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Eugene Dynkin

Eugene Borisovich Dynkin (May 11, 1924 – November 14, 2014) was a Russian-American mathematician recognized for pioneering the modern theory of Markov processes and developing key algebraic tools, including Dynkin diagrams for classifying semisimple Lie algebras.^[1]^[2] Born in Leningrad to a Jewish family, Dynkin faced early adversity when his family was exiled to Kazakhstan at age 11 amid Stalinist purges, yet he advanced to Moscow State University by age 14 and completed a PhD in mathematics there in 1948 under Andrey Kolmogorov.^[3]^[4] His seminal works in the Soviet era, such as Foundations of the Theory of Markov Processes (1959) and Markov Processes (1963), reformulated these stochastic systems as families of measures tied to initial conditions, enabling rigorous analysis of time-homogeneous transitions and influencing fields from physics to finance.^[3]^[1] Emigrating to the United States in 1976 amid restrictions on Soviet Jewish scientists, he joined Cornell University as the A.R. Bullis Professor of Mathematics in 1977, continuing research on probabilistic potential theory and Lie groups while amassing a collection of over 200 video interviews with prominent mathematicians to document the field's history.^[1]^[5] Dynkin's innovations, including Dynkin systems for set algebras and lemmas bridging martingales to stopping times, remain foundational in probability, with his methods underpinning applications in diffusion models and optimal control.^[2]^[3]

Biography

Early Life and Family

Evgenii Borisovich Dynkin was born in 1924 in Leningrad (now Saint Petersburg), Soviet Union, to a family of Jewish origin.^[2]^[1] His father, Boris Solomonovich Dynkin, worked as a lawyer, while his mother was a dentist; both parents originated from Belarus.^[6] In 1935, amid the Stalinist purges following the assassination of Sergei Kirov, Dynkin's family was exiled to Kazakhstan.^[6] His father was arrested and disappeared during the repressions, a fate common for professionals of Jewish background under the regime.^[1]^[7] At age 13, Dynkin and his mother managed to return to European Russia, settling in Moscow two years after the exile.^[2] This period of upheaval marked his early childhood, during which he developed an interest in mathematics despite the hardships of displacement and family loss.^[3]

Education at Moscow University

Dynkin was admitted to the Mechanics and Mathematics Faculty of Moscow State University in 1940 at the age of 16, an achievement he later described as "almost a miracle" given the political and social barriers faced by individuals of Jewish origin during the Stalinist era, compounded by his father's classification as a "people's enemy."^[8] His admission and subsequent academic progress were significantly supported by the probabilist Andrey Kolmogorov, who recognized Dynkin's talent and intervened to facilitate his entry despite these obstacles.^[8] Throughout World War II, Dynkin continued his studies uninterrupted, exempted from military service due to poor eyesight, which allowed him to focus on advanced mathematics amid the wartime disruptions in Moscow.^[8] He attended seminars led by Israel Gelfand on Lie groups, where in 1944 Gelfand tasked him with preparing a survey on semisimple Lie groups, sparking early independent work in algebra that included collaborations and foundational insights into root systems.^[9] Exposure to Kolmogorov's seminar on Markov chains further influenced him, leading to his first publication in 1945 with N.A. Dmitriev on the eigenvalues of stochastic matrices.^[9] Dynkin earned his Master of Science degree from Moscow State University in 1945, having developed key ideas such as the "Dynkin diagram" for classifying semisimple Lie algebras during his student years, which he formalized in a 1947 paper.^[8] Under Kolmogorov's supervision, he completed his Candidate of Sciences (equivalent to Ph.D.) in 1948, marking the culmination of his formal education at the institution.^[9]^[10]

