叶菲姆·泽尔曼诺夫Efim Zelmanov

叶菲姆·泽尔曼诺夫 就读于新西伯利亚国立大学，1977 年获得硕士学位。获得该学位后，他被任命为新西伯利亚国立大学的教职员，在那里任教同时继续自己的研究。1980 年，他在新西伯利亚国立大学获得博士学位，其研究由 Shirshov 和 Bokut 指导。

他为博士学位提交的论文是关于非结合代数的。特别是他的工作通过将有限维 帕斯库尔·约尔丹 代数的经典理论结果推广到无限维 帕斯库尔·约尔丹 代数，完全改变了整个 帕斯库尔·约尔丹 代数主题。泽尔曼诺夫 在 1983 年华沙国际数学家大会的邀请演讲中描述了他在 帕斯库尔·约尔丹 代数上的这项工作。

1980 年，泽尔曼诺夫 被任命为新西伯利亚 苏联科学院 数学研究所的初级研究员。1985 年获得博士学位（教授资格论文（Habilitation））后，他被提升为高级研究员。1986 年，他在 苏联科学院 数学研究所再次晋升，这次成为首席研究员。

1987年，泽尔曼诺夫解决了李代数理论中的一个重大未决问题。他证明了Engel恒等式

$ad(y)^{n} = 0$

蕴含该代数必然是幂零的。这对有限维李代数是经典结果，但泽尔曼诺夫证明该结果对无限维李代数也成立，从而解决了一个重大未决问题。

1990年，泽尔曼诺夫被任命为美国威斯康星大学麦迪逊分校的教授。他一直担任此职至1994年，随后被任命到芝加哥大学。1995年，他在耶鲁大学度过了一年。

上述关于帕斯库尔·约尔丹代数和索菲斯·李代数的结果本可以确保泽尔曼诺夫作为20世纪伟大代数学家之一的地位。然而，在1991年，泽尔曼诺夫继续解决了群论中最基本的结果之一，这个问题在整个20世纪一直困扰着群论学家。他解决了受限伯恩赛德问题。

1994年，泽尔曼诺夫因这项工作在1994年于苏黎世举行的国际数学家大会上被授予菲尔兹奖。让我解释一下受限威廉·伯恩赛德问题的背景，该问题的解决是获得该奖章的主要原因，并解释一下并非群论学家出身的泽尔曼诺夫是如何解决群论中最基本问题之一的。

1902年，威廉·伯恩赛德首次提出，一个有限生成群，如果其中每个元素都有有限阶，那么它是否是有限的。这个问题被称为一般伯恩赛德问题。伯恩赛德问题问的是，对于固定的$d$和$n$，具有$d$个生成元且其中每个元素都满足$x^{n} = 1$的群$B(d, n)$是否是有限的。证明$B(d, 2)$是有限的真的很容易。威廉·伯恩赛德本人证明了$B(d, 3)$是有限的，Sanov证明了$B(d, 4)$是有限的，而马绍尔·哈尔证明了$B(d, 6)$是有限的。

到20世纪30年代，这两个问题都没有取得实质性进展，受限伯恩赛德问题被提出（并由威廉·马格努斯如此命名）。它问的是，对于固定的$d$和$n$，是否存在一个最大的有限$d$生成群，其中每个元素都满足$x^{n} = 1$。这等价于说，受限威廉·伯恩赛德问题的肯定解决将表明只有有限多个有限的factor groups的$B(d, n)$。

一般威廉·伯恩赛德问题在1964年被Golod证明有否定解。1968年，彼得·诺维科夫和谢尔盖·阿迪安表明威廉·伯恩赛德问题对于大的$n$是错误的。对受限威廉·伯恩赛德问题最大的早期贡献是1956年由菲利浦·霍尔和格雷厄姆·希格曼做出的，他们表明，如果Schreier猜想成立，那么受限伯恩赛德问题如果能对所有prime幂$n$证明，就有肯定解。Schreier猜想，即有限单群的外自同构群是soluble，作为有限单群分类的结果被证明是正确的。

威廉·马格努斯 已将限制性 威廉·伯恩赛德 问题在 $n$ 为素数时的情形归结为一个关于满足 Engel 条件的李代数是否局部幂零的问题。Kostrikin 在 1959 年证明了这类李代数确实是局部幂零的。然而 Kostrikin 的证明并不完全令人满意，修正后的版本很久以后才出现。

泽尔曼诺夫 开始研究限制性 威廉·伯恩赛德 问题时，要把在 $n = p$ 情形下已取得的成果推广到 $n = p^{k}$ 情形，存在两大困难。首先，问题尚未化归为具有 Engel 条件的 索菲斯·李 代数。这一点由 泽尔曼诺夫 在 1989 年实现。

泽尔曼诺夫接下来着手证明满足Engel条件的索菲斯·李代数是局部幂零的。他在两篇论文中实现了这一点，第一篇处理奇素数特征，第二篇处理对应于特征2李代数的$n = p^{k}$。Shalev在[3]中写道：-

