roger F. Wheeler. Reprinted from Russian Mathematical Surveys, 49:4 (1994), 183-191; by permission of the London Mathematical Society and the British Library. Copyright 1994 by the London Mathematical Society and The British Library. (Inaccuracies in the translation from Russian are corrected.)

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4. E.B. Dynkin's Seminar on Lie Groups and Lie Algebras

Preface

This volume contains papers of participants of E. B. Dynkin's seminar on Lie groups and Lie algebras, and of students of the participants of this seminar. The seminar was held at Moscow State University from 1957 to 1962. The publication of the volume was timed to coincide with the 70th birthday of Evgenii Borisovich Dynkin.

From my point of view, this seminar was one of the most typical and successful Moscow mathematical seminars. Therefore, it seems natural to publish this book as a volume in "Advances in the Mathematical Sciences," since one of the goals of this series is to inform the Western mathematical community about these seminars. I have written on several occasions about the phenomenon called ``a Moscow mathematical seminar,'' both in general and on the example of one of the most well known, I. Gelfand's seminar. I am glad to have the opportunity to consider one more illuminating example. I must warn the reader that in this preface I present personal recollections, and do not try to reconstruct dates and events very precisely.

If I am not mistaken, the seminar itself was organized in February of 1957. However, it had a long prehistory, which, I think, deserves to be mentioned. In 1945 Dynkin, then a senior (5th year) student organized a ``mathematical circle'' (mathematical club) for high school students. Such circles were a long tradition in Moscow, and many mathematicians participated in them during their school years. The participants of Dynkin's circle later told me that this circle was very successful. E. B. displayed his pedagogical talent, the main characteristic of which is the love and ability to work with high school and junior college students. I saw this both in my student years and also later, when I collaborated with Dynkin in the famous Moscow High School Number 2. In Dynkin's circles and seminars, not only his direct students started their mathematical life, but also many other mathematicians who later worked in other areas of mathematics. One must recall that, although an undergraduate at the time, Dynkin was not a novice in mathematics: in 1944 he discovered simple roots of semisimple complex Lie algebras (this was done during the preparation of a talk on I. M. Gelfand's seminar). Dynkin's circle worked from 1945 to 1947 (Dynkin was a graduate student then) and most of his participants entered the Moscow University.

At that time, the circle transformed continuously into the freshmen seminar on "Selected Problems in Contemporary Mathematics", and continued to work until 1955, at some point changing its title to "Selected Problems in Algebra and Analysisc". Although Dynkin's seminar was always rather broad in its approach to mathematics, it seems that Lie groups and probability were two topics that had a high priority, and it is in these directions that Dynkin oriented his students. Therefore, it can be said that the first version of the Lie groups seminar was a part of the 1947-1955 seminar.

Although Dynkin was interested in both Lie groups and in probability from his student times, at the beginning of the 1950s his work was concentrated mainly around Lie groups. His publications in this area stopped in 1955, and at this moment he presumably switched completely to the probability theory. It is only at the beginning of 1960 that he returned for some time to Lie groups in connection with his work on Brownian motion and the Martin boundary of symmetric spaces. It is interesting to notice that this work lies on the intersection of his two interests.

The outstanding results of E. B. Dynkin on Lie groups are well known. However, I am certain that the survey prepared for this volume by Karpelevich, Onishchik, and Vinberg will allow the reader to understand Dynkin's contribution to this theory even better.

Dynkin organized his Lie group seminar in 1957, when he practically stopped working in this area. This makes the success of this seminar even more remarkable. Here is the sequence of events, as I recollect them. In the fall of 1956, Dynkin announced a one-year course in differential manifolds and Lie groups. At the end of the fall semester he suddenly declared that in the spring semester the course will become a seminar.

Mainly, this was a seminar for second- and third-year undergraduate students. Among the participants I remember A. Kirillov, E. Vinberg, M. Freidlin, M. Shur, and V. Tutubalin (the last three soon switched to probability). Most of us knew each other from our school years, when we used to attend the same mathematical circles and mathematical competitions.

In the spring of 1957, we mainly learned how to read mathematical books, give talks, and solve problems. At the end of the semester, most of the participants prepared an essay, which sometimes could have been considered as a small scientific work.

The situation had changed dramatically in the fall of 1957. Dynkin invited his students of the previous generation to the seminar, F. Berezin and F. Karpelevich among them. A. Onishchik then a graduate student, was an even earlier participant of the seminar. Soon afterward I. Pyatesky-Shapiro, who just returned from Kaluga, where he had worked after graduating from Moscow Pedagogical University, started to attend the seminar. At about the same time the seminar was joined by A. Schwarz who just entered the graduate school, and by G. Tyurina and D. Fuchs, who were in their junior undergraduate year.

This blend of mathematicians of different generations produced wonderful results. It is difficult to overestimate the importance of this for us, the junior participants of the seminar. We found ourselves involved in the process of mathematical research. Problems and results were discussed in a very preliminary format. Here are some examples that are closer to me. Ilya Pyatesky-Shapiro spoke about his plans and results related to automorphic forms and related problems in the theory of bounded symmetric domains. This is how we first heard about Siegel domains. At some point, the speaker asked Onishchik, who had the reputation of an expert on homogeneous complex manifolds, whether it had already been proved that each homogeneous bounded domain is symmetric. The answer was that it was almost proved. After some discussion we came to the conclusion that the remaining part of the proof - either semisimplicity or unimodularity of the group - should not pose a serious obstacle. Soon afterwards (I am tempted to say in just a week) Ilya, somewhat hesitantly, was demonstrating to us an example of a nonsymmetric homogeneous domain in C4.

I also recall team attempts to include the exceptional domains into the general classification scheme for Siegel domains. An exceptional cone for the first domain was soon found, but no one could figure out what to do with the second domain, until Ilya finally found its place among the Siegel domains of the second type related to the light cone. It was no wonder to all of us that soon afterwards Vinberg and I found ourselves involved in the development of the theory of nonsymmetric homogeneous domains.

Another frequently discussed topic was the problem of the boundary of a symmetric space. Pyatesky-Shapiro was interested in the corresponding constructions in his attempts to find compactifications of fundamental domains of discrete groups. Dynkin approached the problem from the point of view of Martin boundaries. Finally, one of the most important chapters in the theory of symmetric spaces, the Karpelevich compactification theory, was developed.

Those were happy years for all of us. The seminar was an important part of our mathematical house. We spent a lot of time together. I can recall joint weekend trips to the country (together with Dynkin's seminar on probability). We have continued our friendly relations for some 40 years, and we all are very grateful today to Evgenii Borisovich Dynkin, who did so much at the start or our mathematical life.

By S. G. Gindikin

Reprinted from the book "E. B. Dynkin's Seminar on Lie Groups and Lie Algebras", American Mathematical Society Translations, Series 2, Volume 169, pages ix-xi; by permission of the American Mathematical Society. Copyright 1995 by the American Mathematical Society.

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