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‘A Good Thing about Mathematical Thinking Is That It Knocks Some Sense into Your Brain’

‘A Good Thing about Mathematical Thinking Is That It Knocks Some Sense into Your Brain’

© iStock

HSE University maintains active cooperation with the Academy of Science institutes, which open joint departments at the university and provide their best experts for teaching and research. HSE News Service talked to Deputy Directors of the RAS Steklov Mathematical Institute (MIRAS) — RAS member Lev Beklemishev and Sergey Gorchinskiy — about the development of mathematics in the modern world, career opportunities for graduates in mathematics, and the specifics of researchers’ work with students.

About Mathematics

What is the position of mathematics among other sciences today? Is it fair to say that it is not only a scientific field, but a kind of art?

Lev Beklemishev: Even within the professional community, there are different perspectives on this. I represent a particular field of mathematics, mathematical logics, in which aspects of the humanities are also important. However, the more traditional fields are mostly related to physics. Academician Vladimir Arnold believed that mathematics is a part of physics, one which doesn’t require any spending on experimental apparatus. Different fields of mathematics have different origins: geometry initially was a science about ground measuring, while probability theory was ‘mathematised’ only around the turn of the 20th century.

Lev Beklemishev
© Mikhail Dmitriev/ HSE University

Speaking about some features of art, every science has it, including mathematics. Useful theorems are beautiful theorems, while ugly ones are most probably useless. Mathematicians often come to their research findings intuitively, and the simplistic image of a mathematician as a person whose main advantage is the ability to perform calculations quickly is far from reality. Of course, mathematicians live in the world of their ideas, but this doesn’t interfere with their ability to navigate reality.

Sergey Gorchinskiy: Prof. Beklemishev and I are involved in different fields of mathematics: I specialize in algebraic and arithmetic geometry, but I totally agree with him. Art is about ‘beautiful’ and ‘ugly’ categories, while science is about ‘proved or not proved’. Long ago, I decided to go into mathematics, because it is the only area of human activity that has objective truth. In physical or chemical experiments, many things depend on the conditions, so it is not as unambiguous as mathematics with its ‘two plus two equals four’ certainty.

Sergey Gorchinskiy
© Mikhail Dmitriev/ HSE University

Why is mathematics called ‘the queen of the sciences’?

LB: Because as early as in ancient times and the middle ages, scientific knowledge evolved as mathematical knowledge, and the language of mathematics was the language of science. And today, it is obvious to everyone that one cannot carry out research in any socially important field without the ability to argue in strict mathematical terms and to use at least some mathematical apparatus.

And irrespectively of formulas, a good thing about mathematical thinking is that it, so to say, knocks some sense into your brain, teaches you to think logically.

SG: I’ll continue the allegory: there are people who have had some sense knocked into their brain with mathematics, and there are those who simply haven’t had good teachers.

How wide is the range of mathematical studies today? Is it possible that scholars working in a certain area of mathematics do not understand what their colleagues are doing in another field of this science?

LB: Unfortunately, this is true. But there are researchers of a universal kind, whose interests cover maybe not all mathematics, but a very wide range, such as Andrey Kolmogorov or Vladimir Arnold. They represent the older generation, and there are fewer and fewer of them.

In the early 20th century, mathematics was continually expanding and branching, while since 1970s and 80s, merging has been the major trend. Those years saw the evolution of new applications of mathematics in physics, algebraic geometry and other areas that deal with different mathematics fields. This is still the dominant trend today.

About Cooperation with HSE University

Why do you think it is necessary for MIRAS to cooperate with universities?

For us, universities are mostly a source of staff, and we believe that the best students, graduates and young researchers should come to us. Staff are essential for mathematical research. Traditionally, most of our employees have been graduates of the MSU Faculty of Mechanics and Mathematics, and many forms of cooperation have been focused on this institution. The MSU FMM still remains our main source of new staff. But when a new, modern Faculty of Mathematics appeared at HSE University, and strong high school graduates started to go there, who then went on to became strong students, we couldn’t ignore it.

How is the cooperation between MIRAS and HSE University organized?

It’s all very natural. Our staff are closely involved in the life of the Faculty of Mathematics: we teach and carry out research, participate in its development and the discussion of various issues. Never in my life have I seen such a democratic academic council as at this faculty. At their sessions, everyone is equal, no matter the positions and degrees, and questions of human resources and workload distribution always cause energetic debate.

The Faculty of Mathematics includes a Joint Department with MIRAS, which is chaired by RAS member Viktor Vasiliev. But the department in this case is a notional concept. The faculty has no departments in the traditional understanding, when a small department team determines its own courses and barely interacts with other departments. The Faculty of Mathematics operates in a different way: all the teachers communicate with each other, participate in joint activities and are not split into specialization groups.

