Translated material
Sergey Lando: "Breakthroughs in mathematics are poorly predictable"
Sergey Lando, Doct. physical. Sciences, professor of the Faculty of Mathematics of the Higher School of Economics, stood at the origins of the Faculty of Mathematics and performed the duties of the dean from the moment the faculty was created in 2007 until the spring of 2015. Lyudmila Sapchenko asked Sergei Konstantinovich about his scientific activity, about what place the mathematical sciences in the modern world occupy, how the faculty was created, what tasks are set before the faculty at present.
- What, in your opinion, are the reasons for increasing interest in mathematics as a science in the modern world?
- First of all, I would like to understand whether such a phenomenon as an increase in interest in mathematics as a science is really observed. It seems that it is, because my conversations, say, with colleagues from the Netherlands demonstrate that there are much more local students, not foreigners who want to study mathematics, up to the point of new faculties, and the number of professor positions is being opened to the universities of this country, not to study mathematics. This growth is quite noticeable in recent decades, so, perhaps, I agree that there is an increase in interest in mathematics.
I can only comment on the reasons for Russian universities. In the last ten years, new interesting young professors who make science appeared at Russian universities. The students who received them also perceived the professor’s interest, then by the chain it spread to future applicants who learned that there are people who want and are able to teach, and that this area of activity could be quite promising.
The development of computer sciences, which are quite close to mathematics and in which mathematical methods are used to a large extent, played a significant role. In addition, in my opinion, it came - although it did not fully penetrate into society - the realization that mathematical education at the university level can give a life perspective and tools for working in various fields, where not so much mathematical theorems are in demand as mathematical methods and a mathematical way of thinking.
This is a fairly wide range of areas, including, in addition to computer sciences and programming, an applied economy, banking and insurance business, which was underdeveloped in Russia and is now rapidly developing, technical activities - up to the design of aircraft and cars. So, I think, this awareness gradually comes to society.
- What mathematical problems are most relevant for Russian and foreign mathematicians ?
- I'm not sure that the term “relevance” is applicable to fundamental scientific problems. Mathematics-researchers usually reflect on the tasks to which they have naturally led the previous research activity, so the relevance of mathematical problems is an individual concept. Another thing is that if someone manages to think of the principal things and make a breakthrough in the tasks that a person worked on, then such a breakthrough attracts the attention of a large number of researchers around the world working on similar problems. At this moment, an outbreak of interest occurs, the amount of work in the area where the breakthrough was performed, and growing an avalanche. However, we can fix the flash, as a rule, a post -fact, when a breakthrough has already perfect and fundamentally new methods have appeared, which perhaps even return to the old classical tasks, to the solution of which there were no obvious approaches.
To predict in advance where this breakthrough will happen and the more so what it will consist of is an ungrateful matter. There are some common considerations that work as a whole, but do not give accurate answers. They allow one way or another to outline the area in which such breakthroughs can be expected, but this area is usually so wide that such a prediction is devoid of practical interest.
At the same time, it is clear that, say, the use of algebraic geometry methods, differential geometry methods in theoretical physics, which has developed violently in recent decades, has not exhausted itself. There will undoubtedly be breakthroughs, but this area, again, is so wide that it is difficult to promise something specific here. Breakthroughs turn out to be breakdowns because they are poorly predictable.
- What are the features of the Russian mathematical school, if any?
-Apparently, a single Russian mathematical school does not exist, and it is unlikely to ever exist, although more local schools in certain areas of mathematics undoubtedly developed. For example, the Moscow Mathematical School, which was primarily implemented at the Mechanical and Mathematics Faculty of Moscow State University in the 1960s, is a concept that is quite entitled to exist. It does not break up into many individual schools, although it was created by several extremely bright mathematicians, each of which has its own students and their followers, but this is really something one and rather integral.
In general, the origin of Russian mathematical schools, their strength is explained in a sense of the weaknesses of the organization of science in the Soviet Union. This weakness is associated with low mobility of the entire population, including scientists; with all kinds of restrictions in hiring; excessive centralization of outstanding researchers in a very small number of places. But the possibility for several decades in a row of the current circle of people to discuss the same issues gave a serious impulse for the development and deep penetration into a number of areas of mathematical activity.
The principles that guided the leaders of mathematical schools are very significant here: to perceive mathematics as a whole, without closing in the framework of a narrow specialization, and in every possible way stimulate the work on the boundaries between different areas of science. And the border zones are the most favorable soil for making breakthroughs.
- Tell us about your research. What are you working on now?
- I do not think that my recent studies deserve such serious attention - simply because for eight years I was the dean, and, of course, my main efforts concentrated on the organization of the faculty, and not on the research process. After I had folded the dean duties a year and a half ago, I am making efforts to return to the research environment, to those tasks that have remained unresolved. I continued to reflect on them throughout my administrative period, but due to objective circumstances not very intensively.
