In his youth, Nikolai Alexandrovich wrote a short novel. By misfortune—or perhaps fortune—the text was lost, and literature never became his profession. Nevertheless, he wrote his academic works brilliantly. His pen was light and swift, and his texts radiated clarity and harmony. He employed a rich and generous language, unafraid of verbosity or adding extra touches of color. He spared no effort in providing explanations, eschewing brevity, yet often achieved extraordinary precision in his formulations. Over the years, his style became increasingly refined, and certain phrases seemed to leap onto the page on their own, guiding the reader toward the core ideas of each specific text. It is particularly painful now that the monograph N. A. Vavilov had planned and partially written, “The Theory of Chevalley Groups”, never came to fruition. With his encyclopedic knowledge and literary mastery, it would undoubtedly have stood alongside the classic works of Steinberg, Carter, and Springer. It simply didn’t happen. However, his series of textbooks for university undergraduates remains, presenting modern algebra in a unique and colorful manner: “Not Quite Naive Set Theory”, “Not Quite Naive Linear Algebra”, and “Concrete Group Theory”.

For N. A., clarity of exposition was not only a mathematical criterion but also an ethical measure of the correctness and significance of results.

5.1 Recollections by Alexei Stepanov

After Sasha Sivatski and I proved the boundedness of the lengths of commutators [a, b], where \(a\in GL _n(R)\) and \(b\in E _n(R)\) [47], Kolya decided to extend this result to all Chevalley groups instead of just the general linear group.

However, in our work with Sivatskiy, in addition to localization, we used the method of transvection decomposition, which is not applicable to all Chevalley groups. So, Kolya said, “If we can’t use transvection decomposition, we’ll use double localization.” After several discussions, I wrote a draft of the text. Kolya returned it to me, deeply disappointed. He said it couldn’t be written like that—it was as if there was no result at all because, even though everything was formally correct, it was completely unclear why it worked. At that point, we parted ways: Kolya focused on finding precise estimates for zero-dimensional rings, while I tried to simplify the proof.

A couple of years(!) later, we returned to my text, which I still hadn’t managed to simplify, though I could now explain the ideas to Kolya without the technical details. He was still dissatisfied, and we postponed writing the paper for another couple of years. Finally, I managed to write down the ideas I had been explaining to him. This time, he said: “Yes, now I see that it can’t be done any simpler.” He rewrote my entire text, inserting his own part, and the paper was ready.

Afterward, Kolya wrote several more papers, refining the technique of double localization to perfection and clearly explaining to all readers how it works and what kinds of problems it can solve.

5.1.1 Reminiscences by Vladimir Khalin

The productivity of Nikolai Alexandrovich was astounding: when inspired, he could write a fully-fledged, thirty-page article in just a few days, and in impeccable English. This is exactly how “The Skies are Falling: Mathematics for Non-Mathematicians” [87] was born. His preparatory materials for the course “Mathematics and Computers,” only a fraction of which made it into the concise book “Mathematica for Non-Mathematicians” [88], reportedly amounted to over a thousand pages, according to Vavilov himself!

To him, mathematics was “...the highest manifestation of human spirit and culture, valuable regardless of any applications.” This belief underscored the unique role of mathematics education in society, which he divided into three fundamentally different levels: pre-university, mathematics for mathematicians, and mathematics for non-mathematicians. Nikolai Alexandrovich believed that “...the most important aspect of teaching mathematics at the elementary level is fostering intellectual honesty: the ability to distinguish between what you understand and what you don’t; between what has a precise meaning and what doesn’t; between what is stated and what is implied; between the possible and the impossible; the true and the false; the proven and the conjectured.” Another equally important aspect was “mental gymnastics,” training the brain to tackle any challenging problems.

At the university level, different goals took precedence—foremost among them, cultivating the mathematical way of thinking: the ability to start from first principles, consider the simplest case, use analogies and metaphors, generalize and specialize, and so forth. Of course, this also included developing a genuine understanding of mathematics and practicing fundamental modes of reasoning.

Professor N. A. Vavilov was the author of a unique approach to teaching mathematics to non-mathematicians at the university level, leveraging symbolic computation systems and computer algebra. Nikolai Alexandrovich believed that “teaching mathematics should intrigue, captivate, and enchant” and advocated a new approach to mathematics education: delegating most routine calculations to computer algebra systems and focusing entirely on the conceptual side of mathematics, emphasizing the most important, useful, intriguing, and exciting layers of mathematics—concepts, ideas, analogies, constructions, and metaphors. He insisted that non-mathematicians should be “taught mathematics as we, mathematicians, understand it—by prioritizing UNDERSTANDING” [87].