他令人惊叹的证明……将惊人的技术能力与来自各个学科的高度原创思想结合在一起。该证明使用了McCrimmon和泽尔曼诺夫先前发展的（二次）帕斯库尔·约尔丹代数的深刻结构理论，以及分次幂和其他工具；它还依赖于Kostrikin和泽尔曼诺夫的联合工作，该工作确立了所谓三明治代数的局部幂零性。虽然索菲斯·李代数长期以来被视为限制伯恩赛德问题背景下的天然试验场，但帕斯库尔·约尔丹代数的出现是前所未有的，且相当令人惊讶。

1993年在爱尔兰戈尔韦举行的群-圣安德鲁斯会议上，我[EFR]是联合组织者之一，泽尔曼诺夫是主要演讲者之一，他就Nil rings methods in the theory of 幂零群做了五场系列讲座。他的讲座结构精美，清晰典范，展示了已取得的成就，并呈现了未来研究可能方向的许多线索。讲座充满幽默，全部以泽尔曼诺夫眼中那种富有感染力的闪亮神采来讲述。

除了 约翰·查尔斯·菲尔兹 奖章外，泽尔曼诺夫 还因其杰出工作获得了其他荣誉。他于 1992 年 1 月获得法兰西学院奖章，并于 1996 年 5 月获得 Andre Aizenstadt 奖。

泽尔曼诺夫于1995年至2002年在耶鲁大学担任教授。在耶鲁的最后一年，他当选为美国国家科学院院士，成为该院数学部最年轻的院士。2002年，他离开耶鲁，前往加利福尼亚大学圣迭戈分校，被任命为丽塔·L·阿特金森数学讲席教授。在他被任命时，加利福尼亚大学圣迭戈分校数学系主任詹姆斯·邦奇说：-

泽尔曼诺夫是世界顶尖数学家之一，他将在提升加利福尼亚大学圣迭戈分校数学系的国际声誉和卓越传统方面发挥重要作用。他也是一位杰出的教师，我期待他成为我们学生的非凡榜样。

加利福尼亚大学圣迭戈分校物理科学部副院长、前数学系主任杰弗里·雷梅尔将泽尔曼诺夫招至加利福尼亚大学圣迭戈分校，他说：-

泽尔曼诺夫在加利福尼亚大学圣迭戈分校的存在确保我们拥有全国领先的代数和表示论研究团队之一。此外，他是一位极出色的讲师和学位论文导师。他将帮助数学系吸引代数学领域最优秀的年轻研究人员和研究生，并将对加利福尼亚大学圣迭戈分校下一代数学学生产生深远影响。

泽尔曼诺夫在加利福尼亚大学圣地亚哥分校占据的办公室，与早先由菲尔兹奖章获得者丘成桐和迈克尔·弗里德曼占据的办公室相同。

然而，泽尔曼诺夫 对数学的贡献远不止他卓越的研究和教学成就，他还担任十多种主要数学期刊的编辑或编委会成员，其中包括 The Annals of Mathematics、The Journal of Algebra 和 The Journal of the American Mathematical Society。

Efim Zelmanov attended Novosibirsk State University, obtaining his Master's degree in 1977. On being awarded this degree he was appointed to the staff at Novosibirsk State University and taught there while continuing with his own research. He received his Ph.D. from Novosibirsk State University in 1980 having had his research supervised by Shirshov and Bokut.

The thesis he presented for his Ph.D. was on nonassociative algebra. In particular his work completely changed the whole of the subject of Jordan algebras by extending results from the classical theory of finite dimensional Jordan algebras to infinite dimensional Jordan algebras. Zelmanov described this work on Jordan algebras in his invited lecture to the International Congress of Mathematicians at Warsaw in 1983.

In 1980 Zelmanov was appointed as a Junior Researcher at the Institute of Mathematics of the USSR Academy of Sciences at Novosibirsk. On the award of his doctorate (habilitation) in 1985, he was promoted to Senior Researcher. He was promoted again at the Institute of Mathematics of the USSR Academy of Sciences in 1986, this time becoming a Leading Researcher.

In 1987 Zelmanov solved one of the big open questions in the theory of Lie algebras. He proved that the Engel identity

$ad(y)^{n} = 0$

implies that the algebra is necessarily nilpotent. This was a classical result for finite dimensional Lie algebras but Zelmanov solved a big open problem when he proved that the result also held for infinite dimensional Lie algebras.

In 1990 Zelmanov was appointed a professor at the University of Wisconsin-Madison in the United States. He held this appointment until 1994 when he was appointed to the University of Chicago. In 1995 he spent the year at Yale University.

The results mentioned above on Jordan algebras and Lie algebras would have guaranteed Zelmanov a place as one of the great algebraists of the 20th century. However, in 1991, Zelmanov went on to settle one of the most fundamental results in the theory of groups which had occupied group theorists throughout the 20th century. He solved the restricted Burnside problem.