Career in the Soviet Union

Dynkin began his professional career at Moscow State University shortly after completing his graduate studies. In 1948, following his Ph.D., he was appointed assistant professor in Andrey Kolmogorov's probability chair, where he contributed to both teaching and research in probability theory and Lie algebras.^[8] By 1951, he had earned the Doctor of Physics and Mathematics degree, a higher doctoral qualification in the Soviet system.^[8] In 1954, Dynkin was promoted to full professor and given the chair of probability theory at Moscow State University, a position he held until 1968. During this period, he supervised graduate students, developed coursework on Markov processes, and actively participated in extracurricular mathematical education, including mentoring at specialized high schools for mathematically gifted youth established in the early 1960s amid Soviet educational reforms.^[8] ^[6] His research output included foundational texts such as Foundations of the Theory of Markov Processes (1959) and Markov Processes (1963), which advanced probabilistic methods while navigating the constraints of Soviet academic publishing.^[8] Dynkin's tenure at Moscow University ended abruptly in 1968, when his position was compulsorily terminated amid broader institutional pressures on prominent Jewish mathematicians during the late Soviet era. He was transferred to the Central Economics and Mathematics Institute (CEMI) of the USSR Academy of Sciences, where he served as a senior research scientist from 1968 to 1976.^[8] ^[2] At CEMI, an institute focused on applied mathematical economics, Dynkin led a research group investigating models of economic growth and equilibrium, adapting his expertise in stochastic processes to interdisciplinary problems despite the shift from pure mathematics.^[8] This reassignment, common for displaced scholars, limited his access to core mathematical circles but allowed continued publication and supervision of junior researchers.^[11]

Emigration to the United States and Cornell Professorship

In 1976, Dynkin decided to emigrate from the Soviet Union after years of professional restrictions, including a compulsory transfer in 1968 from Moscow University to the Central Economic Mathematical Institute of the USSR Academy of Sciences, where he worked until his departure.^[8] The decision was difficult, as it meant leaving behind students, colleagues, and a lifetime of connections in Russia, and applying for exit permission carried significant personal and professional risks under Soviet policies.^[8] Despite his international stature in mathematics, Dynkin had been barred from traveling to the West prior to this point.^[8] Dynkin left the USSR at the end of 1976, transiting through Israel before immigrating to the United States.^[3] In 1977, he joined Cornell University as a professor of mathematics, where he appreciated the supportive environment, including collaborative colleagues, natural surroundings, and resources conducive to research.^[1] At Cornell, he shifted focus toward probability theory while maintaining ties to algebraic structures from his earlier career, mentoring students and contributing to the department's strengths in stochastic processes.^[1] He held the position until retirement, later designated as A. R. Bullis Professor of Mathematics Emeritus.^[8]

Later Years and Death

Dynkin retired from his position as the A.R. Bullis Professor of Mathematics at Cornell University on July 1, 2010, after more than 33 years of service there.^[12]^[6] In the period preceding retirement, he sustained his focus on probability theory, advancing studies in Markov processes, Gaussian random fields, and superprocesses.^[3] That May, he presented reflections on his seventy years in mathematics, underscoring the foundational value of clarity and rigor in the discipline as taught to generations of students.^[13] Post-retirement, Dynkin stayed engaged with mathematics through mentoring—having influenced over 500 descendants in the field—and curating the Dynkin Collection, an archive of oral history interviews preserving insights into probability theory's evolution.^[3] He continued research contributions until his death on November 14, 2014, at Cayuga Medical Center in Ithaca, New York, aged 90.^[1]^[3] Dynkin was survived by his wife, Irene; daughter, Olga Barel; three grandchildren; and seven great-grandchildren.^[3]

Mathematical Contributions

Work in Lie Theory

Dynkin's foundational contributions to Lie theory occurred primarily in the 1940s, focusing on the structure and classification of semisimple Lie algebras. At age 20, in 1944, he developed a method using simple roots to analyze these algebras, introducing graphical representations that encode the relations among simple roots via nodes and weighted edges, forming what are now termed Dynkin diagrams.^[1] These diagrams correspond one-to-one with the Cartan matrices of simple Lie algebras, where the edge multiplicities reflect the off-diagonal entries derived from the Killing form.^[8] Dynkin diagrams enabled a complete classification of finite-dimensional complex semisimple Lie algebras over the complex numbers, identifying the infinite families (A_n, B_n, C_n, D_n, E_6, E_7, E_8, F_4, G_2) through their unique combinatorial structures. He established that irreducible positive definite Cartan matrices are precisely those realizable by these diagrams, providing a criterion for distinguishing semisimple types from non-semisimple ones.^[8] This approach, detailed in his 1947 publications, streamlined earlier classifications by Killing (1880s) and Cartan (1910s–1920s) by reducing the problem to finite graph theory.^[14] Beyond classification, Dynkin examined subalgebras within semisimple Lie algebras g, defining R-subalgebras as those contained in proper regular subalgebras and S-subalgebras as maximal irreducible ones outside regular subalgebras. This dichotomy advanced the understanding of embedded structures and Levi decompositions in Lie algebras.^[6] His techniques extended to representation theory, where Dynkin diagrams facilitate the computation of Weyl group orbits and highest weights for irreducible representations, though his primary innovations lay in algebraic structure rather than explicit module classifications.^[3] Dynkin's Lie theory work, conducted amid Soviet academic constraints, fostered a Moscow school on Lie groups and algebras in the 1950s, influencing subsequent developments in symmetry studies for physics and geometry.^[1] Note that similar diagrams for Coxeter groups were independently found by H.S.M. Coxeter around the same period, highlighting convergent insights into reflection and root systems.^[8]