Starting in 2005, N. A. Vavilov began teaching his signature course, “Mathematics and Computers,” at the Faculty of Economics of Saint Petersburg State University. The course focused on core mathematical ideas rather than specific applications. Here is how Nikolai Alexandrovich himself described this work: “Initially, we introduced new mathematical concepts and ideas, as well as formulated several key statements, sometimes with sketches of proofs. Full proofs were presented only when they were particularly concise and illustrative or contained powerful general ideas useful in many contexts. We then transitioned to algorithms, computer demonstrations, computations, graphics, etc. With active participation and interest from the students, we managed to cover significantly more mathematics—more diverse, interesting, and ultimately more useful mathematics—than would have been possible with a more traditional approach” [88].

Students greatly appreciated N. A. Vavilov’s approach, and even years later, they fondly remember the course. One of them wrote: “Mathematics and Computers was one of the subjects in the ‘Applied Informatics in Economics’ program at SPbU, and the corresponding book left a pleasant aftertaste. The concepts and algorithms described in the book remain highly relevant for solving any tasks requiring mathematics, which is indispensable for analysts or data professionals.” Another added: “Mathematica for Non-Mathematicians is the only university textbook I occasionally revisit even ten years after graduation.”

When it comes to mathematics teaching, N. A. Vavilov delivered an incredible variety of courses throughout his career. Here are just a few of them: “Algebraic Geometry”, “Lie Algebras and Groups”, “Algebraic Groups”, “Hopf Algebras and Galois Theory”, “Kac–Moody Algebras and Groups”, “Category Theory”, “Central Simple Algebras”, “Computer Algebra”, “Non-Commutative Rings”, “Representation Theory of Finite Groups”, “Finite Groups of Lie Type”, “Modular Representations of Finite Groups”, “Unipotent Decompositions”, “Exceptional Objects in Algebra and Geometry”, “Cartan-Type Algebras”, and “The Jacobian Problem”. This list could go on, as it would be easier to name areas of algebra he did not lecture on than to compile a complete list of his courses. Without delving into the specifics of his teaching, one thing stands out: he was a MASTER. He not only delivered material but also had the rare ability to identify those select students who developed a true sense of the subject. These students would later write their theses under his supervision, defend their dissertations, and become researchers—all of which began with his undergraduate algebra courses.

But it did not end there. Every scholar whose name is associated with a school of thought has their own working secrets. For instance, Niels Bohr claimed his success lay in “never being afraid to tell his students he was a fool”. Conversely, Lev Landau “never hesitated to tell his students that they were fools”. N. A. Vavilov’s signature approach was the ability to say, “We”. We can do it, we will prove it, we will undoubtedly achieve the best result, we will write this theorem for all Chevalley groups, and so on. In most cases, this was only *partially* true. But the secret was that he always spoke with conviction. There were no “halves” in his statements. Later, something specific might emerge, but by then, it hardly mattered. He had a remarkable intuition for knowing what to say, when to say it, and to whom.

Vavilov’s student, V. Nesterov, writes: “One of N. A.’s traits during discussions was his ability to strongly motivate others to tackle the given problem. He would captivate you with the problem, highlight its potential, and often point to overarching goals that could be achieved in the future. With his profound knowledge of the history of mathematics, N. A. often cited fascinating and inspiring examples.” And here are observations from A. Luzgarev:

5.1.2 Recollections by Alexander Luzgarev

Nikolay Alexandrovich started teaching us algebra in the second semester, at the beginning of 2000. I was astonished: he delivered an extraordinary amount of information, covering completely different areas of mathematics. It seems he began with the definition of a group, and by the second lecture, Lie algebras had already appeared. We probably understood only a fraction, but we felt his erudition and the broad scope of his material; and I think it was thanks to him that I first began to understand the unity of mathematics. Later, I realized that this was an important part of his method. I remember him saying: “Some believe you should only say things that a person can understand, using words they already know. By this logic, you shouldn’t talk to a baby at all.”

A couple of years later, I started attending his specialized courses and seminars, and he always encouraged even junior students to attend talks where they had no chance of understanding anything. According to Nikolay Alexandrovich, learning mathematics, like learning a foreign language, should happen through “immersion in the environment”. He said: “You listen, and some words, concepts are repeated many times and settle into your subconscious; after a while, you are no longer afraid of them, and a bit later, you suddenly start to understand everything.”

