Efim Zelmanov

Efim Zelmanov
Efim Zelmanov
Efim Zelmanov
Born	Efim Isaakovich Zelmanov 7 9, 1955
Birthplace	Khabarovsk, Russian SFSR, Soviet Union
Nationality	Russian-American
Occupation	Mathematician, professor
Known for	Solution of the restricted Burnside problem, contributions to nonassociative algebra and group theory
Awards	Fields Medal (1994)

Efim Isaakovich Zelmanov (Template:Lang-ru; born 7 September 1955) is a Russian-American mathematician whose work on combinatorial problems in nonassociative algebra and group theory placed him among the most significant algebraists of the late twentieth century. Born in Khabarovsk in the Soviet Far East, Zelmanov rose to international prominence through his solution of the restricted Burnside problem, a question that had occupied group theorists for decades. For this achievement, he was awarded the Fields Medal at the International Congress of Mathematicians in Zürich in 1994, one of the highest honors in mathematics. Over the course of his career, Zelmanov has held positions at several leading universities, including the University of Wisconsin–Madison, the University of Chicago, Yale University, and the University of California, San Diego. In more recent years, he has served as a chair professor at the Southern University of Science and Technology (SUSTech) in Shenzhen, China, where he has continued his research while also engaging in undergraduate teaching. He is a member of both the United States National Academy of Sciences and the Chinese Academy of Sciences.

Early Life

Efim Isaakovich Zelmanov was born on 7 September 1955 in Khabarovsk, a city in the far eastern part of the Russian Soviet Federative Socialist Republic, near the Chinese border. He grew up during the Soviet era, a period in which the Soviet Union maintained a strong tradition of mathematical education and research. The Soviet mathematical school, with its rigorous training programs and emphasis on problem-solving, provided a formative environment for Zelmanov's early intellectual development.

Details of Zelmanov's family background and childhood have not been widely documented in public sources. What is known is that he demonstrated mathematical aptitude from an early age and pursued advanced study in the subject. He eventually moved westward within the Soviet Union to attend university, enrolling at Novosibirsk State University, which was situated in Akademgorodok, the famous academic town in Siberia that served as a hub for Soviet scientific research. Novosibirsk State University had a distinguished reputation in mathematics and the natural sciences, and it was there that Zelmanov began his formal mathematical training.

The intellectual atmosphere of Akademgorodok, with its concentration of research institutes and universities, exposed Zelmanov to a community of leading Soviet mathematicians. This environment would prove instrumental in shaping his research interests, particularly in algebra, which would become the central focus of his career.

Education

Zelmanov received his undergraduate education at Novosibirsk State University, where he studied mathematics and developed a deep interest in algebraic structures. He subsequently pursued graduate studies at Leningrad State University (now Saint Petersburg State University), one of the most prestigious universities in the Soviet Union and a center of mathematical research with a long and distinguished tradition in algebra.

At Leningrad State University, Zelmanov completed his doctoral work, focusing on problems in algebra that would lay the groundwork for his later breakthroughs. His training at these two institutions gave him a thorough grounding in the methods of both nonassociative algebra and group theory, fields in which he would make his most celebrated contributions. The combination of the problem-solving tradition at Novosibirsk and the deep algebraic expertise available at Leningrad equipped Zelmanov with the technical tools necessary for tackling some of the most challenging open problems in algebra.

Career

Early Research and the Restricted Burnside Problem

Zelmanov's early research focused on problems in nonassociative algebra, particularly in the theory of Jordan algebras. His work in this area produced significant results that established his reputation within the algebraic community. However, it was his solution of the restricted Burnside problem that brought him to international attention and ultimately earned him the Fields Medal.