Work in Probability Theory

Dynkin's research in probability theory gained prominence after 1954, when he shifted focus from algebra to stochastic processes, building on his earlier doctoral work under Andrei Kolmogorov.^[5] His contributions emphasized rigorous foundations for Markov processes, treating them as consistent families of probability measures indexed by initial time and state, with the Markov property expressed through conditional independence.^[15] This approach facilitated the study of subprocesses and transition functions, enabling precise conditions for process existence and uniqueness.^[8] A cornerstone of his work was the semigroup theory of Markov processes, where he characterized them via the infinitesimal generator of the associated transition semigroup, linking algebraic structures to probabilistic dynamics.^[8] In his seminal monograph Markov Processes (Russian edition 1963; English translation 1965), Dynkin explored potentials, harmonic functions, and excessive measures, demonstrating how Markov processes yield solutions to boundary value problems in potential theory.^[6] He proved the strong Markov property under general conditions, using martingales to analyze path regularity and decomposition.^[8] Dynkin introduced Dynkin's formula, which for a Feller process with generator L and bounded stopping time \tau states that \mathbb{E}^x [f(X_\tau)] = f(x) + \mathbb{E}^x \left[ \int_0^\tau Lf(X_s) \, ds \right] for suitable functions f, providing a probabilistic interpretation of elliptic partial differential equations Lf = g.^[1] This result, derived from Itô's lemma and optional stopping, connected stochastic analysis to classical analysis, influencing applications in diffusion theory and boundary problems.^[16] Earlier, Dynkin advanced measure-theoretic tools with Dynkin systems (λ-systems), collections of sets closed under complements and countable disjoint unions containing the whole space, proving that a Dynkin system containing a π-system generates the σ-algebra if the π-system is stable under finite intersections (π-λ theorem).^[17] This theorem underpins uniqueness proofs in probability, such as Kolmogorov's extension theorem, and the Doob-Dynkin lemma, which characterizes random variables measurable with respect to a sub-σ-algebra.^[2] At Cornell from 1976 onward, Dynkin extended these ideas to infinite-dimensional processes and their links to partial differential equations, fostering research on Dirichlet problems and Feynman-Kac representations.^[5] His early probabilistic insights, including complete characterizations of sufficient statistics for one-dimensional families (1949-1950), highlighted minimal data for inference, influencing statistical decision theory.^[15] Overall, Dynkin's probability oeuvre unified algebraic abstraction with empirical process behavior, establishing Markov theory as a bridge to analysis.^[2]

Interconnections and Broader Impact

Dynkin's investigations established foundational links between the algebraic framework of semisimple Lie algebras and the analytic structures of Markov processes, where root systems and Weyl groups from Lie theory parallel the state spaces and transition mechanisms in probabilistic models.^[1]^[18] These interconnections, first systematically explored by Dynkin in the 1950s, enabled the application of algebraic classification techniques—such as his systems of simple roots—to decompose complex stochastic systems into irreducible components, simplifying proofs of existence and uniqueness for solutions to stochastic differential equations.^[6] The broader ramifications of Dynkin's algebraic contributions extend to representation theory and mathematical physics, where Dynkin diagrams serve as standardized tools for enumerating finite-dimensional representations of Lie groups, influencing developments in conformal field theory and integrable systems as of the late 20th century.^[6] In probability, his formulations underpinned the theory of controlled Markov processes, providing rigorous criteria for optimal control problems that have informed applications in operations research and queueing theory since the 1960s.^[19] These dual advancements underscore Dynkin's role in fostering interdisciplinary methodologies, with his probabilistic tools—such as Dynkin's formula for martingale expectations—facilitating convergence results in diffusion approximations that remain central to stochastic modeling in engineering and finance.^[20]