N. A. loved working outside of classrooms. During walks, in cafes, or over friendly conversations, he could endlessly veer into his favorite topics, and everyone who worked with him knew this well. But when the blend of cultural and culinary musings was exhausted, the time for mathematics would arrive, and all the prior prelude suddenly seemed perfectly fitting. Alexander Luzgarev writes:

By the third or fourth year of university, some of the classes on the Faculty of Mathematics and Mechanics were held on Kamskaya Street, near the Smolensk Cemetery. It just so happened that my classes often ended at the same time as the lectures Nikolay Alexandrovich gave to the younger students. We would walk together from Kamskaya Street to the Vasileostrovskaya metro station and then take the metro—he would go to Sennaya Ploshchad, while I continued further to Baltiyskaya.

This happened once or twice a week for several months. During the walks, he would constantly talk: about mathematics, history, or details of his meetings with foreign mathematicians. I had no idea what to say; it was practically a monologue. Soon, Igor Pevzner and I asked him for topics for term papers, then for our theses, and later we entered graduate school and defended our dissertations—all while attending almost every specialized course and seminar Nikolay Alexandrovich taught.

But what had the greatest influence on me were those very first algebra lessons with Nikolay Alexandrovich in my first year, and those frequent walks with him around Vasilievsky Island.

In time, numerous recollections of N. A. by his friends, colleagues, and students will undoubtedly be compiled. Let us include here an excerpt from the vivid memories of N. A. Vavilov’s student, Viktor Petrov. These recollections are also dedicated to the distinctive style of teaching mathematics, in which N. A. Vavilov was a virtuoso.

5.1.3 Viktor Petrov recalls

“The classes were conducted in an absolutely inimitable style. Algebra practice, of course, implies solving computational problems, but perhaps only about one-tenth of the time was dedicated to that. The main focus was on enlightenment in the broadest sense of the word. And not just in mathematics, but also in linguistics (the actual number of cases in the Russian language, Indo-European languages, the Nostratic theory, the structure of Chinese characters...), and in philosophy (Nikolai Alexandrovich’s favorite work was the treatise “Zhuangzi”, and his favorite quote from it was: “A white horse is not a horse, an abelian group is not a group”). The structure of his exposition most resembled Borges’ “The Garden of Forking Paths”: having mentioned a concept or answered a question, Nikolai Alexandrovich would immediately start discussing a new topic. For instance, the definition of a maximal ideal would come up—and immediately we’d hear about Stone’s theorem on Boolean algebras, ultrafilters, hyperreal numbers, model theory... His delivery was emotional; Nikolai Alexandrovich gesticulated, used intonation, and employed tautological phrases for emphasis (“Exactly, precisely this way”). It was evident that he was overflowing with knowledge and impressions he wanted to share, because, as the saying goes, “Out of the abundance of the heart, the mouth speaks.”

Fig. 5

Luminy, Marseille, 2015. N. Vavilov and V. Petrov

The remarks of N. A. Vavilov’s closest student, Alexei Stepanov, resonate in harmony.

5.1.4 Alexei Stepanov recalls

In the late 1990 s and early 2000 s, Kolya and I were in Bielefeld. I was going through one of my creative slumps, which Kolya decided to end by proving a result with me that, as he assured me, would definitely work. We met in his office, and Kolya confidently declared that we were going to prove that the arrangement of subgroups between \(E _n(R)\) and \(GL _n(F)\) is standard, where F is the field of fractions of the ring R. “Well, at least for factorial rings, we’ll prove this without any problems,” he assured me.

We began with the case of a polynomial ring \(R=K\hspace[x,y]\) over a field K. Using the usual method of transvection decomposition, we quickly derived some transvections from an arbitrary matrix in \(GL _n(F)\). Kolya said, “See, just a little more, and we’ll prove everything.” However, I was slightly unnerved by the fact that in all the transvections we extracted, the numerators contained more variables than the denominators. That is, if the matrix consisted of rational functions of degree 0 (i.e., the degree of the numerator equaled the degree of the denominator), we could only extract rational functions of positive degree. That’s where we left it.

After thinking it over at home, I realized that if we took the ring \(A=K\hspace[x,y,y/x]\subseteq F\) and factored it by the ideal generated by x and y, we would obtain a ring isomorphic to K[z], while R would transform into K.

Disheartened, I shared this with Kolya. He said, “Even better—proving the result for the pair \(R\subseteq F\) will automatically yield the result for the pair \(K \subseteq K[z]\).” When I asked whether he believed in the latter result, he replied, “Not really, but it’s definitely true for the pair \(R \subseteq F\)!”