Students of the HSE Faculty of Mathematics make up a considerable share of those who study at MIRAS Research and Education Centre. This year, I am supervising a thesis by one HSE doctoral student, four theses by fourth-year students, and several term papers by junior students.

SG: We cooperate not only with the HSE Faculty of Mathematics but with other HSE departments as well. I personally do not have a teaching position (which doesn’t mean that I don’t have students and doctoral students), but I’m a researcher at the International Laboratory for Mirror Symmetry and Automorphic Forms. A few years ago, I also worked at the Laboratory of Algebraic Geometry and Its Applications. Our MIRAS colleagues also work at these laboratories part-time.

The launch of the Laboratory of Algebraic Geometry and Its Applications was very beneficial for us: we immediately started organizing joint conferences, which has turned out to be very convenient. They mostly take place in the MIRAS building, while participants from other cities and countries stay at an HSE hotel on Vavilova street, which is close to the Institute. Another important factor has been the opportunity to attract student interns to the laboratory and pay them for their work.

What can you say about HSE University students?

LB: When they do mathematics, their eyes sparkle, and it is always a pleasure to teach such students.

About Work and Salaries

What do mathematicians do in today’s Russia and how do they make their living? What academic career options are possible and preferable?

LB: The ideal career for those who want to specialise in theoretical mathematics is academic research. You can calmly do what you love at an academic institute and sometimes, teach. If a mathematician does absolutely no teaching, it is undoubtedly bad for them.

The start of a mathematician’s career is their thesis defence. For some time before the defence, they are in a certain environment, and they already understand what kind of environment it is, what institute has it, what research is done there, and who they want to work with. Today, there is no shortage of jobs for such people in Moscow. There is MIRAS, the Institute for Information Transmission Problems (intersection of mathematics and other sciences), as well as physics institutes, where mathematicians are also in demand, the Institute of Theoretical Physics in Chernogolovka and the RAS Institute of Physics, where mathematics is rather an applied field.

Another option is for those who want to work with school students, for example developing Olympiads. This is well-respected work and those who do it are often strong mathematicians themselves. They understand what real mathematics is, but put their effort into something else.

And of course, there is the traditional trajectory, to teach at a university. A contemporary university teacher must combine the role of a teacher who can explain complicated things to students and a researcher who works independently. Academic publications are among the key requirements for university teachers today.

Unfortunately, teachers are often overloaded with their teaching duties. With the number of class hours that are common for many universities, it is impossible to carry out research productively.

SG: There also other career opportunities, beyond teaching and research, and about 90% of graduates pursue these opportunities. This is mostly IT, and the most popular profession for professional mathematicians is working as a programmer. The other sphere that includes a lot of mathematics is banking, and the third is insurance. There are many examples of people who are professionally involved in financial mathematics, trade on the stock markets, create their companies and hire mathematicians.

Do mathematicians often pursue careers abroad?

LB: Yes, they do, to continue their education and work in academia, since the international market of such jobs with good salaries is bigger than the domestic one. But there are opposite examples, when holders of a PhD degree, post-docs come back to Russia. I worked in different countries for long periods myself, and my longest stay was in Utrecht, the Netherlands, where I had a permanent position, but came back to Russia 15 years ago.

About the Steklov Mathematical Institute

Could you tell us more about MIRAS?

The history of MIRAS dates back to 1921. Today it is the country’s best and most prestigious mathematical institute. Its glory is built on its scholars: portraits of our renowned predecessors are in the gallery on the ninth floor. Our library, in addition to its mathematics books, contains a lot of fascinating archive materials. It has a bookshelf that used to belong to Pafnuty Chebyshev, a great Russian mathematician.

We not only carry out research, but teach at Moscow’s best universities, as well as at our MIRAS Research and Education Centre. In the evenings, we also offer free special courses and workshops for students from any universities in different fields of mathematics. If a student takes exams (which is optional) we inform their university about it, and they get the associated credits. For students, this gives them additional courses in contemporary fields of mathematics by people who deal with it professionally, and for us, it is important that young people get to know MIRAS and get used to its atmosphere.

The institute has a Laboratory of mathematics popularization and propaganda chaired by Nikolay Andreev. MIRAS team consists of 120 scholars, including over 30 members of the Russian Academy of Sciences. About one-third are young researchers.

Nikolay Andreev
© Mikhail Dmitriev/ HSE University

Are there HSE University graduates among your full-time employees?

Not yet, because the HSE Faculty of Mathematics only opened in 2007, and its first graduates are only now starting to defend their doctoral theses. But many of them also work with us part-time, on grants or on temporary contracts, and we consider them as our employees as well.

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