These tasks relate to attempts to establish a connection between the two theories - the theory of invariants of the final order of nodes and hooks that have a material nature, and the complex of Gromov -Witten invariates in nature. The first of these theories was initiated by Viktor Anatolyevich Vasiliev in the late 1980s, the second one several years older. There are many indirect evidence in favor of the internal proximity of these two theories, but the exact statements are fragmented and do not form a whole picture.
- How in demand are graduates of the faculty of mathematics in the labor market?
- Judging even by a fluent analysis of the future life of our graduates, it seems to me that the goals that were set when creating the faculty have been largely achieved. We did not at all strive to ensure that all our graduates or at least their lion's share became mathematicians. It is important for us that they further find life paths in which knowledge of mathematics and mathematical methods helps them. On the other hand, it is significant that there are enough graduates in science to ensure the ability of our area of activity to self -development.
Both, in my opinion, is happening. A significant part of the graduates of our undergraduate enters the mathematical magistracy and graduate school - both our faculty and other best universities in the world. At the same time, many, including strong, graduates who are interested in doing other things go to such directions as mathematical biology, linguistics, economics, information technologies. It can be seen that the knowledge and experience acquired by the children are not just deposited by them just in case, but really help them in further life. Employers are not disappointed in the fact that they took our graduates, because these are people who know and trained to work, and, most importantly, they are interested in it.
- What does mathematical education give a person?
- It depends on what level of mathematical education we are talking about. If we are talking about school mathematical education, then this is one thing, and if we are talking about higher mathematical education, then I would single out the need to get to the bottom and highlight the significant that is in every subject and that plays a key role in solving a particular problem. This is a difficult skill, but it is vital in various fields of activity, including in everyday, domestic existence.
The development of mathematics at the university level is designed not to teach to reproduce proof of theorems or come up with new theorems (this is a lesson for a small number of graduates), how much to see this very essence. This can be learned on the most diverse material, but mathematics in a sense is especially good for this. Various systems for teaching mathematics, if they do not roll into formalism and do not strive to darken this essence, very beneficial affects the development of a person, as practice shows.
- This year you were the chairman of the organizing committee of the Moscow Olympiad of schoolchildren. What are your impressions?
- School mathematical Olympiads are a very big and important thing. Their organization is a goal passing through the red thread through all the structures of school Olympiads-the excitement in schoolchildren of interest in mathematics through beautiful tasks, beautiful combinations, beautiful solutions, that is, the report of the beauty that once led to mathematics. Until the sports start in the Olympics does not prevail when interest in the task, the desire to understand how it is solved, dominates the desire to speak better than an opponent, until this moment the Olympiad play a significant and positive role.
With the Moscow Mathematical Olympiad, I have very long-standing connections, I began to participate in its implementation, in my opinion, in 1973, that is, more than forty years ago. For several years I was organizing quite tightly and was one of the most active members of the organizing committee. This Olympiad, in my opinion, retained the main virtues and features laid down by our predecessors - those who came up with the Moscow Olympiad in ten years before us. It is very important that everyone is allowed to participate, all schoolchildren, regardless of whether they had any victories at other Olympiads.
This year, the Moscow Mathematical Society, which traditionally offers the candidacy of the chairman of the organizing committee, nominated me for this role. This is a very honorable task; It was interesting to give a lecture to schoolchildren at the closure of the Olympiad, to interact with the guys who come up with tasks. As a rule, these are young people, students. It can be seen that they are important and interesting what they do. I was happy to return to this atmosphere. In general, it is, although time -consuming, but a useful, significant and fascinating business.
- Who first expressed the idea of creating the Faculty of Mathematics of the Higher School of Economics? In what composition was this discussed?
- I do not know who generated the initial idea, but in the spring of 2007, the leadership of the Higher School turned to the leadership of an independent Moscow University (NMU) with a proposal to transform the university into the faculty of mathematics of the Higher School of Economics. A meeting took place, at which the first vice -rector Vadim Valerievich Radaev and Vice -Rector Sergey Yuryevich Roshchin participated from the HSE. NMU, of course, was represented by the rector Yuli Sergeyevich Ilyashenko and, as I recall, all the vice -rectors - Alexei Bronislavovich Sosinsky, Mikhail Anatolyevich Tsfasman, Viktor Vladimirovich Furin and me. At this meeting, the question was first raised about the creation of the faculty, and the leadership of the NMU began to ponder this proposal.
-In connection with the accession to the tower, some fears arose?
- The reputation of HSE in the educational community at that time was ambiguous and remains that until today. The tower belongs and attributes various initiatives in the field of secondary and higher education, as well as the reorganization of science, which many members of the educational community are considered as an extremely negative impact on the entire system of education and science in Russia. The fears of the NMU leadership were associated with this negative connotation, and it took several months to clarify the reasons for this attitude and understanding of its sources. Based on the results of the work, we came to the conclusion that the prevailing opinion for the most part does not have objective grounds. After that, it was decided to cooperate with the HSE in the creation of the new faculty. At the same time, it seemed to us important to maintain the independence of NMU.