The Burnside problem, originally posed by William Burnside in 1902, asks whether a finitely generated group in which every element has finite order must necessarily be a finite group. This question, in its general form, was shown to have a negative answer through the construction of counterexamples by Evgeny Golod and Igor Shafarevich in 1964, and later by Pyotr Novikov and Sergei Adian. However, the restricted Burnside problem—which asks whether there are only finitely many finite groups with a given number of generators and a given exponent, up to isomorphism—remained open and was considered one of the central questions in group theory.

Zelmanov's approach to the restricted Burnside problem drew on techniques from the theory of Lie algebras and Jordan algebras, creating an unexpected bridge between nonassociative algebra and group theory. His proof, completed in the early 1990s, established a positive answer to the restricted Burnside problem for all prime-power exponents and, combined with earlier work by A. I. Kostrikin on the prime case, resolved the problem completely. The solution required deep and original methods that connected several areas of algebra in novel ways.

This achievement was recognized by the International Mathematical Union when Zelmanov was awarded the Fields Medal at the 1994 International Congress of Mathematicians in Zürich, Switzerland. The Fields Medal citation noted his contributions to the solution of the restricted Burnside problem and his work on combinatorial problems in nonassociative algebra and group theory.

Move to the United States

Following the dissolution of the Soviet Union, Zelmanov, like many other prominent Soviet mathematicians, relocated to the West. He accepted academic positions in the United States, beginning a career that would take him through several of the country's leading research universities.

Zelmanov held a professorship at the University of Wisconsin–Madison, where he continued his research in algebra. He subsequently moved to the University of Chicago, another institution with a strong tradition in mathematics. He later joined the faculty at Yale University, further establishing his presence in the American mathematical community.

University of California, San Diego

Zelmanov eventually joined the University of California, San Diego (UCSD), where he held a position in the Department of Mathematics for a number of years. At UCSD, he continued his research while also supervising doctoral students and contributing to the university's graduate program in mathematics. His presence at UCSD helped strengthen the university's profile in algebra and related areas of mathematics.

During his time at American universities, Zelmanov's research extended beyond the restricted Burnside problem to encompass a range of topics in algebra, including the theory of pro-p groups, Lie algebras, and infinite-dimensional algebras. He also contributed to the study of growth in groups and algebras, an area with connections to both pure algebra and other branches of mathematics.

Southern University of Science and Technology

In a significant career move, Zelmanov accepted a chair professorship at the Southern University of Science and Technology (SUSTech) in Shenzhen, China. SUSTech, a relatively young university founded in 2011, has sought to attract internationally recognized scholars to build its research capacity, and Zelmanov's appointment represented a notable success in this effort.

At SUSTech, Zelmanov has been active both in research and in teaching. In a 2022 interview with CGTN, he expressed his hope that more talented individuals would join the university, highlighting the institution's potential for growth in mathematics and the sciences. By 2025, at the age of 70, Zelmanov was reported to remain engaged in teaching undergraduate students in Shenzhen, reflecting a commitment to mathematical education at all levels.

Zelmanov's role at SUSTech has also positioned him as a bridge between the American and Chinese mathematical communities. He holds membership in both the United States National Academy of Sciences and the Chinese Academy of Sciences, a dual affiliation that reflects his engagement with mathematical institutions in both countries.

Public Lectures and Broader Engagement

In addition to his research and teaching, Zelmanov has been active as a public speaker and commentator on issues related to mathematics, science, and technology. In October 2025, he participated in a University Lecture Series event at the University of Macau, where he and other mathematicians discussed the essence and future of mathematics in modern society.

In November 2025, Zelmanov spoke at Beijing Jiaotong University, where he discussed mathematics' role in modern society and the importance of international scientific cooperation. In an interview with China News Network, he expressed confidence that China would produce Fields Medal winners in the future, citing the rapid development of Chinese mathematics.

Zelmanov has also weighed in on topics at the intersection of mathematics and technology. In an October 2025 interview reported by Network World, he commented on the state of quantum computing, stating: "As of today we have no quantum computer. It does not exist." He cautioned about what he characterized as hype surrounding quantum computing in the marketplace, drawing on his expertise in the mathematical foundations of cryptography.