Recognition and Legacy

Awards and Honors

Dynkin was awarded the Prize of the Moscow Mathematical Society in 1951 for his contributions to the structure theory of semisimple Lie algebras.^[2]^[3] In recognition of his foundational work in Lie theory and probability, he received the Leroy P. Steele Prize for Lifetime Achievement from the American Mathematical Society in 1993.^[21]^[1] Dynkin was elected a Fellow of the American Academy of Arts and Sciences in 1978 and a member of the National Academy of Sciences of the United States in 1985.^[1]^[3]

Influence on Students and Mathematics

Dynkin mentored numerous students throughout his career, supervising over 30 doctoral candidates and fostering a lineage exceeding 500 mathematical descendants according to academic genealogy records.^[10]^[3] In the Soviet era, he organized a mathematical circle for high school students at Moscow State University from 1944 to 1946, guiding early talents such as F. I. Karpelevich, A. A. Yushkevich, and A. V. Skorokhod.^[22] These efforts evolved into specialized undergraduate seminars on Lie groups and probability theory in the 1950s, where he encouraged independent problem-solving and collaboration, leading to publications like Ya. G. Sinai's first paper derived from a seminar challenge.^[22] He founded the probability theory seminar at Moscow State University in 1954–1955, with core participants including R. L. Dobrushin, N. N. Chentsov, and A. V. Skorokhod, which emphasized Markov processes and shaped the Moscow school of probability.^[22]^[23] Notable doctoral students from this period included N. V. Krylov, A. V. Skorokhod, M. I. Freidlin, and R. Z. Khasminskii, many of whom advanced stochastic processes and ergodic theory.^[22]^[23] Similarly, his Lie groups seminar, still active under successors E. B. Vinberg and A. L. Onishchik, influenced algebraic research.^[23] Dynkin's informal teaching style prioritized depth over rote learning, producing researchers who extended his foundational ideas in Markov processes and Lie algebras.^[22]^[8] After emigrating to the United States in 1976, Dynkin continued this tradition at Cornell University from 1977 until his retirement in 2010, conducting graduate seminars that preserved Moscow-era methods and mentored American and international students.^[22]^[23] His lecturing was renowned for clarity and engagement, transforming apprentices' understanding of probability and analysis across generations.^[3]^[8] This pedagogical legacy amplified his mathematical influence, as students like Skorokhod and Krylov—active into the 2000s—applied Dynkin-inspired techniques to superprocesses, Gaussian fields, and nonlinear partial differential equations.^[22]^[3] Overall, Dynkin's emphasis on rigorous exploration ensured his concepts, such as the strong Markov property and Dynkin formulas, permeated modern probability and algebra through successive cohorts.^[8]

Publications

Monographs and Books

Dynkin authored eight research monographs, primarily focused on probability theory and stochastic processes, alongside three popular educational books.^[6] His foundational contributions to Markov process theory were systematically presented in early works such as Foundations of the Theory of Markov Processes (1959), which provided axiomatic underpinnings for defining Markov processes via families of measures and transition functions, influencing subsequent developments in stochastic analysis.^[2] This was complemented by Markov Processes (1963), offering rigorous treatments of subprocesses, infinitesimal generators, and resolvents, with applications to diffusion and potential theory.^[2] ^[8] A Dover edition of Theory of Markov Processes (reprinted 2006 from earlier originals) further elaborated on trajectory properties, stopping times, and optimal control in Markov chains and diffusions, emphasizing logical foundations over empirical simulations. In controlled settings, Dynkin explored Controlled Markov Processes (1979), addressing dynamic programming and value functions for stochastic optimization problems.^[16] Later monographs shifted toward measure-valued processes, exemplified by An Introduction to Branching Measure-Valued Processes (1994, CRM Monograph Series, American Mathematical Society), which constructed superprocesses as limits of branching particle systems and linked them to solutions of nonlinear parabolic PDEs via probabilistic representations.^[24] These works prioritized analytical rigor, deriving causal structures from generator semigroups and resolvent equations rather than heuristic approximations, and remain cited for bridging probability with PDE theory.^[25] Dynkin's monographs, often originating from Soviet-era publications with Western translations, underscore his role in formalizing abstract stochastic frameworks verifiable through operator theory.^[26]