“Did you worry about the future fate of NMU?”
- It was clear that part of the NMU teachers would go into the tower and gradually would pay less attention to activities at an independent university, and that as a result of the center of gravity of mathematical training of researchers can move from NMU in the Higher School. It must be said that this happened, but this did not lead to a significant weakening of the NMU, because new strong people came to the places of teachers who left in the Higher School. Many of the Mattakha employees continue to teach at NMU, and the contingent of students of the independent was replenished at the expense of students of the tower engaged there in the evenings.
- How did the creators of the faculty react to the task of competition with traditional mathematical centers?
- When creating the faculty, we did not set ourselves the task of competing with traditional mathematical centers, such as the MHMAT MSU or Matmeh of St. Petersburg State University. Rather, as our main goal, we see competition with the leading foreign centers of mathematical training and with the mechmatian period of its heyday in the 1960s. We would like to achieve the same level or exceed it.
- What were the initial goals before the new faculty?
- From the very beginning, the Faculty of Mathematics was created as a research faculty. For us, this meant that all teachers of mathematical disciplines that we take to work should also conduct research at the world level. Therefore, when hiring for the teaching and research qualities of candidates, strict requirements are made equally.
Of course, it was fundamentally important for us that the students of the faculty were the best graduates of Russian schools, focused on further study of mathematics. As for graduate school and magistracy, here for us a fundamental task is, in addition to the best graduates of Russian universities, to attract strong students from abroad. We believe that we have something to offer: the level of education at the faculty is not inferior to the one that can be obtained at the best world universities.
The solution of all these complex problems falls on the shoulders of all employees of the faculty, but first of all, of course, on the shoulders of the new dean. It seems to me that Vladlen Anatolyevich Timorin was in its place, and, for my part, I will try to make every effort to help him continue to develop the faculty of mathematics.
- As of today, are these problems solved?
- It is better for me to judge this, since my opinion is addicted, but the perception is distorted, but I would say that we have advanced very much in solving these problems.
- What are the main achievements of the faculty for all the time of existence?
-First of all, this is a formed team of teachers and researchers, among whose members are 14 invited speakers of international congresses of mathematicians, including plenary speakers, and this is one of the highest levels of professional recognition of mathematicians in the world. In addition, the faculty is steadily attracting students of a very high level. Due to the intense individual work of the teaching staff with students at the faculty, it was possible to create a real research atmosphere. In my opinion, this is our main achievement.
- What prospects do you see for the further development of the faculty?
- To increase the level of research activity: no matter how high it is, there are always new borders that need to be taken. We need to attract students of magistracy and graduate students more to scientific activity, despite the fact that the parameters of this process are now also good. Нам нужно повышать международную привлекательность факультета: к нам и сейчас приезжают сильные студенты из-за рубежа, однако это ручеек, а не поток, то есть к нам не стоит очередь из выпускников лучших вузов, желающих пройти у нас магистерское или аспирантское обучение. This is our main task.
— Что этому препятствует?
— Препятствий много. Например, очевидно, что мы пока еще не так широко известны в мире, как мы бы этого хотели. Есть внешние препятствия для привлечения иностранных студентов — поездка в Россию на два или четыре года воспринимается скорее как экстремальное приключение, а не как естественный вариант продолжения образования. Мы считаем, что привлекательные стороны факультета способны перевесить препятствия, и наша задача — эти стороны усиливать. Какие-то вещи происходят независимо от наших действий, например, за последние полтора года жизнь в Москве для иностранцев очень сильно подешевела.
— Какие мероприятия факультет планирует провести в предстоящем году?
— Мы постоянно проводим научные мероприятия. Факультет и ассоциированные с ним лаборатории организуют самостоятельно и участвуют в организации большого количества международных конференций. Ежегодно их больше десятка. На конференции приглашаются и молодые люди — еще не получившие степень выпускники вузов либо только защитившиеся. Мы надеемся в том числе и через эти конференции донести до заинтересованной молодежи информацию о том, что в Москве имеются возможности для продолжения обучения и для совершенствования исследовательских навыков.
Кроме того, ежегодно ВШЭ организует зимнюю школу, куда приглашаются и иностранцы, потенциальные студенты магистратуры. Через эти школы информация доводится не только до непосредственных участников, но и до их сокурсников. Преподаватели активно участвуют в зарубежных конференциях, мы стараемся всякий раз использовать эту возможность, чтобы с помощью преподавателей принимающего университета доводить до его студентов необходимую информацию. Но, как я уже сказал, ручеек еще не превратился в поток, и до этого далеко.
- Thank you for the interview!