Involvement in Wartime Education

In 2024, reporting by Frontline magazine highlighted Zelmanov's involvement in educational efforts related to the Russo-Ukrainian War. The article noted that Zelmanov's engagement with wartime education initiatives challenged conventional boundaries between science and politics, suggesting that scientists have a role to play beyond their research during times of conflict.

Personal Life

Zelmanov has maintained a relatively private personal life. Publicly available information indicates that he has held citizenship in both Russia and the United States, reflecting his background as a Soviet-born mathematician who built much of his career in the American academic system before later working in China.

His later career at SUSTech in Shenzhen indicates a willingness to engage with new academic environments and institutions. In interviews, he has spoken about the importance of mathematics education being accessible to all, not only to those perceived as geniuses, reflecting a belief in broad mathematical literacy. At the age of 70, as reported in 2025, he continued to be active in both teaching and research, maintaining a schedule that included undergraduate instruction as well as public lectures across multiple countries in Asia.

Recognition

Zelmanov's most prominent honor is the Fields Medal, awarded in 1994 at the International Congress of Mathematicians in Zürich for his solution of the restricted Burnside problem and his work on combinatorial problems in nonassociative algebra and group theory. The Fields Medal, often described as the equivalent of a Nobel Prize in mathematics, is awarded every four years to mathematicians under the age of forty who have made outstanding contributions to the field.

In 2011, the University of Alberta announced that Zelmanov would receive an honorary Doctor of Science degree, recognizing his contributions to mathematics.

Zelmanov has been invited to deliver distinguished lectures at numerous institutions worldwide. He was a visitor to the London Mathematical Society, an event noted in the society's newsletter, underscoring his standing in the international mathematical community. He was also a speaker in the Turán Memorial Lectures at the Alfréd Rényi Institute of Mathematics in Hungary, a lecture series named in honor of Pál Turán and reserved for mathematicians of the highest distinction.

In 2016, the University of Lincoln recognized Zelmanov's achievements, further adding to his list of institutional honors.

His membership in the United States National Academy of Sciences and the Chinese Academy of Sciences reflects recognition from the scientific establishments of two major nations.

Zelmanov was also noted in the Notices of the American Mathematical Society in 2001, which reported on his activities and contributions to the mathematical profession.

Legacy

Efim Zelmanov's solution of the restricted Burnside problem stands as one of the landmark achievements in twentieth-century algebra. The problem, which had resisted the efforts of group theorists for decades, was resolved through Zelmanov's innovative use of techniques from the theory of Lie algebras and Jordan algebras, demonstrating unexpected connections between different branches of algebra. This work not only answered a long-standing question but also opened new avenues of research in the study of groups, Lie algebras, and their interrelations.

Beyond the restricted Burnside problem, Zelmanov's contributions to the structure theory of Jordan algebras, pro-p groups, and infinite-dimensional Lie algebras have had a lasting impact on the field. His methods have influenced subsequent generations of algebraists and have been incorporated into the standard toolkit of researchers working in these areas.

Zelmanov's career trajectory—from the Soviet mathematical tradition through American research universities to a position at a rapidly developing Chinese institution—mirrors broader patterns in the globalization of mathematics during the late twentieth and early twenty-first centuries. His willingness to engage with emerging academic institutions, as demonstrated by his move to SUSTech, reflects a commitment to nurturing mathematical talent in new settings.

His public statements on topics such as quantum computing and mathematical education suggest an engagement with the broader implications of mathematical research for technology and society. His emphasis on the accessibility of mathematics—that it is "not only for geniuses, but for everyone"—reflects a pedagogical philosophy that extends beyond the research frontier to the cultivation of mathematical thinking in the next generation.

As a doctoral advisor, Zelmanov has trained a number of mathematicians who have gone on to make their own contributions to algebra and related fields, extending his intellectual influence through the mathematical community.