Although Ilia Alekseev is a third-year student in Mathematics at St Petersburg University, he has a number of research publications. He is engaged in research carried out by the new University’s Laboratory of Modern Algebra and Applications and manages research competitions for senior school students. The most talented of them work directly with him. 

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At the 2019 Intel International Science and Engineering Fair (Intel ISEF), his student Ruslan Magdiev was given the First Place Award from the American Mathematical Society for the project ‘Geodesics in the Discrete Heisenberg Group’. The award is a so-called small Nobel Prize of Mathematics. Ilia Alekseev describes: how they collaborated within the project; what challenges a person who wins international competitions may face when applying to the University; and how the research competitions differ from the Olympiads.

How did you start to work with Ruslan on the research projects?

For a start, Ruslan and I studied at the same school in St Petersburg, namely the LCME (Laboratory of Continuous Mathematical Education). It follows the University model. Its students work directly with University members on the research projects and engage in the research competitions. Among the University members are those who have recently graduated from the LCME: students and early-career researchers, who are willing to gain some practical experience in teaching on a voluntary basis.

In my first year at the University, I started to help my school manage a number of the competitions and tournaments. A year later I realised that I could work directly with the pupils on the research projects. I became a tutor of a group of four, including Ruslan. A year later, my students prepared three team and individual research projects. They introduced and discussed them at the Baltic Scientific and Engineering Research Competition. It serves as a preliminary round for the Intel International Science and Engineering Fair (Intel ISEF), which is a global research competition for senior pupils. Mathematicians, the winners of the Baltic Competition, and the LCME’s students regularly win awards from the American Mathematical Society. Ruslan is the first one who has won the First Place Award.  The other three of my students from the LCME, who also won the Baltic Competition, are also on the list of winners.

Do students choose a particular topic they would like to work on? What is the role of the supervisor in this process? 

Choosing the right topic is a hard nut to crack. On the one hand, pupils can take initiative and lead the way. On the other hand, it is up to the supervisor to assess the potential of the research and make a final decision. More often than not, we offer the topic. What students genuinely love may be far from having research potential or beyond what they can manage. Sometimes the project will not be a success as it may lead to nowhere. What is important is to select a research topic that would spark imaginative and inventive insights and solutions, lead to societal change, and could be managed by school pupils.

How did you select the research topic for Ruslan that brought him the award?

It is an interesting story. My research supervisor is Andrei Malutin, who also works at the Chebyshev Laboratory and is a leading research associate at the St Petersburg Department of V.A. Steklov Mathematical Institute (PDMI RAS). He focuses on group theory which I am also interested in. This is what I have been working on with my students. Andrei Malutin and his research supervisor Anatoly Vershik, who is an outstanding mathematician at the Chebyshev Laboratory and an academician, studied the absolute (exit boundary) of a group. They published a series of articles on the infinite geodesics in the discrete Heisenberg group. They also tried to explore finite geodesics. Yet much is to be done. Last September, they suggested that I delve deeper into the matter. I was very short of time then as I had my other projects and study. I told my students about it and forgot it.

In a month, Ruslan told me he had solved the problem. He indeed succeeded in solving the problem by taking a different perspective. He focused on geometry, although the problem was more algebraic in nature. This idea translated into successful research. By January we gained a detailed insight into the matter and published a paper. By May, while constantly involved in a range of competitions, we found a solution to the problem. Thus, Ruslan made an in-depth analysis of the finite geodesics in the discrete Heisenberg group. In other words, he found a truly remarkable solution to what had baffled the greatest minds.

How much time does it take to solve a problem? Apart from the research projects, both Ruslan and his fellow students study at school and take exams. The workload is overwhelming, indeed.

We met every week and spent two or three hours exploring ideas and getting new perspectives. Mostly, I talked about a particular topic in mathematics. I tried to help them unlock their potential and enable them to explore the links between their research and real mathematics. Obviously, what they do goes far beyond our meetings: they engage in self-study and regularly report to me. It does take more than two or three hours a week.

Are universities involved in the competitions for pupils?

Both Baltic Competition and Intel ISEF have university members as their jury members. A number of the universities are partners of these competitions and offer benefit to those who are applying for study there. At the ISEF, it is mostly American universities that offer such benefit to the American pupils as it is much easier to manage. Yet, neither me, nor my students have had such experience. Among our pupils who won the ISEF are successful applicants to the European universities, for example, in Poland. The ISEF award is definitely your privilege.

Unfortunately, the University is not a partner of the Baltic Competition, although quite a number of the universities in St Petersburg are. The Competition is supported by the large companies which collaborate with the University: Gazprom Neft, BIOCAD, and St Petersburg city administration. The companies offer internships for the most successful candidates, while the universities give an advantage to those who are applying for study. The winners don’t need to have admissions tests as winning the prize is an outstanding achievement. When applying for study at the University, the advantage is only given to those who win certain Olympiads. If you are a winner of the Baltic Competition, you still have to take admissions tests. Although the jury in Mathematics mostly comprises University members and they do appreciate your research, nevertheless there is no advantage when applying. It may sound quite paradoxical.  

Are the Olympiads lower in rank or status than the research competitions? 

I wouldn’t say so. The competitions differ from the Olympiads in two ways.  First, it is a different mode of working. If we liken it to sports, the Olympiads are like sprinting, while the research competitions are more of a marathon. What you are expected to show at the Olympiads is how quick you are at maths. When preparing for the research competitions, you are like a real researcher: it does take time to explore ideas, generate and discuss hypothesis, run experiments. It is completely different from what you are expected to do in the Olympiads.

Second, we use different tools for assessment. It is a key factor here. The Olympiads are more reliable evidence that you are smart and university is for you. The results can be easily calculated and checked as we have a rigorous assessment scheme. The Olympiads are like the Unified State Examination (USE). Everything is checked and ranked. On the contrary, a research competition is not about calculating how smart you are and what project is going to have an impact. It is the jury that decides who the winner is. We have little evidence, if any, whether a pupil made a research project independently or not. It may be quite obvious though for the jury to see who came up with their own ideas and developed them into the project. Yet there is no rigorous algorithm or evaluation methods for that. The main thing is whether we trust the decision of the jury or we don’t. Still, the paradox is that the jury in Mathematics of the Baltic Competition mostly consists of academic experts from the University but its winners have to take our admissions test.

In other words, the main thing is whether the University regards the jury’s decision as reliable evidence?

Yes. As I see, we can solve this problem by having more of our academic experts in the jury, not necessarily in the jury of all research competitions. We can start with the Baltic Competition. Actually, it is in the making.

Yet the winners, regardless of all the difficulties, choose the University for their study?

I don’t have any figures but among my friends who have won research competitions there is an overwhelming majority of students who opted for the University. It doesn’t have to be necessarily mathematics. As winning the research competitions doesn’t give you any advantage in applying for study, you have to find proper Olympiads. Ruslan, for example, has won the Olympiad and successfully passed the USE. Still, the Olympiad he won was not a guarantee of successful application for the field of study he chose. For him, as for many other students who have failed to score to be offered a place, there is just one way out: to choose another field of study in mathematics and after their first year at university to be transferred to the field of study they are interested in.

Without doubt, everyone who wins the Olympiads is incredibly talented and smart. And everyone is offered a place at universities. Yet it is incredibly arduous and they push themselves to the limit. By the end of your eleventh year at school you are supposed to complete the research project, take part in numerous competitions, Olympiads, and take the USE.  It is extremely arduous.