The contradiction was resolved only a couple of years later, when I proved that for the pair \(K\subseteq K[z]\), there is no standard arrangement of subgroups, and hence none for the pair \(R\subseteq F\) either. I must say, the ideas used in this work ended my slump, so Kolya’s original goal was achieved. In this way, Nikolai Alexandrovich’s scientific optimism often served as a serious stimulus for my work.

All of Nikolai Alexandrovich’s students have their own stories about his scientific generosity.

Sometime around 2005–2007, I found myself in a mild depression. It was a slightly delayed midlife crisis, compounded by the need to write up results on subring subgroups. I lacked the necessary knowledge and found the task uninteresting because, ideologically, everything was already clear and (in my opinion) beautiful, but technically challenging.

And, as usual, I was broke. Actually, worse than usual because the invitations to Bielefeld had stopped (their SFB funding had ended, or something like that).

Then, in 2007, Kolya came up to me and said: “I’ve written our joint article. Take a look—maybe you’ll add something or suggest corrections”.

I was stunned and asked, “What joint article?”

He replied, “Well, don’t you remember? We discussed the generalized commutator formula in your office in Bielefeld. So, I’ve written down the results of our discussion. Everything worked out just as you said.”

I vaguely remembered the discussion, but I had no memory of contributing anything substantive.

However, no amount of persuasion that this article should have a single author had any effect. So, I fixed a couple of typos and agreed. The article was published in the *Vestnik of St. Petersburg State University* under the title “Standard Commutator Formulas”.

A couple of years later, another article, “Once More on the Standard Commutator Formula”, was published with roughly the same level of my involvement.

In addition to his generosity, Kolya had a healthy dose of scientific pragmatism. Many people probably remember this period—it was a time when Russian universities almost stopped paying salaries, and we survived on grants, primarily foreign ones.

To secure these grants, it wasn’t enough to achieve good results; you needed a lot of publications. Regarding this, Kolya used to say, “Every day—an observation, three observations—a lemma, three lemmas—a theorem, three theorems—an article. That’s how you work!”

Of course, it was a joke. But Kolya wrote many articles, which secured grants that enabled many of his students and colleagues to pursue science without going hungry.

5.1.5 Vladimir Nesterov remembers

In 1991 (in my fourth year), I approached N. A. Vavilov to ask for a diploma project. N. A. offered me a choice of three problems. I intuitively chose the task of describing subgroups generated by pairs of short root unipotent subgroups in a Chevalley group. As far as I know, at that time, N. A. had two graduate students: the Bulgarian Alan Kharebov (they co-authored one paper) and another woman whose name I cannot recall.

In the spring of 1992, I defended my diploma, and in the summer of 1992, Vavilov left for Europe, and I didn’t see him again until 2001. We corresponded by mail. In 1994, I sent him the text of my dissertation and a paper for a preprint, which was later published in Bielefeld. I must say that N. A. did an enormous amount of work correcting my paper, not only adding deep remarks to the introduction but also refining the presentation and improving the English.

Thanks to N. A., the journal version of the paper was published in *Doklady RAN* in 1995. N. A. wrote all sorts of glowing reviews (clearly exaggerated), which allowed me to secure grants for several consecutive years. N. A. also made great efforts to ensure I could attend a school-conference on Crete in 1995, where my trip was fully funded. In the early 2000 s, I finally published the results from my dissertation, and N. A. submitted a grant proposal for further research on toral generation. We were awarded a substantial grant.

Here’s an interesting story. In 2016 or 2017, Nikolai Alexandrovich and I met at the cafe *Sladkoezhka* on Marata Street. After discussing scientific matters, N. A. told me about the dialects of the Spanish language, and we headed home. But in the metro, I realized that my wallet had been stolen at the cafe. N. A. immediately called home and said he would be delayed. We went together to the police station, where we waited in line for over half an hour. N. A. stayed until I finished writing a statement, and only then did we part ways.

It so happened that much of my learning from N. A. occurred remotely. Yet, I began to truly understand many of my works only thanks to certain phrases N. A. included in his feedback or suggested adding to the introduction. It’s amazing how just a few words can deepen your understanding and allow you to see things in a broader context.

Under N. A.’s guidance, I initially worked on generation by short root unipotent subgroups in Chevalley groups. At his suggestion, I described subsystem subgroups in the group of type \(F_4\), generated by short root subgroups. We then shifted to generation by microweight tori in Chevalley groups. Together, we developed a reduction theorem and described subgroups generated by pairs of microweight tori in \(GL (5)\) and \(GL (6)\), which do not embed into \(GL (4)\). Currently, my student from China is working on the general and most challenging case of \(GL (4)\).”