Selected Papers and Editions

Dynkin's prolific output included over 200 research papers spanning Lie theory, probability, and related fields. A curated selection of his most influential works appears in the 2000 volume Selected Papers of E. B. Dynkin with Commentary, edited by A. A. Yushkevich, G. M. Seitz, and A. L. Onishchik, published by the American Mathematical Society. This compilation organizes papers into sections on Lie groups and algebras (Part I) and Markov processes, superprocesses, and partial differential equations (Part II), with commentaries by experts such as G. M. Seitz on Dynkin's role in Lie theory and assessments of his boundary theory contributions.^[27]^[28] Seminal papers in the Lie algebras section include those from 1946 and 1947 that developed Dynkin diagrams—a combinatorial tool simplifying the classification of semisimple Lie algebras via Cartan matrices—and explored subalgebra structures, influencing subsequent classifications and representations.^[8] These works built on Cartan-Killing theory, providing explicit criteria for semisimple subalgebras and embeddings. In probability, the volume highlights early boundary theory papers, such as those on Martin boundaries for discrete-state Markov processes (circa 1960s), which connected stochastic processes to potential theory and harmonic functions.^[6]^[29] Later selections cover Dynkin's 1980s–1990s papers on superprocesses, modeling nonlinear partial differential equations via branching Markov processes, with applications to measure-valued diffusions and Dirichlet problems. These established probabilistic interpretations of PDE solutions, impacting modern stochastic analysis. No major edited editions of others' works by Dynkin are noted, though his seminars influenced collaborative volumes like Lie Groups and Lie Algebras: E. B. Dynkin's Seminar (1995).^[6]^[30]