In 1994 Zelmanov was awarded a Fields Medal for this work at the International Congress of Mathematicians in Zürich in 1994. Let me explain the background to the restricted Burnside problem, the solution of which was the main reason for the award of the Medal, and also explain how Zelmanov, not a group theorist by training, came to solve one of the most fundamental questions in group theory.

In 1902 Burnside first asked whether a finitely generated group in which every element has finite order, is finite. This problem is known as the General Burnside problem. The Burnside problem asks whether, for fixed $d$ and $n$, the group $B(d, n)$ having $d$ generators and in which every element satisfies $x^{n} = 1$, is finite. It is really easy to show the $B(d, 2)$ is finite. Burnside himself showed that $B(d, 3)$ is finite, Sanov showed $B(d, 4)$ is finite and Marshall Hall showed $B(d, 6)$ is finite.

By the 1930s no real progress had been made on either of these problems and the Restricted Burnside problem was formulated (and so named by Magnus). It asks whether, for fixed $d$ and $n$, there is a largest finite $d$ generator group in which every element satisfies $x^{n} = 1$. This is equivalent to saying that a positive solution to the Restricted Burnside problem would show that there are only finitely many finite factor groups of $B(d, n)$.

The General Burnside problem was shown to have a negative solution by Golod in 1964. In 1968 Novikov and Adian showed that the Burnside problem was false for large $n$. The greatest early contribution to the Restricted Burnside problem was by Hall and Higman in 1956 where they showed that, if the Schreier conjecture holds, then the Restricted Burnside problem has a positive solution if it could be proved for all prime powers $n$. The Schreier conjecture, that the outer automorphism groups of finite simple groups are soluble, was shown to be true as a consequence of the classification of finite simple groups.

Magnus had reduced the case of the Restricted Burnside problem for $n$ prime to a question about whether Lie algebras satisfying an Engel condition are locally nilpotent. Kostrikin, in 1959, proved that such Lie algebras were indeed locally nilpotent. However Kostrikin's proof is not entirely satisfactory and a corrected version only appeared much later.

When Zelmanov began to work on the Restricted Burnside problem there were two major difficulties in pushing what had been achieved for $n = p$ to $n = p^{k}$. Firstly, there was no reduction of the problem to Lie algebras with the Engel condition. This Zelmanov achieved in 1989.

Zelmanov next set about proving that a Lie algebra with an Engel condition was locally nilpotent. This he achieved in two papers, the first dealing with odd prime characteristic and the second dealing with $n = p^{k}$ which corresponds to Lie algebras of characteristic 2. Shalev writes in [3]:-

His stunning proof ... combines an amazing technical capability with highly original ideas from various disciplines. The proof uses a deep structure theory for (quadratic) Jordan algebras, previously developed by McCrimmon and Zelmanov, as well as divided powers and other tools; it also relies on the joint work of Kostrikin and Zelmanov, which establishes the local nilpotency of the so-called sandwich algebras. While Lie algebras have long been considered a natural playground in the context of the Restricted Burnside problem, the appearance of Jordan algebras is unprecedented and quite surprising.

At the Groups-St Andrews conference at Galway, Ireland in 1993, of which I [EFR] was a joint organiser, Zelmanov was one of the main speakers and he gave a series of five lectures on Nil rings methods in the theory of nilpotent groups. His lectures were beautifully constructed, models of clarity, showing what had been achieved and presenting many glimpses of possible directions for future research. Filled with humour, they were all delivered with Zelmanov's infectious twinkle in his eyes.

In addition to the Fields Medal, Zelmanov has received other honours for his outstanding work. He received the Collège de France Medal in January 1992 and the Andre Aizenstadt Prize in May 1996.

Zelmanov held a professorship at Yale University from 1995 to 2002. During his last year at Yale he was elected to the National Academy of Sciences, becoming the youngest member in the Academy's mathematics division. In 2002 he left Yale and went to the University of California, San Diego, where he was appointed to the Rita L Atkinson Endowed Chair in Mathematics. At the time of his appointment, James Bunch, chair of mathematics at the University of California, San Diego, said:-

Professor Zelmanov is one of the top mathematicians in the world and he will play an important role in furthering the international reputation and tradition of excellence of UCSD's mathematics department. He is also an outstanding teacher and I expect him to be an exceptional role model for our students.

Jeffrey Remmel, associate dean of the UCSD's Division of Physical Sciences and the former chair of mathematics who recruited Zelmanov to UCSD said:-

Professor Zelmanov's presence at UCSD ensures that we have one of the leading research groups in algebra and representation theory in the country. In addition, he is a superb lecturer and thesis advisor. He will help the mathematics department attract the best young researchers and graduate students in the field of algebra and will have a profound effect on the next generation of mathematics students at UCSD.

Interestingly, Zelmanov occupied the same office at the University of California, San Diego, as had earlier been occupied by Fields Medal winners Shing-Tung Yau and Michael Freedman.

Zelmanov's contribution to mathematics goes far beyond his remarkable research and teaching achievements, however, being an editor or on the editorial board of more than ten major mathematics journals, including The Annals of Mathematics, The Journal of Algebra and The Journal of the American Mathematical Society.