References

[1]
Mathematician Eugene Dynkin dies at 90 - Cornell Chronicle
Nov 20, 2014 · Born in Leningrad in 1924 as Evgenii Borisovich Dynkin, he suffered greatly under the oppressive Stalinist regime. His family was exiled to ...Missing: biography | Show results with:biography
[2]
Obituary: Eugene Dynkin 1924–2014
Feb 16, 2015 · Eugene Dynkin, the AR Bullis Professor of Mathematics Emeritus at Cornell University, died November 14, 2014, in Ithaca, NY. He was 90.
[3]
[PDF] Eugene B. Dynkin - Cornell eCommons
Nov 14, 2014 · Evgenii Borisovich Dynkin was born in Leningrad (now St. Petersburg) in 1924. When he was. 11, his family was exiled to Kazakhstan and, ...Missing: biography | Show results with:biography
[4]
Eugene Dynkin - Random Services
Eugene Dynkin was born on May 11, 1924 in Leningrad. He attended Moscow University, beginning at age 14 and graduating wih a PhD in 1948.Missing: Borisovich achievements
[5]
Guide to the Eugene B. Dynkin papers, 1950-2014.
Professor at Cornell University from 1977-2014. Lie groups were the main ... Principal efforts were devoted to Markov processes and their connections with ...
[6]
Evgenii (Eugene) Borisovich Dynkin - Mathnet.RU
His works in this field were mostly concerned with the theory of Markov processes, where he is regarded as one of the founders. His fundamental monograph Markov.<|control11|><|separator|>
[7]
Dynkin, Cornell math titan, dies in Ithaca
Nov 24, 2014 · Born in Leningrad in 1924, his family suffered under the Stalinist regime, first through exile to Kazakhstan and the disappearance of his father ...Missing: early life biography
[8]
Evgenii B Dynkin (1924 - 2014) - Biography - MacTutor
Eugene Borisovich Dynkin. Quick Info. Born: 11 May 1924. Leningrad, USSR (now St Petersburg, Russia); Died: 14 November 2014. Ithaca, New York, USA. Summary ...Missing: achievements | Show results with:achievements
[9]
Eugene Dynkin - Scientific Library
While Dynkin was a student at Moscow University, he attended Israel Gelfand's seminar on Lie groups. In 1944, Gelfand asked him to prepare a survey on the ...
[10]
Eugene B. (Evgenii Borisovitsh) Dynkin
Eugene B. (Evgenii Borisovitsh) Dynkin, Biography, MathSciNet, Ph.D. Lomonosov Moscow State University 1948 Russia, Dissertation: Mathematics.
[11]
Soviet Mathematical Economics
CEMI served as a temporary home for some bright mathematicians, including. Eugene Dynkin, Gennadi Henkin, Anatole Katok, and Boris Mityagin.6 Some of them.Missing: B. positions era
[12]
[PDF] MATH MATTERS - Cornell Mathematics
Eugene Dynkin will retire July 1, 2010, after spending more than 30 years here. Reinventing probability at Cornell will be an interesting chal- lenge for the ...
[13]
Eugene Dynkin: Seventy Years in Mathematics - Cornell Video
May 26, 2010 · Eugene B. Dynkin, the A.R. Bullis Professor of Mathematics ... University Relations is no longer accepting submissions to CornellVideo.
[14]
Visualizing Lie Subalgebras using Root and Weight Diagrams
In 1947, Eugene Dynkin simplified the process of classifying complex semi-simple Lie algebras by using what became known as Dynkin diagrams [9]. As pointed out ...
[15]
The Dynkin Festschrift: Markov Processes and their Applications
Free in-store returnsHe created the fundamental concept of a Markov process as a family of measures corresponding to var ious initial times and states and he defined time ...<|separator|>
[16]
Markov Processes and Related Problems of Analysis
The theory of Markov Processes has become a powerful tool in partial differential equations and potential theory with important applications to physics.
[17]
[PDF] 1. Dynkin systems
If D is a Dynkin system containing C then D also contains the σ-algebra σ(C) generated by C. www.probability.net.<|separator|>
[18]
In Memory of E. B. Dynkin (11.05.1924--14.11.2014) - SIAM.org
Dynkin's contributions in algebra and theory of probability, as well as his teaching in the USSR and in the USA, have been recognized with numerous distinctions ...
[19]
[PDF] Stochastic models and algorithms dedicated to the 60th birthday of ...
Oct 1, 2022 · During his studies Eugene was significantly influenced by Professor ... Dynkin the now classic monograph “Controlled Markov Processes.
[20]
New Publications Offered by the AMS, Volume 46, Number 8
Eugene Dynkin is a rare example of a contemporary mathematician who has achieved outstanding results in two quite different areas of research: algebra and ...<|control11|><|separator|>
[21]
AMS :: Browse Prizes and Awards - American Mathematical Society
Steele Prize for Lifetime Achievement (1993 - present) 1993 Eugene B. Dynkin. To Eugene B. Dynkin for his foundational contributions to Lie algebras and ...
[22]
Cornell Math - A History of Dynkin's School
Jul 28, 2004 · The seminar on probability theory began during the 1954–1955 academic year. The initial core included R. L. Dobrushin, F. I. Karpelevich, N. N. ...
[23]
https://www.mathnet.ru/eng/rm9715
[24]
AMS eBooks: CRM Monograph Series
Eugene B Dynkin, Cornell University, Ithaca, NY. Publication: CRM Monograph Series Publication Year: 1994; Volume 6 ISBNs: 978-0-8218-0269-4 (print); 978-1 ...
[25]
An Introduction to Branching Measure-Valued ... - Google Books
This is the first monograph devoted to the theory of BMV processes. Dynkin first constructs a large class of BMV processes, called superprocesses, by passing to ...
[26]
[PDF] Markov Processes. - Harvard Mathematics Department
Professor Evgenii Borisovich Dynkin, who teaches at Moscow Uni- versity, has written two earlier textbooks on Markov processes and is also the author of a ...
[27]
Selected Papers of E. B. Dynkin with Commentary - AMS Bookstore
His work in probability is part of both the historical and the modern development of the topic. This volume presents Dynkin's scientific contributions in both ...
[28]
Selected papers of E.B. Dynkin with commentary | Semantic Scholar
Part I: Lie groups and Lie algebras: Dynkin and Lie theory by G. M. Seitz On the work of E. B. Dynkin in the theory of Lie groups by F. I. Karpelevich, ...<|separator|>
[29]
E. B. Dynkin, “Boundary theory of Markov processes (the discrete ...
Abstract: The paper contains a detailed account of the theory of Martin boundaries for Markov processes with a countable number of states and discrete time.
[30]
Lie Groups and Lie Algebras: E. B. Dynkin's Seminar
In celebration of E.B. Dynkin's 70th birthday, this book presents current papers by those who participated in Dynkin's seminar on Lie groups and Lie algebras in ...